The Gibbs phenomenon in generalized Padé approximationG. Németh and G. Páris Citation: Journal of Mathematical Physics 26, 1175 (1985); doi: 10.1063/1.526521 View online: http://dx.doi.org/10.1063/1.526521 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/26/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On Padé approximants to virial series J. Chem. Phys. 129, 044509 (2008); 10.1063/1.2958914 Fourier series and the Gibbs phenomenon Am. J. Phys. 60, 425 (1992); 10.1119/1.16895 Padé Approximants Comput. Phys. 6, 82 (1992); 10.1063/1.4823047 Correction terms for Padé approximants J. Math. Phys. 16, 918 (1975); 10.1063/1.522598 Properties of Generalizations to Padé Approximants J. Math. Phys. 10, 1875 (1969); 10.1063/1.1664773

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The Gibbs phenomenon in generalized Pade approximation G. Nemeth and G. Paris Hungarian Academy of Sciences, P. O. B. 49, H-1525 Budapest, Hungary

(Received 30 May 1984; accepted for publication 21 September 1984)

The Gibbs phenomenon in generalized Pade approximation is discussed, and with the aid of some rational approximants the Gibbs constants are determined. In addition, the steepest of the rational approximants is calculated.

I. INTRODUCTION

If one approximates a discontinuous function by poly­nomials (or by Fourier series) it leads to an unusual proper­ty-the Gibbs phenomenon. The polynomials do not con­verge to the function near the discontinuity. The maximal value of the error is called the Gibbs constant. For example, it is well known that when we approximate the function sgn (x) in ( - 1, + 1) by Fourier series the Gibbs constant is

2 i1T sin t G = - - dt - 1 = 0.178 979 7 .... 1T 0 t

Another important property of the approximation is the steepness. We call the value of the derivative of the approxi­mant at the discontinuity the steepness. For the function sgn (x) the steepness is (411T)(n + 1), for an n-term Fourier approximation. It is noted that both properties in a multidi­mensional generalization can appear in more difficult analy­tical features (rapid behavior of the trajectory in nonlinear system, strange attractors, etc.). It is desirable to obtain an approximation for which the Gibbs constant is as small as possible and the steepness is as high as possible.

Zygmund I proved that one can decrease the Gibbs con­stant by Cesaro's method of summing series, but as experi­mentally shown by Arfken2 this method halves the steep­ness.

In this paper we consider some rational functions and we show that in our case the generalized Pade approximants have Gibbs constants smaller than G and their steepness is higher than Cn.

The paper is arranged as follows. In Sec. II we consider the generalized Pade approximation in the sense ofCheney3; in Sec. III we treat the same problem using the method of Clenshaw and Lord.4 We provide proofs of the results of the previous sections in Sec. IV, and in Sec. V we present some calculations of the steepness following Cesaro's method of summing series.

II. APPROXIMANTS FOR sgn (x) BY CHENEY'S METHOD

Here and further we apply to a series representation for the function sgn (x) in the form

{-I, -1~X<O} 00

sgn(x)= +1, O<x~1 =n~oCnT2n+dx),

(1)

where T2n + I (x) is the Chebyshev polynomial and Cn = (411T)( - 1)"/(2n + 1). The rationals

R () = ~~o Pi T2i+ I (x) nm X , , ~7=oqiT2i(X)

n,m = 0,1,2, ... , (2)

which satisfy the relation

ttoqiT2i(X)}Sgn(x) - i~OPiT2i+ I (x)

= O(T2n+2m+3(X)), (3)

are called the generalized Pade approximants.3 The 0 term in (3) means a function for which the series in Ti(x) begins with the term T 2n + 2m + 3 (X).

Next we shall list our main results. The solution of problem (3) in explicit form is

3F2( - m, - n + !,n + m + 2;M;x2) Rnm(x) =Anmx ,

, '3F2( - n, - m - !,n + m + ~;P;X2) (4)

where the steepness A is

4 n! r (m + ~) r (n + m + 2) A =-.-. (5)

n,m ..[ii m! r(n + 1) r(n + m +~)

For n = 0 we can get the classic result. In this case the ap­proximating polynomial is

(6)

Its error function takes the highest maximum at the point x = aim, m-oo. This value is the Gibbs constant

G = (411T)a IF2(!;M; - a 2) - 1.

