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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Fonctions Monogenes
William T. Ross
University of Richmond
Fonctions Monogenes University of Richmond
![Page 2: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/2.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Fonctions Monogenes University of Richmond
![Page 3: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/3.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Fonctions Monogenes University of Richmond
![Page 4: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/4.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Fonctions Monogenes University of Richmond
![Page 5: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/5.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
How do we assign a number to
1− 1 + 1− 1 + 1− · · ·?
Fonctions Monogenes University of Richmond
![Page 6: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/6.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 1:
1− 1 + 1− 1 + 1− 1 · · ·
Let sn be the n-th partial sum
sn =
0, n even;1, n odd.
and so (via Cauchy’s definition of convergence of series) this seriesdiverges.
Fonctions Monogenes University of Richmond
![Page 7: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/7.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 1:
1− 1 + 1− 1 + 1− 1 · · ·
Let sn be the n-th partial sum
sn =
0, n even;1, n odd.
and so (via Cauchy’s definition of convergence of series) this seriesdiverges.
Fonctions Monogenes University of Richmond
![Page 8: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/8.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 1:
1− 1 + 1− 1 + 1− 1 · · ·
Let sn be the n-th partial sum
sn =
0, n even;1, n odd.
and so (via Cauchy’s definition of convergence of series) this seriesdiverges.
Fonctions Monogenes University of Richmond
![Page 9: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/9.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 1:
1− 1 + 1− 1 + 1− 1 · · ·
Let sn be the n-th partial sum
sn =
0, n even;1, n odd.
and so (via Cauchy’s definition of convergence of series) this seriesdiverges.
Fonctions Monogenes University of Richmond
![Page 10: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/10.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 2: Cesaro
1− 1 + 1 · · · = limn→∞
s0 + s1 + · · ·+ sn
n + 1
= limn→∞
1 + 0 + 1 + 0 + · · · 1n + 1
=1
2
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 2: Cesaro
1− 1 + 1 · · · = limn→∞
s0 + s1 + · · ·+ sn
n + 1
= limn→∞
1 + 0 + 1 + 0 + · · · 1n + 1
=1
2
Fonctions Monogenes University of Richmond
![Page 12: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/12.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 2: Cesaro
1− 1 + 1 · · · = limn→∞
s0 + s1 + · · ·+ sn
n + 1
= limn→∞
1 + 0 + 1 + 0 + · · · 1n + 1
=1
2
Fonctions Monogenes University of Richmond
![Page 13: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/13.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 2: Cesaro
1− 1 + 1 · · · = limn→∞
s0 + s1 + · · ·+ sn
n + 1
= limn→∞
1 + 0 + 1 + 0 + · · · 1n + 1
=1
2
Fonctions Monogenes University of Richmond
![Page 14: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/14.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 3: Abel
1− 1 + 1− 1 + 1 · · ·= lim
x→1−(1− x + x2 − x3 + · · · )
= limx→1−
1
1 + x
=1
2
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 3: Abel
1− 1 + 1− 1 + 1 · · ·= lim
x→1−(1− x + x2 − x3 + · · · )
= limx→1−
1
1 + x
=1
2
Fonctions Monogenes University of Richmond
![Page 16: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/16.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 3: Abel
1− 1 + 1− 1 + 1 · · ·= lim
x→1−(1− x + x2 − x3 + · · · )
= limx→1−
1
1 + x
=1
2
Fonctions Monogenes University of Richmond
![Page 17: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/17.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 3: Abel
1− 1 + 1− 1 + 1 · · ·= lim
x→1−(1− x + x2 − x3 + · · · )
= limx→1−
1
1 + x
=1
2
Fonctions Monogenes University of Richmond
![Page 18: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/18.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 4: CalletLet m < n
1 + x + x2 + · · · xm−1
1 + x + x2 + · · · xn−1
=1− xm
1− xn
= (1− xm)(1 + xn + x2n + x3n + · · · )= 1− xm + xn − xm+n + x2n − · · ·
Evaluate both sides at x = 1 to get
1− 1 + 1− 1 + · · · =m
n
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 4: CalletLet m < n
1 + x + x2 + · · · xm−1
1 + x + x2 + · · · xn−1
=1− xm
1− xn
= (1− xm)(1 + xn + x2n + x3n + · · · )= 1− xm + xn − xm+n + x2n − · · ·
Evaluate both sides at x = 1 to get
1− 1 + 1− 1 + · · · =m
n
Fonctions Monogenes University of Richmond
![Page 20: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/20.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 4: CalletLet m < n
1 + x + x2 + · · · xm−1
1 + x + x2 + · · · xn−1
=1− xm
1− xn
= (1− xm)(1 + xn + x2n + x3n + · · · )= 1− xm + xn − xm+n + x2n − · · ·
Evaluate both sides at x = 1 to get
1− 1 + 1− 1 + · · · =m
n
Fonctions Monogenes University of Richmond
![Page 21: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/21.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 4: CalletLet m < n
1 + x + x2 + · · · xm−1
1 + x + x2 + · · · xn−1
=1− xm
1− xn
= (1− xm)(1 + xn + x2n + x3n + · · · )= 1− xm + xn − xm+n + x2n − · · ·
Evaluate both sides at x = 1 to get
1− 1 + 1− 1 + · · · =m
n
Fonctions Monogenes University of Richmond
![Page 22: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/22.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 4: CalletLet m < n
1 + x + x2 + · · · xm−1
1 + x + x2 + · · · xn−1
=1− xm
1− xn
= (1− xm)(1 + xn + x2n + x3n + · · · )= 1− xm + xn − xm+n + x2n − · · ·
Evaluate both sides at x = 1 to get
1− 1 + 1− 1 + · · · =m
n
Fonctions Monogenes University of Richmond
![Page 23: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/23.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 4: CalletLet m < n
1 + x + x2 + · · · xm−1
1 + x + x2 + · · · xn−1
=1− xm
1− xn
= (1− xm)(1 + xn + x2n + x3n + · · · )= 1− xm + xn − xm+n + x2n − · · ·
Evaluate both sides at x = 1 to get
1− 1 + 1− 1 + · · · =m
n
Fonctions Monogenes University of Richmond
![Page 24: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/24.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Answer # 4: CalletLet m < n
1 + x + x2 + · · · xm−1
1 + x + x2 + · · · xn−1
=1− xm
1− xn
= (1− xm)(1 + xn + x2n + x3n + · · · )= 1− xm + xn − xm+n + x2n − · · ·
Evaluate both sides at x = 1 to get
1− 1 + 1− 1 + · · · =m
n
Fonctions Monogenes University of Richmond
![Page 25: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/25.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
How do we assign a number S to
a0 + a1 + a2 + · · ·?
