CENTRAL RESEARCH INSTITUTE FOR PHYSICSBUDAPEST

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Т К ' I S S . < S J ß

G, NÉMETH/

G, PARIS

T H E G I B B S P H E N O M E N O N IN G E N E R A L I Z E D P A D E A P P R O X I M A T I O N

сHungarian Academy o f Sciences

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

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2017

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KFKI-1984-50

T HE G I B B S P H E N O M E N O N IN G E N E R A L I Z E D P A D E A P P R O X I M A T I O N

G. NÉMETH, G. PARIS

Central Research Institute for Physics H-1525 Budapest 114, P.O.B.49, Hungary

HU ISSN 0368 5330 ISBN 963 372 227 6

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A B S T R A C T

The Gibbs phenomenon in generalized Pádé approximation is discussed, and with the aid of some rational approximants the Gibbs constants are determined.

In addition, the steepness of the rational approximants is calculated.

А Н Н О Т А Ц И Я

Исследуется явление Гиббса при обобщенной аппроксимации Падэ с помощью рациональных дробей. Определяется постоянная Гиббса и крутизна для разных аппроксимаций.

KI VONAT

Az általánosított Padé-közelitések Gibbs-jelenségét vizsgáljuk racionális törtek segítségével. Meghatározzuk a közelítések Gibbs állandóját és a mere

dekségüket .

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1. I n t r o d u c t i o n

If one a p p r o x i m a t e s a d i s c o n t i n u o u s f u n c t i o n b y p o l y  n o m i a l s /or Ъ у Fou r i e r s e r i e s / it leads to a n u n u s u a l p r o  p e r t y - the G i b b s p h e n o m e n o n . The p o l y n o m i a l s do n o t c o n  v e r g e to the f u n c t i o n n e a r the d i s c o n t i n u i t y . The m a x i m a l v a l u e of the e r r o r is c a l l e d G i b b s c o n s tant. For e x a mp l e , it is wel l k n o w n that w h e n we a p p r o x i m a t e the f u n c t i o n sgn o o in ( - i f" M ) b y F o u r i e r s e r i e s the G i b b s c o n s t a n t is

~ J u t - 1 -■* .

A n o t h e r i m p o r t a n t p r o p e r t y of the a p p r o x i m a t i o n is the s t e e p n e s s . We call the v a l u e of the d e r i v a t i v e of the a p p r o x i m a n t a t the d i s c o n t i n u i t y the s t e e p n e s s . For the f u n c t i o n sgn

00

the s t e e p n e s s is ii-(n-M) , for an n - t e r m F o u r i e r a p p r o x i m a t i o n . It is n o t e d that b o t h p r o p e r t i e s in a m u l t i d i m e n s i o n a l g e n e r a l i z a t i o n can a p p e a r in m o r e d i f f i  c u l t a n a l y t i c a l feature / r a p i d b e h a v i o u r of the t r a j e c t o r y in n o n l i n e a r system, s t r a n g e a t t r a c t o r s , e t c . / . It is d e s i r a b l e to o b t a i n a n a p p r o x i m a t i o n for w h i c h the G i b b s c o n s t a n t is as small as p o s s i b l e a n d the s t e e p n e s s is as h i g h as p o s s i b l e .

Z y g m u n d M p r o v e d tha t one can d e c r e a s e the G i bbs c o n s t a n t b y C e s a r o ’s m e t h o d of s u m m i n g ser i e s , b u t as e x  p e r i m e n t a l l y s h own by A r f k e n D 0 this m e t h o d h a l v e s the s t e e p n e s s .

In this paper we c o n s i d e r some r a t i o n a l f u n c t i o n s and we s h o w that in our case the g e n e r a l i z e d P á d é a p p r o x i m a n t s have Gibbs c o n s t a n t s s m a l l e r t h a n G a n d t h e i r s t e e p n e s s is h i g h e r than Cn.

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The p a p e r is a r r a n g e d as follows. In s e c t i o n 2 we c o n s i d e r the g e n e r a l i z e d Pádé a p p r o x i m a t i o n in the sense of C h e n e y [з] ; in S e c t i o n 3 we treat the same p r o b l e m u s i n g the m e t h o d of C l e n s h a w a n d L o r d M - We p r o v i d e proofs of the r e s u l t s of the p r e v i o u s s e c t i o n s in S e c t i o n

k , and in S e c t i o n 5 we p r e s e n t some c a l c u l a t i o n s of the s t e e p n e s s f o l l o w i n g C e s a r o ’s m e t h o d of s u m m i n g series.