Differentiating by a we get an equation for a:

oFI(;~; - a 2) = sin 2al2a = O.

Its first zero is a = 1T 12. The previous series considered in integral form gives the classical result

G =..±. t sin 2au du _ 1 = ~ {1T sin t dt _ 1. 1T Jo 2u 1T Jo t

The steepness is

Ao,m = (411T) (m + 1). (7)

Second, we consider the case m = 0, the reciprocal polyno­mial case. In this case the approximants are

Rn,o(x) = An,ox/3F2( - n,n +~, - !;P;X2). (8)

Its error function takes its maximum at the point x = Pin n-oo. By elementary calculations one can prove that Pis the root of the equation

Jo(2{3) = 0, where Jo(x) is the Bessel function. From its first root we get

P = 1.202412 778 8 ... ,

therefore the Gibbs constant is

_ ({2P JI(u) )-1 Go, I - Vo -u- du - 1 = 0.051 356067 .... (9)

1175 J. Math. Phys. 26 (6), June 1985 0022-2488/85/061175-04$02.50 © 1985 American Institute of Physics 1175

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That is, in this case the Gibbs constantis approximately 5 %. The steepness is

An.o = 2n!(n + 1)!/F(n + !)F(n +~) = 2(n + l)an,

where an ;:::; 1 for moderate and large values of n. The most interesting case is n = m. The approximants are

~2( - n, - n + pn + 2;~,~;X2) Rn.n(x) =An.n x 3 2' (10)

3F2( - n, - n - pn + };;p;X )

where

A = 2(2n + 1) F(2n + 2). n.n .[iT F (2n + ~)

The error function takes its maximal value at the point x = rln3/2, n-oo. The constant ris the root of the equation

00 (_ 2r)k L =0, k=O kl2Hlk

and its value r = 0.951020874 .... The Gibbs constant is given by the formula

GI,I = ~ oF2(;~,Pr) _ 1 = 0.008 148 902 .... /iii oF2(;!,1;2r)

(11)

Semerdjiev and Nedelchev5 performed a numerical experi­ment for determining G 1,1 enabling them to state that G 1,1

does not exceed 2 %, The steepest is

An.n = (4..j21.[iT) n312bn,

where bn ;:::; 1 for moderate (n > 10) and large values of n.

III. APPROXIMATIONS FOR sgn(x) BY THE CLENSHAW­LORD METHOD

Again, from series representation (1) we determine the rationals Sn.m (x) by the method of Clenshaw and Lord4

:

S () = I.~oriT2i+ 1 (x)

nm X • . I.~ = OSi T2i (x)

(12)

The coefficients ri and Si can be determined from the equa­lity

sgn(x) -Sn.m(x) = o (T2n+ 2m + 3 (X)). (13)

Our result is 4

Sn.m(x) = - (m + 1)(2n + 1)x 1T'

4F3( - m,m + 2, - n + !,n + ~;M~;X2) X 2'

4F3( - n,n + 1, - m - !,m + ~;!,I,I;X ) (14)

First we consider the reciprocal polynomial approximants (m=O)

S () = (4hr)(2n + 1)x n,O X 2

4F3( - n,n + 1, - M;p,l;x ) (15)

Its error function takes the maximal value at the point x = lJln, n-oo, where lJ is the root ofthe equation,

Jo(lJ) = WdlJ),

and lJ = 0.940 770 564 .... Its Gibbs constant is

1176 J. Math. Phys .• Vol. 26. No.6. June 1985

G1•0 = 0.082 417 272 .... (16)

The steepness is 411r(2n + 1). The case n = m presents powerful approximants. Here

4 Sn.n(x) = - (n + 1)(2n + 1)x

11'

The error function takes its maximal value at the point x = TJln2, n-oo. The value ofTJ is the root of the equation

ber(2&) =0,

its first root is TJ = 1.014541 594 .... The Gibbs constant is

G1•I = ~ oF3(;~'~'~;TJ; _ 1 = 0.049325286.... (18) 11' oF3(;P,I;TJ)

The steepness is (4111')(n + 1 )(2n + 1). This is the highest val­ue in all cases.