Cauchy
Sdef= lim
n→∞(a0 + a1 + · · ·+ an)
Abel
SAdef= lim
x→1−(a0 + a1x + a2x2 + · · · )
Cesaro
SCdef= lim
n→∞
s0 + s1 + · · ·+ sn
n + 1
TheoremIf S exists, then S = SA = SC .
Fonctions Monogenes University of Richmond
![Page 26: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/26.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
How do we assign a number S to
a0 + a1 + a2 + · · ·?
Cauchy
Sdef= lim
n→∞(a0 + a1 + · · ·+ an)
Abel
SAdef= lim
x→1−(a0 + a1x + a2x2 + · · · )
Cesaro
SCdef= lim
n→∞
s0 + s1 + · · ·+ sn
n + 1
TheoremIf S exists, then S = SA = SC .
Fonctions Monogenes University of Richmond
![Page 27: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/27.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
How do we assign a number S to
a0 + a1 + a2 + · · ·?
Cauchy
Sdef= lim
n→∞(a0 + a1 + · · ·+ an)
Abel
SAdef= lim
x→1−(a0 + a1x + a2x2 + · · · )
Cesaro
SCdef= lim
n→∞
s0 + s1 + · · ·+ sn
n + 1
TheoremIf S exists, then S = SA = SC .
Fonctions Monogenes University of Richmond
![Page 28: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/28.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
How do we assign a number S to
a0 + a1 + a2 + · · ·?
Cauchy
Sdef= lim
n→∞(a0 + a1 + · · ·+ an)
Abel
SAdef= lim
x→1−(a0 + a1x + a2x2 + · · · )
Cesaro
SCdef= lim
n→∞
s0 + s1 + · · ·+ sn
n + 1
TheoremIf S exists, then S = SA = SC .
Fonctions Monogenes University of Richmond
![Page 29: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/29.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
How do we assign a number S to
a0 + a1 + a2 + · · ·?
Cauchy
Sdef= lim
n→∞(a0 + a1 + · · ·+ an)
Abel
SAdef= lim
x→1−(a0 + a1x + a2x2 + · · · )
Cesaro
SCdef= lim
n→∞
s0 + s1 + · · ·+ sn
n + 1
TheoremIf S exists, then S = SA = SC .
Fonctions Monogenes University of Richmond
![Page 30: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/30.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Uses of divergent series
Theorem (du Bois Reymond - 1873)
If f is continuous on [0, 2π] with f (0) = f (2π) with FS
(∗) a0
2+∞∑
n=1
an cos(nθ) + bn sin(nθ),
then the FS of f need not converge pointwise.
Fonctions Monogenes University of Richmond
![Page 31: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/31.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Theorem (Poisson)
If f ∈ C [0, 2π] with FS (*), then
f (θ) = limr→1−
a0
2+∞∑
n=1
rnan cos(nθ) + rnbn sin(nθ)
Fonctions Monogenes University of Richmond
![Page 32: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/32.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Theorem (Fejer - 1904)
If f ∈ C [0, 2π], then σn(f )→ f uniformly.
σn(f , θ) =s0(f , θ) + · · ·+ sn(f , θ)
n + 1
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Theorem (Fatou - 1906)
If f ∈ L1[0, 2π] with FS (*), then
f (θ) = limr→1−
a0
2+∞∑
n=1
rnan cos(nθ) + rnbn sin(nθ)
a.e.
Fonctions Monogenes University of Richmond
![Page 34: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/34.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Theorem (Lebesgue - 1920)
If f ∈ L1[0, 2π], then σn(f )→ f a.e.
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Generalized Analytic Continuation
We want to explore ways in which
f |Ωj , j = 1, 2, · · ·
can be related beyond analytic continuation.
I Ω is a disconnected open set in CI Ωj : j = 1, 2, · · · are the connected components of Ω.
I f ∈ Mer(Ω).
Fonctions Monogenes University of Richmond
![Page 36: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/36.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Generalized Analytic Continuation
We want to explore ways in which
f |Ωj , j = 1, 2, · · ·
can be related beyond analytic continuation.
I Ω is a disconnected open set in CI Ωj : j = 1, 2, · · · are the connected components of Ω.
I f ∈ Mer(Ω).