2. A p p r o x i m a n t s for sgn (X ) b y C h e n e y ’s m e t h o d

He r e a n d f u r t h e r we a p p l y a s e r i e s r e p r e s e n t a t i o n for the f u n c t i o n s g n (x) in the form

(x)

- Т / **н * Т 1 Ь 5 с л п > ) ,**

1 f. \ j Q <, d h=0

/ 1/

w h ere T i h H (x ) is the C h e b y s h e v p o l y n o m i a l and The r a t i o n a l s

c - .

31 pi Talc, W

R „ „ 0 0 = - Т Г ---- -- — г . V /2/

■C*0

w h i c h s a t i s f y the r e l a t i o n

»n

(|л > ) ~ 2 _ p*

л /

are c a l l e d the g e n e r a l i z e d Pádé a p p r o x i m a n t s j_3j. The 0 - term in /3/ m e a n s a f u n c t i o n for w h i c h the s e r i e s in Т^(л) b e g i n s w i t h the t e r m i . n t 2 m h 3 .

N e x t we shall l i s t our m a i n resul t s . The s o l u t i o n of p ro b l e m /3/ in e x p l i c i t fo r m is

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3

Y’ i u i

л»

Him Г / „ _ 1

> 21-,х^;

5 ( r ri/'rr, l»nfwif^ y i ' Ь л 2}

Л/

W h e r e the s t e e p n e s s A is

A

^{ii_ n i}

Пт<~А)

r(nvi>H-2)

' b «nnS xx' шпГр(п^) r(r.frn»1)

F o r n = 0 we can get the c l a s s i c result. In this case the a p p r o x i m a t i n g p o l y n o m i a l is

R ü j V n W ® А о , л > * Л | . f / )

/

6

/

Its e r r o r f u n c t i o n t a k e s the h i g h e s t m a x i m u m at the p o int у » ~ I m ~ ? O o . Th i s v a l u e is the G i b b s c o n s t a n t

G - =

**Ы г * 2) н**

D i f f e r e n t i a t i n g b y X we g e t an e q u a t i o n for X

■ f ( ■ 1 > - ь , 1 } - Ь ^уьЛх q

Its f i r s t zero is t/ = -2- . The p r e v i o u s s e r i e s c o n s i d e r e d in i n t e g r a l form g i v e s the c l a s s i c a l r e s u l t .

Г - h

I

^I ^I^{, *2}

G = n , - n r - - 1 - X

The s t e e p n e s s is

A

^{Ot fY\ ~} ⁽^VT"^*

t l)

/ 7/

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к

Second, we c o n s i d e r the case m=0, the r e c i p r o c a l p o l y  n o m i a l case. In t h i s case the a p p r o x i m a n t s are

Its e r r o r f u n c t i o n t a kes its m a x i m u m a t the p o i n t X = Т Г ri->oö. B y e l e m e n t a r y c a l c u l a t i o n s one c a n prove tha t ( b>

is the r o o t of the e q u a t i o n

w h e r e J 0 (x) is the B e s s e l f u n c tion. F r o m its f i r s t r o o t we ge t

T h e r e f o r e the G i h b s c o n s t a n t is

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Tha t is, in this c a s e the G i b b s c o n s t a n t is a p p r o x i m a t e  ly 5 °/>. The s t e e p n e s s is

' A 0 , 0 ‘

II .'lltl)’-

1 1 )

= ^ ( n t i ) a n ;

w he r e Q n $ z \ for m o d e r a t e a n d large v a l u e s of A . The m o s t i n t e r e s t i n g c a s e is r » = m . The a p p r o x i m a n t s are

•t? u y - A у

^ » , r i W - ” v r , ____ 1 (• « О

' A ° /

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w h o re Д - 2.(£пН) Г^3п»д.>

Г ( 2 п г ^ )

The e r r o r f u n c t i o n t a k e s its m a x i m a l v a l u e at the p o i n t I n — > 0 0 • The c o n s t a n t is the roo t of the e q u a t i o n

СО о

t l f í i - 0

r 0 t't ( % ) ( , . '

a n d its v a l u e rj T - 0 . 9 5 1 0 2 0 8 7 ^ . . ., The G i b b s c o n s t a n t is g i v e n b y the f o r m u l a

G = Щ Ъ \ и -o.mnmxL....

