Summability methods bc acFrom Wikipedia, the free encyclopediaContents1 BochnerRiesz mean 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Borel summation 32.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Borels exponential summation method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Borels integral summation method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Borels integral summation method with analytic continuation . . . . . . . . . . . . . . . . 42.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Equivalence of Borel and weak Borel summation . . . . . . . . . . . . . . . . . . . . . . . 42.2.3 Relationship to other summation methods . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Uniqueness theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.1 Watsons theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.2 Carlemans theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.1 The geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.2 An alternating factorial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.3 An example in which equivalence fails . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Existence results and the domain of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5.1 Summability on chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5.2 The Borel polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.3 A Tauberian Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Cauchy principal value 113.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11iii CONTENTS3.2 Distribution theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.1 Well-denedness as a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.2 More general denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Chapter 1BochnerRiesz meanThe BochnerRiesz mean is a summability method often used in harmonic analysis when considering convergenceof Fourier series and Fourier integrals. It was introduced by Salomon Bochner as a modication of the Riesz mean.Dene()+={, if > 00, otherwise.Let f be a periodic function, thought of as being on the n-torus, Tn, and having Fourier coecientsf(k) for k Zn. Then the BochnerRiesz means of complex order , BRf of (where R > 0 and Re() > 0 ) are dened asBRf() =kZn|k|R(1 |k|2R2)+f(k)e2ik.Analogously, for a function f on Rnwith Fourier transformf() , the BochnerRiesz means of complex order ,SRf (where R > 0 and Re() > 0 ) are dened asSRf(x) =||R(1 ||2R2)+f()e2ixd.For >0 andn=1 ,SR andBR may be written as convolution operators, where the convolution kernel is anapproximate identity. As such, in these cases, considering the almost everywhere convergence of BochnerRieszmeans for functions inLpspaces is much simpler than the problem of regular almost everywhere convergenceof Fourier series/integrals (corresponding to =0 ). In higher dimensions, the convolution kernels become morebadly behaved (specically, for n12, the kernel is no longer integrable) and establishing almost everywhereconvergence becomes correspondingly more dicult.Another question is that of for which and which p the BochnerRiesz means of an Lpfunction converge in norm.This is of fundamental importance for n 2 , since regular spherical normconvergence (again corresponding to = 0) fails in Lpwhen p = 2 . This was shown in a paper of 1971 by Charles Feerman.[1] By a transference result, the Rnand Tnproblems are equivalent to one another, and as such, by an argument using the uniformboundedness principle,for any particular p (1, ) , Lpnorm convergence follows in both cases for exactly those where (1 ||2)+ isthe symbol of an Lpbounded Fourier multiplier operator. For n=2 , this question has been completely resolved,but for n 3 , it has only been partially answered. The case of n = 1 is not interesting here as convergence followsfor p (1, ) in the most dicult = 0 case as a consequence of the Lpboundedness of the Hilbert transform andan argument of Marcel Riesz.12 CHAPTER 1. BOCHNERRIESZ MEAN1.1 References[1] Feerman, Charles (1971). The multiplier problemfor the ball. Annals of Mathematics 94 (2): 330336. doi:10.2307/1970864.1.2 Further readingLu, Shanzhen (2013). Bochner-Riesz Means on Euclidean Spaces (First ed.). World Scientic. ISBN 978-981-4458-76-4.Grafakos, Loukas (2008). Classical Fourier Analysis (Second ed.). Berlin: Springer. ISBN 978-0-387-09431-1.Grafakos, Loukas (2009). Modern Fourier Analysis (Second ed.). Berlin: Springer. ISBN978-0-387-09433-5.Stein, Elias M. & Murphy, Timothy S. (1993). Harmonic Analysis: Real-variable Methods, Orthogonality, andOscillatory Integrals. Princeton: Princeton University Press. ISBN 0-691-03216-5.Chapter 2Borel summationBorel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classicaldivergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leer, who was the recognized lordof complex analysis. Mittag-Leer listened politely to what Borel had to say and then, placing his hand upon thecomplete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'.Mark Kac, quoted by Reed & Simon (1978, p. 38)In mathematics, Borel summation is a summation method for divergent series, introduced by mile Borel (1899).It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum forsuch series. There are several variations of this method that are also called Borel summation, and a generalization ofit called Mittag-Leer summation.2.1 DenitionThere are (at least) three slightly dierent methods called Borel summation. They dier in which series they can sum,but are consistent, meaning that if two of the methods sum the same series they give the same answer.Throughout let A(z) denote a formal power seriesA(z) =k=0akzkand dene the Borel transform of A to be its equivalent exponential seriesBA(t) k=0akk!tk.2.1.1 Borels exponential summation methodLet An(z) denote the partial sumAn(z) =nk=0akzk.A weak form of Borels summation method denes the Borel sum of A to belimtetn=0tnn!An(z).34 CHAPTER 2. BOREL SUMMATIONIf this converges at z C to some a(z), we say that the weak Borel sum of A converges at z, and write akzk=a(z) (wB) .2.1.2 Borels integral summation methodSuppose that the Borel transform converges for all real numbers to a function growing suciently slowly that thefollowing integral is well dened (as an improper integral), the Borel sum of A is given by0etBA(tz) dt.If the integral converges at z C to some a(z), we say that the Borel sum of A converges at z, and write akzk=a(z) (B) .2.1.3 Borels integral summation method with analytic continuationThis is similar to Borels integral summation method, except that the Borel transform need not converge for all t, butconverges to an analytic function of t near 0 that can be analytically continued along the positive real axis.2.2 Basic properties2.2.1 RegularityThe methods (B) and (wB) are both regular summation methods, meaning that whenever A(z) converges (in thestandard sense), then the Borel sum and weak Borel sum also converge, and do so to the same value. i.e.k=0akzk= A(z) < akzk= A(z)(B,wB).Regularity of (B) is easily seen by a change in order of integration: if A(z) is convergent at z,thenA(z) =k=0akzk=k=0ak(0ettkdt) zkk!=0etk=0ak(tz)kk!dt,where the rightmost expression is exactly the Borel sum at z.Regularity of (B) and (wB) imply that these methods provide analytic extensions to A(z).2.2.2 Equivalence of Borel and weak Borel summationAny series A(z) that is weak Borel summable at z C is also Borel summable at z. However, one can constructexamples of series which are divergent under weak Borel summation, but which are Borel summable. The followingtheorem characterises the equivalence of the two methods.Theorem ((Hardy 1992, 8.5)).Let A(z) be a formal power series, and x z C, then:1. If akzk= a(z) (wB) , then akzk= a(z) (B) .2. If akzk= a(z) (B) , and limtetBA(zt) = 0, then akzk= a(z) (wB) .2.3. UNIQUENESS THEOREMS 52.2.3 Relationship to other summation methods(B) is the special case of Mittag-Leer summation with = 1.(wB) can be seen as the limiting case of generalized Euler summation method (E,q) in the sense that as q the domain of convergence of the (E,q) method converges up to the domain of convergence for (B).[1]2.3 Uniqueness theoremsThere are always many dierent functions with any given asymptotic expansion. However there is sometimes a bestpossible function, in the sense that the errors in the nite-dimensional approximations are as small as possible in someregion. Watsons theorem and Carlemans theorem show that Borel summation produces such a best possible sum ofthe series.2.3.1 Watsons theoremWatsons theorem gives conditions for a function to be the Borel sum of its asymptotic series. Suppose that f is afunction satisfying the following conditions:f is holomorphic in some region |z|