Fonctions Monogenes University of Richmond
![Page 37: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/37.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Generalized Analytic Continuation
We want to explore ways in which
f |Ωj , j = 1, 2, · · ·
can be related beyond analytic continuation.
I Ω is a disconnected open set in CI Ωj : j = 1, 2, · · · are the connected components of Ω.
I f ∈ Mer(Ω).
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Generalized Analytic Continuation
We want to explore ways in which
f |Ωj , j = 1, 2, · · ·
can be related beyond analytic continuation.
I Ω is a disconnected open set in CI Ωj : j = 1, 2, · · · are the connected components of Ω.
I f ∈ Mer(Ω).
Fonctions Monogenes University of Richmond
![Page 39: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/39.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Fonctions Monogenes University of Richmond
![Page 40: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/40.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Overview of GAC
I Define six types of continuationsI analytic continuation (the ’gold standard’)I pseudocontinuationI Goncar continuationI continuation via formal mult. of seriesI Bochner-Bohnenblust continuationI continuation via overconvergence
I how these compare to analytic continuation
I how they prove themselves useful
I how they can be unified
Fonctions Monogenes University of Richmond
![Page 41: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/41.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Overview of GAC
I Define six types of continuationsI analytic continuation (the ’gold standard’)I pseudocontinuationI Goncar continuationI continuation via formal mult. of seriesI Bochner-Bohnenblust continuationI continuation via overconvergence
I how these compare to analytic continuation
I how they prove themselves useful
I how they can be unified
Fonctions Monogenes University of Richmond
![Page 42: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/42.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Overview of GAC
I Define six types of continuationsI analytic continuation (the ’gold standard’)I pseudocontinuationI Goncar continuationI continuation via formal mult. of seriesI Bochner-Bohnenblust continuationI continuation via overconvergence
I how these compare to analytic continuation
I how they prove themselves useful
I how they can be unified
Fonctions Monogenes University of Richmond
![Page 43: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/43.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Overview of GAC
I Define six types of continuationsI analytic continuation (the ’gold standard’)I pseudocontinuationI Goncar continuationI continuation via formal mult. of seriesI Bochner-Bohnenblust continuationI continuation via overconvergence
I how these compare to analytic continuation
I how they prove themselves useful
I how they can be unified
Fonctions Monogenes University of Richmond
![Page 44: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/44.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Overview of GAC
I Define six types of continuationsI analytic continuation (the ’gold standard’)I pseudocontinuationI Goncar continuationI continuation via formal mult. of seriesI Bochner-Bohnenblust continuationI continuation via overconvergence
I how these compare to analytic continuation
I how they prove themselves useful
I how they can be unified
Fonctions Monogenes University of Richmond
![Page 45: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/45.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Overview of GAC
I Define six types of continuationsI analytic continuation (the ’gold standard’)I pseudocontinuationI Goncar continuationI continuation via formal mult. of seriesI Bochner-Bohnenblust continuationI continuation via overconvergence
I how these compare to analytic continuation
I how they prove themselves useful
I how they can be unified
Fonctions Monogenes University of Richmond
![Page 46: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/46.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Overview of GAC
I Define six types of continuationsI analytic continuation (the ’gold standard’)I pseudocontinuationI Goncar continuationI continuation via formal mult. of seriesI Bochner-Bohnenblust continuationI continuation via overconvergence
I how these compare to analytic continuation
I how they prove themselves useful
I how they can be unified
Fonctions Monogenes University of Richmond
![Page 47: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/47.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Overview of GAC
I Define six types of continuationsI analytic continuation (the ’gold standard’)I pseudocontinuationI Goncar continuationI continuation via formal mult. of seriesI Bochner-Bohnenblust continuationI continuation via overconvergence
I how these compare to analytic continuation
I how they prove themselves useful
I how they can be unified
Fonctions Monogenes University of Richmond
![Page 48: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/48.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Overview of GAC
I Define six types of continuationsI analytic continuation (the ’gold standard’)I pseudocontinuationI Goncar continuationI continuation via formal mult. of seriesI Bochner-Bohnenblust continuationI continuation via overconvergence
I how these compare to analytic continuation
I how they prove themselves useful
I how they can be unified
Fonctions Monogenes University of Richmond
![Page 49: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/49.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Overview of GAC
I Define six types of continuationsI analytic continuation (the ’gold standard’)I pseudocontinuationI Goncar continuationI continuation via formal mult. of seriesI Bochner-Bohnenblust continuationI continuation via overconvergence
I how these compare to analytic continuation
I how they prove themselves useful
I how they can be unified
Fonctions Monogenes University of Richmond
![Page 50: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/50.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Theorem (Poincare - 1883)
Let e iθn be a dense sequence in T = |z | = 1 and cn be anabs. summable sequence in C \ 0. Define
f (z) =∞∑
n=1
cn
1− e−iθnz∈ Hol(C \ T).
Then f does not have an AC across any e iθ.
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
f (z) =∞∑
n=1
cn
1− e−iθnz
Still...
f (re iθ)− f (1
re iθ) =
∫1− r 2
|re iθ − e it |2dµ(e it), dµ =
∑n>1
cnδe iθn .
Fatou’s theorem (1906):
limr→1−
∫1− r 2
|re iθ − e it |2dµ(e it) =
dµ
dθ(e iθ) a.e.
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
f (z) =∞∑
n=1
cn
1− e−iθnz
Still...
f (re iθ)− f (1
re iθ) =
∫1− r 2
|re iθ − e it |2dµ(e it), dµ =
∑n>1
cnδe iθn .