IV. PROOFS

First we will prove formula (4). Let us consider a more generalized series expansion for sgn (x) like (1):

2k () 2k I() ~ T2J+ dx) x sgnx = '!k i=-oF(k+j+~)F(k-j+!)'

k = 0,1,2, ....

Next, multiplying it by numbers qk (k = O,I, ... ,n), then summing these equations, we get

CtoqkX2k )Sgn (x)

4 00 (-IY n klHlk =- L -. -T2j + l (x) L qk .

11' j = 0 21 + 1 k = 0 (~ + A (! -A We want to determine the coefficients in such a manner that the following equations are satisfied:

n k!Hlk L qk =0, k=O (~+AH-A

j= m + I,m + 2, ... ,m +n. In this case the numerator polynomial will be

.i. ~ (- IY T. (x) ~ k!Hlk ~2' 1 2}+1 ~qk(3 ')(1 .) 11'j=0;'+ k=O '1+1 k'l-1 k

To solve the previous equations let us suppose for a moment that

qk = (- n)d - m - !)k(n + m + ~)klkfHlk' Consider now the sum

S= ± (-n)k(-m-!)k(n+m+~)k k=O k!(~ +A(!-A

= 3F2( - n, - m - !,n + m + ~;~ + j,! - j; I). S is a Saalschiitz-type hypergeometric sum and therefore it is summable by factorial functions. Really,6

S = (1 + m - 11n ( - 1 - m - n - 11n

X [I! - 11n( -! - j - n)n] -I.

G. Nemeth and G. paris 1176

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It is not difficult to see that all products differ from zero except the first one. Further, when j runs from m + 1 to m + n then 1 + m - j runs from 0 to - n + 1 by - 1. Therefore, S = 0 for all j (j = m + 1, ... ,m + n). We have thus proved the form of the denominator polynomial. To get the explicit form of the numerator polynomial we apply the value of S forj = O,I, ... ,m:

Z=.! i (:-IY T2j+dx) 1rj=oq+l

X (1 + m -lIn ( - 1 - m - n -lIn .

H - lIn ( - ! - n - lIn Taking the power form of the Chebyshev polynomial

(l/X)T2H 1 (x) = (2j + 1)( - IY zPI( - jJ + q;X2),

we get

Z= (n + m)!r(n + m + 2).!x

!!lnmnm!r(m + 2) 1r

m

XL (2j + 1) 2FI (-jJ + q;X2) j=O

( - m)j(n + m + 2)j(! - n)j X .

( - n - m)j(m + 2)j(1 + n)j

Let us transform Z to the power form in x 2:

Z=.!x (n+m)!(n+m+ I)! 1r !!lnmnm!(m + I)!

m 2" ( - m);(n + m + 2);(! - n)/ XL (-4x)' w, /=0 (- n - m)/(m + 2);(1 + n)/

where

W= mi; (2i + l)j j=O j!

X ( - m + i )j(n + m + i + 2)A - n + i )jH + i)j . ( - n - m + i )j(m + 2 + i )j(1 + n + i)A + i)j

The sum Wis an sF4 hypergeometric function which one can sum by theorem of Dougall7

:

W=( _1)m

r(i +m + 2)r( -! -i)rH + n +i)n!( - n -m); X .

r(2i + 2)r( -! - m)r(1 + n + m)(n + m)!

By elementary calculations we get the required result:

Z=--±"'~ rim +1) r(n +m +2) x

Iii m! r (n +!) r (n + m + 1)

~ (- m);( - n + !);(n + m + 2); 2"

X~ x~ /=0 z1(1);(1);

TABLE I. Gibbs constant and the steepness.