ш , L

^{i f )} ^{/ и /}

Semerd,jiev a n d N e d e l c h e v 1 Я p e r f o r m e d a n u m e r i c a l e x  p e r i m e n t f o r d e t e r m i n i n g G, , e n a b l i n g t h e m to state that

, 1 »1

^ does n o t e x c e e d 2 °f>.

The s t e e p n e s s is

0 /\,n f JT d )

w h e r e for m o d e r a t e a n d large v a l u e s oft! ■

3. A p p r o x i m a t i o n s f o r sgn|V) b y C l e n s h a w - L o r d ’s m e t h o d A g a i n , from s e r i e s r e p r e s e n t a t i o n / 1 / we d e t e r m i n e the r a t i o n a l e S (x)

n,m t /

S n . m W “ " I F

1 Sc liiW

/12/ -C-Ö

b y the m e t h o d of C l e n s h a w a n d L o r d W - The c o e f f i c i e n t s r^ a n d s^ c a n be d e t e r m i n e d f r o m the e q u a l i t y

( X ) —

S ^ GO — I

^г»

1

+

2

гпг.з(х^ ^{/13 /}

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6

O u r r é s ü l t Is

( x ) — ~ ( т и ) ['М4)-Х

Í (~n

/ i V

F i r s t we c o n s i d e r the r e c i p r o c a l p o l y n o m i a l a p p r o x i m a n t s /ш=0/

V X n v Q X _____________

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Its e r r o r f u n c t i o n t a kes the m a x i m a l v a l u e at the p o i n t x = j : t n o o w h e r e cf is the ro o t of the e q u a t i o n ,

a n d & = 0 . 9 4 0 7 7 0 5 6 4 ... » I t s Gibbs c o n s t a n t is

G x 0 = 0 . 0 8 2 4 1 7 2 7 2 ... , /16 /

The s t e e p n e s s is •

The case W=*T\ p r e s e n t s p o w e r f u l a p p r o x i m a n t s . Here

S ,

^{/ Л} ^{u .} ^, J k ( - n r ” 4 i n, } í i n * 2 ; í i h i , ,

**« W - S N M * • ¥V , 4 i ~Г Т**

⁷

'Г

⁷

о - /17/

Л ( Г ° ' I / 4 *1' ‘‘i* г» ' Ъ x ✓

The e r r o r f u n c t i o n t a k e s its m a x i m a l v a l u e at the p o i n t X - Tyi ) П — > 0 0 . The v a l u e of 77 is the ro o t of the e q u a  tion

= - 0 /

its f i r s t r o o t is í j = I . O I4 5 4 1 5 9 4 ... . The G i bbs c o n s t a n t is

o h f (■ ‘L i i i V )

zS, c ÜL L i _ x

jI X()í i’b'í j'l'1)

- 1 = о.<М)9:йИ*е..

/18/

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»

The s t e e p n e s s is 4-fy> + t y f i r ) r f ) • Thi s is the h i g h e s t value in all cases.

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;l . Proofs

First wo wil l prove f o r m u l a /4/. Let us c o n s i d e r a more g e n e r a l i z e d s e r i e s e x p a n s i o n for sgn ( x ) l i k e / 1 / :

Next, m u l t i p l y i n g it h y n u m b e r s ( " k «0, О then s u m m i n g these e q u a t i o n s , we get

We w a n t to d e t e r m i n e the c o e f f i c i e n t s in such a m a n n e r that the f o l l o w i n g e q u a t i o n s are s a t i s f i e d

q é - H h b — = tmt-i Ivn t2y... m t o .

In this case the n u m e r a t o r p o l y n o m i a l w i l l he

JC Í r ^ Í T

To solve the p r e v i o u s e q u a t i o n s let us suppose for a m o  m e n t that