Fatou’s theorem (1906):
limr→1−
∫1− r 2
|re iθ − e it |2dµ(e it) =
dµ
dθ(e iθ) a.e.
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
f (z) =∞∑
n=1
cn
1− e−iθnz
Still...
f (re iθ)− f (1
re iθ) =
∫1− r 2
|re iθ − e it |2dµ(e it), dµ =
∑n>1
cnδe iθn .
Fatou’s theorem (1906):
limr→1−
∫1− r 2
|re iθ − e it |2dµ(e it) =
dµ
dθ(e iθ) a.e.
Fonctions Monogenes University of Richmond
![Page 54: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/54.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Actually] lim
z→ζ
|z|<1
f (z) = ] limz→ζ
|z|>1
f (z) a.e. ζ ∈ T
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Pseudocontinuation
Let f ∈ Mer(D),F ∈ Mer(De). If
] limz→ζ
|z|<1
f (z) = ] limz→ζ
|z|>1
F (z) a.e. ζ ∈ T
then f and F are pseudocontinuations of each other.
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Theorem (Lusin-Privalov)
If f ∈ Mer(D) and
] limz→ζ
f (z) = 0 ∀ ζ ∈ E ⊂ T
with |E | > 0, then f ≡ 0.
Corollary
If f ∈ Mer(D) has a PC F ∈ Mer(De) and f has an AC across e iθ,then this AC must be F .
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Theorem (Lusin-Privalov)
If f ∈ Mer(D) and
] limz→ζ
f (z) = 0 ∀ ζ ∈ E ⊂ T
with |E | > 0, then f ≡ 0.
Corollary
If f ∈ Mer(D) has a PC F ∈ Mer(De) and f has an AC across e iθ,then this AC must be F .
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Example
log(1− z) does not have a pseudocontinuation F/G ∈ Mer(De).
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of pseudocontinuations
I cyclic vectors for the backward shift operator
Bf =f − f (0)
z
on the Hardy space (Douglas-Shapiro-Shields - 1970)
I rational approximation with pre-assigned poles (Walsh 1935,Tumarkin 1966)
I Darlington synthesis problem (Douglas-Helton, Arov - 1971)
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of pseudocontinuations
I cyclic vectors for the backward shift operator
Bf =f − f (0)
z
on the Hardy space (Douglas-Shapiro-Shields - 1970)
I rational approximation with pre-assigned poles (Walsh 1935,Tumarkin 1966)
I Darlington synthesis problem (Douglas-Helton, Arov - 1971)
Fonctions Monogenes University of Richmond
![Page 61: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/61.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of pseudocontinuations
I cyclic vectors for the backward shift operator
Bf =f − f (0)
z
on the Hardy space (Douglas-Shapiro-Shields - 1970)
I rational approximation with pre-assigned poles (Walsh 1935,Tumarkin 1966)
I Darlington synthesis problem (Douglas-Helton, Arov - 1971)
Fonctions Monogenes University of Richmond
![Page 62: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/62.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of pseudocontinuations
I cyclic vectors for the backward shift operator
Bf =f − f (0)
z
on the Hardy space (Douglas-Shapiro-Shields - 1970)
I rational approximation with pre-assigned poles (Walsh 1935,Tumarkin 1966)
I Darlington synthesis problem (Douglas-Helton, Arov - 1971)
Fonctions Monogenes University of Richmond
![Page 63: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/63.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Gap theorems
Theorem (Hadamard - 1892)
The gap series ∑n
2−nz2n
does not have an AC across any e iθ
Theorem (H. S. Shapiro - 1968)
The gap series ∑n
2−nz2n
does not have a PC across any arc of T
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Gap theorems
Theorem (Hadamard - 1892)
The gap series ∑n
2−nz2n
does not have an AC across any e iθ
Theorem (H. S. Shapiro - 1968)
The gap series ∑n
2−nz2n
does not have a PC across any arc of T
Fonctions Monogenes University of Richmond
![Page 65: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/65.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Gap theorems
Theorem (Hadamard - 1892)
The gap series ∑n
2−nz2n
does not have an AC across any e iθ
Theorem (H. S. Shapiro - 1968)
The gap series ∑n
2−nz2n
does not have a PC across any arc of T
Fonctions Monogenes University of Richmond
![Page 66: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/66.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Theorem (Fabry - 1898)
The gap series ∑n
2−nzn2
does not have an AC across any e iθ
Theorem (Aleksandrov - 1997)
The gap series ∑n
2−nzn2
does not have a PC across any arc of T
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Theorem (Fabry - 1898)
The gap series ∑n
2−nzn2
does not have an AC across any e iθ
Theorem (Aleksandrov - 1997)
The gap series ∑n
2−nzn2
does not have a PC across any arc of T
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Borel series
Theorem (Borel - 1917)
Let zn be a seq. of points in C with |zn| → 1 and
|cn| ≤ exp(exp(−n2),
and define
f (z) =∞∑
n=1
cn
z − zn
If f |D has an analytic continuation f to a neighborhood Uw with|w | > 1, then
f (z) =∞∑
n=1
cn
z − zn, z ∈ Uw .
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Borel series
Theorem (Borel - 1917)
Let zn be a seq. of points in C with |zn| → 1 and
|cn| ≤ exp(exp(−n2),
and define
f (z) =∞∑
n=1
cn
z − zn
If f |D has an analytic continuation f to a neighborhood Uw with|w | > 1, then
f (z) =∞∑
n=1
cn
z − zn, z ∈ Uw .