Cheney'S method

Fourier Reciprocal series polynomial Rational

18% 5.1% 0.8% (4/lT)n 2n (40j2/,fii) n3 / 2

1177 J. Math. Phys., Vol. 26, No.6, June 1985

Proof of the form of Gibbs constants for the cases m = 0 and m = n can be obtained by elementary analysis. Here we omit the details. The proof of the results of Sec. III is analo­gous with the previous one.

v. CESARO'S METHOD OF SUMMING SERIES FOR sgn (x) AND THE STEEPNESS

It is well known that if we have a series

"" L av , 1'=0

its Cesaro's sum is defined by the formula

C a __ ~ (-n)v n ~ av •

1'=0 (-n-a)v

Here a is a positive parameter. It is well known that if a = 1, C~ is Fejer's arithmetic mean and in this case the Gibbs phenomenon does not occur. (The case a = 0 gives the origi­nal series.)

Next we will prove that if a > ao = 0.439 551 2893 ... , then, again, the Gibbs phenomenon does not occur.

Consider again the series (1), thus av = (4/1T)[( - It/(2v + 1)] T2v+ dx).

By short, elementary calculation we get

Cal ) =.! n + 1 + a n (- n)j(n + a + 2)j(!)j 2j nX xL x.

1r 1 + a j=O (~)j(I + aI2)j(~ + al2)j

Its error function has the maximum at the point x = sin, n- 00. The maximum is

Ga(s)=.!_s- f _1_ (_S2)k -1 1r 1 + a k=O 2k + 1 (1 + a12)d1 + al2)k

= 2. t (1 _ t )a sin 2st dt - 1. 1r Jo t

By determining the value s we get the equation

f (_r)k =0, k=O (1 + a12)d1 + al2)k

or in integral form

f (1 - t)a cos 2st dt = 0,

The solutions a and s of the equation G a (s) and of the pre­vious equation are

a = 0.439 551 2893 ... ,

s = 2.025 782 092 ... .

Note: Gronwall8 also determined the values a and s, but the stated precision of his results is incorrect. The steepness in Cesaro's method is (4/1r)(n + 1 + a)/(I + a). For a = 1, the steepness is (4/1r)(n + 2)/2. It is halved corresponding to

Method of Clenshaw and Lord

Reciprocal polynomial

8.2% (8/1r)n

Rational

4.9% (8/lT) n2

Cesaro's sum

18%s;Gs;0% (4/lT)n/(1 + a)

G. NlImeth and G. Paris 1177

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a = o. Thus, we have proved that Cesaro's method of sum­ming series decreases the Gibbs constant, but it also de­creases the steepness.

VI. CONCLUSIONS

As a means of summarizing our results, we have listed in Table I the Gibbs constants and their steepness corre­sponding to the methods used.

ACKNOWLEDGMENT

The authors are indebted to Dr. M. Huszar for his con­tinued interest and for helpful discussions on hypergeome­tric identities.

1178 J. Math. Phys., Vol. 26, No.6, June 1985

I A. Zygmund, Trigonometric Series (Cambridge U. P., Cambridge, 1959). 2G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1968).

lE. W. Cheney, Introduction to Approximation Theory (McGraw-Hill, New York, 1966).

'C. W. Clenshaw and K. Lord, "Rational Approximations from Chebyhev Series," in Studies in Numerical Analysis, edited by B. K. P. Scaife (Aca­demic, London, 1974), pp. 95-113.

'C. Semerdjiev and C. N ede1chev, Diagonal Pade Approximants for a Dis­continuous Function Given by its Fourier Series (Plovdiv U. P., Plovdiv, 1977), Vol. 15, p. 427 (in Bulgarian).

6 A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.

7J. Dougall, Proc. Edinburgh Math. Soc. 25, 114 (1907). ST. H. Gronwall, Ann. Math. 31, 232 (1930).

G. Nemeth and G. Paris 1178

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