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Borel series
Theorem (Borel - 1917)
Let zn be a seq. of points in C with |zn| → 1 and
|cn| ≤ exp(exp(−n2),
and define
f (z) =∞∑
n=1
cn
z − zn
If f |D has an analytic continuation f to a neighborhood Uw with|w | > 1, then
f (z) =∞∑
n=1
cn
z − zn, z ∈ Uw .
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
f (z) =∞∑
n=1
cn
z − zn.
Theorem (Walsh - 1930)
Same conclusion as Borel but with
limn→∞
n√|cn| = 0.
Theorem (Goncar - 1967)
Same conclusion as Borel but with
lim supn→∞
n√|cn| < 1.
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
f (z) =∞∑
n=1
cn
z − zn.
Theorem (Walsh - 1930)
Same conclusion as Borel but with
limn→∞
n√|cn| = 0.
Theorem (Goncar - 1967)
Same conclusion as Borel but with
lim supn→∞
n√|cn| < 1.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
f (z) =∞∑
n=1
cn
z − zn.
Theorem (Walsh - 1930)
Same conclusion as Borel but with
limn→∞
n√|cn| = 0.
Theorem (Goncar - 1967)
Same conclusion as Borel but with
lim supn→∞
n√|cn| < 1.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Wolff, Denjoy, Beurling, etc. gave examples of Borel series
f (z) =∞∑
n=1
cn
z − zn
withf |D ≡ 0 but f |De 6≡ 0.
In these examples,lim supn→∞
n√|cn|= 1.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Wolff, Denjoy, Beurling, etc. gave examples of Borel series
f (z) =∞∑
n=1
cn
z − zn
withf |D ≡ 0 but f |De 6≡ 0.
In these examples,lim supn→∞
n√|cn|= 1.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Goncar continuation
I Let A = compact set which separates the plane
I Ωj : j = 1, 2, · · · = connect. comp. of Ac
I Rn = rat. funct., poles in A, deg(Rn) 6 n
DefinitionIf f ∈ Hol(Ac), we say f |Ωj is a Goncar continuation of f |Ωk ifthere are rational functions (as above) with
supK⊂Ωk∪Ωj
limn→∞
supz∈K|f (z)− Rn(z)|1/n < 1
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Goncar continuation
I Let A = compact set which separates the plane
I Ωj : j = 1, 2, · · · = connect. comp. of Ac
I Rn = rat. funct., poles in A, deg(Rn) 6 n
DefinitionIf f ∈ Hol(Ac), we say f |Ωj is a Goncar continuation of f |Ωk ifthere are rational functions (as above) with
supK⊂Ωk∪Ωj
limn→∞
supz∈K|f (z)− Rn(z)|1/n < 1
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Goncar continuation
I Let A = compact set which separates the plane
I Ωj : j = 1, 2, · · · = connect. comp. of Ac
I Rn = rat. funct., poles in A, deg(Rn) 6 n
DefinitionIf f ∈ Hol(Ac), we say f |Ωj is a Goncar continuation of f |Ωk ifthere are rational functions (as above) with
supK⊂Ωk∪Ωj
limn→∞
supz∈K|f (z)− Rn(z)|1/n < 1
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Goncar continuation
I Let A = compact set which separates the plane
I Ωj : j = 1, 2, · · · = connect. comp. of Ac
I Rn = rat. funct., poles in A, deg(Rn) 6 n
DefinitionIf f ∈ Hol(Ac), we say f |Ωj is a Goncar continuation of f |Ωk ifthere are rational functions (as above) with
supK⊂Ωk∪Ωj
limn→∞
supz∈K|f (z)− Rn(z)|1/n < 1
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Goncar continuation
I Let A = compact set which separates the plane
I Ωj : j = 1, 2, · · · = connect. comp. of Ac
I Rn = rat. funct., poles in A, deg(Rn) 6 n
DefinitionIf f ∈ Hol(Ac), we say f |Ωj is a Goncar continuation of f |Ωk ifthere are rational functions (as above) with
supK⊂Ωk∪Ωj
limn→∞
supz∈K|f (z)− Rn(z)|1/n < 1
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Example
Iflim sup
n
n√|cn| < 1
andf =
∑n
cn
z − e iθn,
then f |De is a G-continuation of f |D.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
An‘Abelian’ theorem
Theorem (Goncar - 1967)
If f |Ωj is a G-continuation of f |Ωk and suppose g = f |Ωj has anAC g along some path to a nbh Uw of w ∈ Ωk . Theng |Uw = f |Uw .
Again, although G-continuation is a generalization of AC, it iscompatible with AC.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
An‘Abelian’ theorem
Theorem (Goncar - 1967)
If f |Ωj is a G-continuation of f |Ωk and suppose g = f |Ωj has anAC g along some path to a nbh Uw of w ∈ Ωk . Theng |Uw = f |Uw .
Again, although G-continuation is a generalization of AC, it iscompatible with AC.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
An‘Abelian’ theorem
Theorem (Goncar - 1967)
If f |Ωj is a G-continuation of f |Ωk and suppose g = f |Ωj has anAC g along some path to a nbh Uw of w ∈ Ωk . Theng |Uw = f |Uw .
Again, although G-continuation is a generalization of AC, it iscompatible with AC.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of G-continuation
I Cyclic vectors for linear operators with a spanning set ofeigenvectors, corresponding to distinct eigenvalues (Sibilev)
I Common cyclic vectors for certain sets of normal operators(Sibilev, R-Wogen)
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of G-continuation
I Cyclic vectors for linear operators with a spanning set ofeigenvectors, corresponding to distinct eigenvalues (Sibilev)
I Common cyclic vectors for certain sets of normal operators(Sibilev, R-Wogen)
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of G-continuation
I Cyclic vectors for linear operators with a spanning set ofeigenvectors, corresponding to distinct eigenvalues (Sibilev)
I Common cyclic vectors for certain sets of normal operators(Sibilev, R-Wogen)
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via formal mult. of series
f = a0 + a1z + a2z2 + · · · ∈ Hol(D)
F
G∈ Mer(De)
F = A0 +A1
z+
A2
z2+ · · · G = B0 +
B1
z+
B2
z2+ · · ·
DefinitionF/G is a formal series continuation of f if
B0 +
B1
z+
B2
z2+ · · ·
a0 + a1z + a2z2 + · · ·
= A0 +
A1
z+
A2
z2+ · · ·
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via formal mult. of series
f = a0 + a1z + a2z2 + · · · ∈ Hol(D)
F
G∈ Mer(De)
F = A0 +A1
z+
A2
z2+ · · · G = B0 +
B1
z+
B2
z2+ · · ·
DefinitionF/G is a formal series continuation of f if
B0 +
B1
z+
B2
z2+ · · ·
a0 + a1z + a2z2 + · · ·
= A0 +
A1
z+
A2
z2+ · · ·
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via formal mult. of series
f = a0 + a1z + a2z2 + · · · ∈ Hol(D)
F
G∈ Mer(De)
F = A0 +A1
z+
A2
z2+ · · · G = B0 +
B1
z+
B2
z2+ · · ·
DefinitionF/G is a formal series continuation of f if
B0 +
B1
z+
B2
z2+ · · ·
a0 + a1z + a2z2 + · · ·
= A0 +
A1
z+
A2
z2+ · · ·
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via formal mult. of series
f = a0 + a1z + a2z2 + · · · ∈ Hol(D)
F
G∈ Mer(De)
F = A0 +A1
z+
A2
z2+ · · · G = B0 +
B1
z+
B2
z2+ · · ·
DefinitionF/G is a formal series continuation of f if
B0 +
B1
z+
B2
z2+ · · ·
a0 + a1z + a2z2 + · · ·
= A0 +
A1
z+
A2
z2+ · · ·
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via formal mult. of series
f = a0 + a1z + a2z2 + · · · ∈ Hol(D)
F
G∈ Mer(De)
F = A0 +A1
z+
A2
z2+ · · · G = B0 +
B1
z+
B2
z2+ · · ·
DefinitionF/G is a formal series continuation of f if
B0 +
B1
z+
B2
z2+ · · ·
a0 + a1z + a2z2 + · · ·
= A0 +
A1
z+
A2
z2+ · · ·
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via formal mult. of series
f = a0 + a1z + a2z2 + · · · ∈ Hol(D)
F
G∈ Mer(De)
F = A0 +A1
z+
A2
z2+ · · · G = B0 +
B1
z+
B2
z2+ · · ·
DefinitionF/G is a formal series continuation of f if
B0 +
B1
z+
B2
z2+ · · ·
a0 + a1z + a2z2 + · · ·
= A0 +
A1
z+
A2
z2+ · · ·
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
An ‘Abelian’ theorem for FS continuation
Theorem (R-Shapiro - 2000)
FS cont. are compatible with AC: If f has a FS continuation F/Gand f has an AC f to a nbh U of e iθ, then f = F/G on U.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Example
log(1− z) does not have a FS continuation.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Example
The Borel series ∑n
cn
z − zn, zn ∈ De
can have a FS continuation but not a PC.
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Example
The gap series ∑n
2−nz2n
does not have a FS continuation.
Note: It does not have an AC (Hadamard) nor a PC (Shapiro).
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of FS continuation
X is a B-space of analytic functions on D.
B : X → X , Bf =f − f (0)
z.
[f ] := spanBnf : n = 0, 1, 2, · · · 6= X .
L ∈ [f ]⊥ \ 0.
fL(λ) := L(f
· − λ)/
L(1
· − λ), λ ∈ De \ poles
Then fL is a formal series continuation of f .
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of FS continuation
X is a B-space of analytic functions on D.
B : X → X , Bf =f − f (0)
z.
[f ] := spanBnf : n = 0, 1, 2, · · · 6= X .
L ∈ [f ]⊥ \ 0.
fL(λ) := L(f
· − λ)/
L(1
· − λ), λ ∈ De \ poles
Then fL is a formal series continuation of f .
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of FS continuation
X is a B-space of analytic functions on D.
B : X → X , Bf =f − f (0)
z.
[f ] := spanBnf : n = 0, 1, 2, · · · 6= X .
L ∈ [f ]⊥ \ 0.
fL(λ) := L(f
· − λ)/
L(1
· − λ), λ ∈ De \ poles
Then fL is a formal series continuation of f .
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of FS continuation
X is a B-space of analytic functions on D.
B : X → X , Bf =f − f (0)
z.
[f ] := spanBnf : n = 0, 1, 2, · · · 6= X .
L ∈ [f ]⊥ \ 0.
fL(λ) := L(f
· − λ)/
L(1
· − λ), λ ∈ De \ poles
Then fL is a formal series continuation of f .
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of FS continuation
X is a B-space of analytic functions on D.
B : X → X , Bf =f − f (0)
z.
[f ] := spanBnf : n = 0, 1, 2, · · · 6= X .
L ∈ [f ]⊥ \ 0.
fL(λ) := L(f
· − λ)/
L(1
· − λ), λ ∈ De \ poles
Then fL is a formal series continuation of f .
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of FS continuation
X is a B-space of analytic functions on D.
B : X → X , Bf =f − f (0)
z.
[f ] := spanBnf : n = 0, 1, 2, · · · 6= X .
L ∈ [f ]⊥ \ 0.
fL(λ) := L(f
· − λ)/
L(1
· − λ), λ ∈ De \ poles
Then fL is a formal series continuation of f .
Fonctions Monogenes University of Richmond
![Page 106: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/106.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Applications of FS continuation
X is a B-space of analytic functions on D.
B : X → X , Bf =f − f (0)
z.
[f ] := spanBnf : n = 0, 1, 2, · · · 6= X .
L ∈ [f ]⊥ \ 0.
fL(λ) := L(f
· − λ)/
L(1
· − λ), λ ∈ De \ poles
Then fL is a formal series continuation of f .
Fonctions Monogenes University of Richmond
![Page 107: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/107.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Bochner and Bohnenblust continuation
Let A(n)n∈Z be a two-sided almost periodic sequence. Forexample
A(n) =∞∑
m=1
cme inθm , n ∈ Z,∞∑
m=1
|cm| <∞.
f1(z) :=∞∑
n=0
A(n)zn, |z | < 1,
f2(z) := −∞∑
n=1
A(−n)
zn, |z | > 1.
Theorem (B-B - 1934)
If f1 has an AC across e iθ, then it must continue to f2.
Fonctions Monogenes University of Richmond
![Page 108: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/108.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Bochner and Bohnenblust continuation
Let A(n)n∈Z be a two-sided almost periodic sequence. Forexample
A(n) =∞∑
m=1
cme inθm , n ∈ Z,∞∑
m=1
|cm| <∞.
f1(z) :=∞∑
n=0
A(n)zn, |z | < 1,
f2(z) := −∞∑
n=1
A(−n)
zn, |z | > 1.
Theorem (B-B - 1934)
If f1 has an AC across e iθ, then it must continue to f2.
Fonctions Monogenes University of Richmond
![Page 109: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/109.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Bochner and Bohnenblust continuation
Let A(n)n∈Z be a two-sided almost periodic sequence. Forexample
A(n) =∞∑
m=1
cme inθm , n ∈ Z,∞∑
m=1
|cm| <∞.
f1(z) :=∞∑
n=0
A(n)zn, |z | < 1,
f2(z) := −∞∑
n=1
A(−n)
zn, |z | > 1.
Theorem (B-B - 1934)
If f1 has an AC across e iθ, then it must continue to f2.
Fonctions Monogenes University of Richmond
![Page 110: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/110.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Bochner and Bohnenblust continuation
Let A(n)n∈Z be a two-sided almost periodic sequence. Forexample
A(n) =∞∑
m=1
cme inθm , n ∈ Z,∞∑
m=1
|cm| <∞.
f1(z) :=∞∑
n=0
A(n)zn, |z | < 1,
f2(z) := −∞∑
n=1
A(−n)
zn, |z | > 1.
Theorem (B-B - 1934)
If f1 has an AC across e iθ, then it must continue to f2.
Fonctions Monogenes University of Richmond
![Page 111: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/111.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Bochner and Bohnenblust continuation
Let A(n)n∈Z be a two-sided almost periodic sequence. Forexample
A(n) =∞∑
m=1
cme inθm , n ∈ Z,∞∑
m=1
|cm| <∞.
f1(z) :=∞∑
n=0
A(n)zn, |z | < 1,
f2(z) := −∞∑
n=1
A(−n)
zn, |z | > 1.
Theorem (B-B - 1934)
If f1 has an AC across e iθ, then it must continue to f2.
Fonctions Monogenes University of Richmond
![Page 112: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/112.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Let φ be an almost periodic function on R. For example,
φ(x) =∞∑
n=1
cne iλnx , λn ∈ R,∞∑
n=1
|cn| <∞.
f (z) =
∫ ∞0
φ(t)e−tzdt, <z > 0,
F (z) =
∫ 0
−∞φ(t)e−tzdt, <z < 0,
Theorem (B-B - 1934)
If f has an analytic continuation across iy , then this AC must beequal to F .
Fonctions Monogenes University of Richmond
![Page 113: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/113.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Let φ be an almost periodic function on R. For example,
φ(x) =∞∑
n=1
cne iλnx , λn ∈ R,∞∑
n=1
|cn| <∞.
f (z) =
∫ ∞0
φ(t)e−tzdt, <z > 0,
F (z) =
∫ 0
−∞φ(t)e−tzdt, <z < 0,
Theorem (B-B - 1934)
If f has an analytic continuation across iy , then this AC must beequal to F .
Fonctions Monogenes University of Richmond
![Page 114: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/114.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Let φ be an almost periodic function on R. For example,
φ(x) =∞∑
n=1
cne iλnx , λn ∈ R,∞∑
n=1
|cn| <∞.
f (z) =
∫ ∞0
φ(t)e−tzdt, <z > 0,
F (z) =
∫ 0
−∞φ(t)e−tzdt, <z < 0,
Theorem (B-B - 1934)
If f has an analytic continuation across iy , then this AC must beequal to F .
Fonctions Monogenes University of Richmond
![Page 115: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/115.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Let φ be an almost periodic function on R. For example,
φ(x) =∞∑
n=1
cne iλnx , λn ∈ R,∞∑
n=1
|cn| <∞.
f (z) =
∫ ∞0
φ(t)e−tzdt, <z > 0,
F (z) =
∫ 0
−∞φ(t)e−tzdt, <z < 0,
Theorem (B-B - 1934)
If f has an analytic continuation across iy , then this AC must beequal to F .
Fonctions Monogenes University of Richmond
![Page 116: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/116.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Let φ be an almost periodic function on R. For example,
φ(x) =∞∑
n=1
cne iλnx , λn ∈ R,∞∑
n=1
|cn| <∞.
f (z) =
∫ ∞0
φ(t)e−tzdt, <z > 0,
F (z) =
∫ 0
−∞φ(t)e−tzdt, <z < 0,
Theorem (B-B - 1934)
If f has an analytic continuation across iy , then this AC must beequal to F .
Fonctions Monogenes University of Richmond
![Page 117: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/117.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Suppose
φ(x) =∞∑
n=1
cne iλnx , λn ∈ R,∞∑
n=1
|cn| <∞.
Then
f (z) =∞∑
n=1
cn
z − iλn, <z > 0,
F (z) =∞∑
n=1
cn
z − iλn, <z < 0.
Thenlim
x→0+f (x + iy) = lim
x→0−F (x + iy), a.e. y
Fonctions Monogenes University of Richmond
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Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via overconvergence
X is a B-space of analytic functions on D with
X = span 1
z − λ: |λ| > 1 E = span 1
z − λ: λ ∈ A 6= X .
If f ∈ E , then there are fn ∈ Rat(A) with fn → f in norm.
Theorem (Shapiro - 1968)
fn → Sf ucs of De \ A.
So Sf is a ‘continuation’ of f .
Theorem (R-Shapiro - 2000)
If f has an AC continuation across e iθ, it must continue to Sf .
Fonctions Monogenes University of Richmond
![Page 119: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/119.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via overconvergence
X is a B-space of analytic functions on D with
X = span 1
z − λ: |λ| > 1 E = span 1
z − λ: λ ∈ A 6= X .
If f ∈ E , then there are fn ∈ Rat(A) with fn → f in norm.
Theorem (Shapiro - 1968)
fn → Sf ucs of De \ A.
So Sf is a ‘continuation’ of f .
Theorem (R-Shapiro - 2000)
If f has an AC continuation across e iθ, it must continue to Sf .
Fonctions Monogenes University of Richmond
![Page 120: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/120.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via overconvergence
X is a B-space of analytic functions on D with
X = span 1
z − λ: |λ| > 1 E = span 1
z − λ: λ ∈ A 6= X .
If f ∈ E , then there are fn ∈ Rat(A) with fn → f in norm.
Theorem (Shapiro - 1968)
fn → Sf ucs of De \ A.
So Sf is a ‘continuation’ of f .
Theorem (R-Shapiro - 2000)
If f has an AC continuation across e iθ, it must continue to Sf .
Fonctions Monogenes University of Richmond
![Page 121: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/121.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via overconvergence
X is a B-space of analytic functions on D with
X = span 1
z − λ: |λ| > 1 E = span 1
z − λ: λ ∈ A 6= X .
If f ∈ E , then there are fn ∈ Rat(A) with fn → f in norm.
Theorem (Shapiro - 1968)
fn → Sf ucs of De \ A.
So Sf is a ‘continuation’ of f .
Theorem (R-Shapiro - 2000)
If f has an AC continuation across e iθ, it must continue to Sf .
Fonctions Monogenes University of Richmond
![Page 122: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/122.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via overconvergence
X is a B-space of analytic functions on D with
X = span 1
z − λ: |λ| > 1 E = span 1
z − λ: λ ∈ A 6= X .
If f ∈ E , then there are fn ∈ Rat(A) with fn → f in norm.
Theorem (Shapiro - 1968)
fn → Sf ucs of De \ A.
So Sf is a ‘continuation’ of f .
Theorem (R-Shapiro - 2000)
If f has an AC continuation across e iθ, it must continue to Sf .
Fonctions Monogenes University of Richmond
![Page 123: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/123.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via overconvergence
X is a B-space of analytic functions on D with
X = span 1
z − λ: |λ| > 1 E = span 1
z − λ: λ ∈ A 6= X .
If f ∈ E , then there are fn ∈ Rat(A) with fn → f in norm.
Theorem (Shapiro - 1968)
fn → Sf ucs of De \ A.
So Sf is a ‘continuation’ of f .
Theorem (R-Shapiro - 2000)
If f has an AC continuation across e iθ, it must continue to Sf .
Fonctions Monogenes University of Richmond
![Page 124: Fonctions Monog enes - University of Richmondwross/PDF/SouthAlabama-Seminar.pdfFonctions Monog enesDivergent seriesGACPCGCFSBBOverconvergence Fonctions Monog enes William T. Ross University](https://reader033.vdocuments.site/reader033/viewer/2022042122/5e9ce99034ccfd6e4c346617/html5/thumbnails/124.jpg)
Fonctions Monogenes Divergent series GAC PC GC FS BB Overconvergence
Continuation via overconvergence
X is a B-space of analytic functions on D with
X = span 1
z − λ: |λ| > 1 E = span 1
z − λ: λ ∈ A 6= X .
If f ∈ E , then there are fn ∈ Rat(A) with fn → f in norm.
Theorem (Shapiro - 1968)
fn → Sf ucs of De \ A.
So Sf is a ‘continuation’ of f .
Theorem (R-Shapiro - 2000)
If f has an AC continuation across e iθ, it must continue to Sf .
Fonctions Monogenes University of Richmond