„ _ ( ~ ” ) ь

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8

C o n s i d e r n o w the sum

S is a Salsiitz type h y p e r g e o m e t r i e sum a n d t h e r e f o r e it

It is n o t d i f f i c u l t to see th a t all p r o d u c t s d i f f e r from

in+1 to m + n then 1+m-j runs f r o m 0 to -n + 1 b y -1. T h e r e  fore S = 0 for all j / j=m+l, ... m+n/. We h a v e thus p r o v e d the f o r m of the d e n o m i n a t o r p o l y n o m i a l . To get the e x  p l i c i t form of the n u m e r a t o r p o l y n o m i a l we a p p l y the v a lue of S for j«0,l, ..., m:

is s u m m a b l e b y f a c t o r i a l f u n c t i o n s . R e a l l y

zero e x c e p t the f i rst one. Furt h e r , w h e n t runs from

T a k i n g the p o w e r form of the C h e b y s h e v p o l y n o m i a l

we ge t

3 гу- ч ^ и,^ / Ь ) ; 2/

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.г

Let us t r a n s f o r m Z to the p o w e r for m in X

where

VvJ

^{Z -} ^j« ( _ п ^ п - 9 Д п ) + 2 + ^ ( I fnf9j { ^ H ) j

The sum W is an h y p e r g e o m e t r i c f u n c t i o n w h i c h one can sum b y t h e o r e m of D o u g a l l L v L

P(; n ' _

r ( 2 U a ) Г(-^ -гл)Г(^:tn+ ю ) ("n+ m)\

By e l e m e n t a r y c a l c u l a t i o n s we get the r e q u i r e d r e s u l t

7 _ k. üL r(nt-mf2) X “ Г-т);/ - n t n ^ );

2i

<h \fjT mi f P ( n f x * . ( 4( 97^0*

P r o o f of the form of G i b b s c o n s t a n t s for the c a ses m = 0 and m = n one can be o b t a i n e d b y e l e m e n t a r y a n a l y s i s . Here we o m i t the details. The p r o o f of the r e s u l t s of S e c t i o n 3 is a n a l o g o u s w i t h the p r e v i o u s one.

5. C e s a r o ’s m e t h o d of s u m m i n g s e r i e s for sgn ^ a n d the stee pne ss

It is w e l l k n o w n that if we have a s e ries IM

Z- av

V=c '

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10

its C e s a r o ’s sum is d e f i n e d b y the f o r m u l a

51 .

V = o (rn-o0i/

He r e (X is a p o s i t i v e p a r a m e t e r . It is w e l l - k n o w n tha t if

c 4 ~'1 , C n

is F e j é r ’s a r i t h m e t i c m e a n a n d in this case the G ib b s p h e n o m e n o n does n o t occur. /The case ex — 0 g ives the o r i g i n a l s e r i e s . /

N e x t we w i l l prove that if С*ч>(Х0= 0. ^3855123*33,** >

then, again, the Gibbs p h e n o m e n o n does n o t occur.

C o n s i d e r a g a i n the s e r i e s /l/, thus

B y short, e l e m e n t a r y c a l c u l a t i o n we get

Г (J\* — ^ у у f-n)i Ц

jf i f « 4 ^

$ Its e r r o r f u n c t i o n has the m a x i m u m at the p o i n t X - jy / О The m a x i m u m is

Oc

z <

5 Г 14«. £ ;

1 l1 / У f f d i s t |4_ f - 1 =

■5

f B y d e t e r m i n i n g the v a l u e 5 we get the e q u a t i o n

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11

or in inte g r a l form

\ (Л-tf СеоЫ-сМ^О,

**0

The s ol u t i o n s <X a n d S of the e q u a t i o n &*<S) a n d of the p r e v i o u s e q u a t i o n are

# = ^оД 3 9 5 ? Ш 9 3 , . < ; S = 2.025782.Q 92.,.. «

N o t e . G r o n w a l l L8! a l s o d e t e r m i n e d the v a l u e s oC a n d S , b u t the s t a t e d p r e c i s i o n of his r e s u l t s are i n c o r r e c t . The s t e e p n e s s in C e s a r o ’s m e t h o d is ÍL 4 . F o r & — Ú.

the s te e p n e s s is a- . It is h a l v e d c o r r e s p o n d i n g to o<.~ 0 . Thus, we h a v e p r o v e d that C e s a r o ’s m e t h o d of s u m m i n g series d e c r e a s e s the Gibbs c o n s t a n t , b u t it a l s o d e c r e a s e s the s t e e p n e s s .

C o n c l u s i o n s

As a m e a n s of s u m m a r i z i n g our r e s ul t s , we have l i s t e d in Table 1 the G i bbs c o n s t a n t s a n d t h e i r s t e e p n e s s c o r r e s  p o n d i n g to the m e t h o d s used.

F o u r i e r series

C h e n e y ’s m e t h o d M e t h o d of C l e n s h a w a n d L o r d

C e s a r o ’ s sum r e c i p r o c a l

p o l y n o m i a l r a t i o n a l re c i p r o c a l

p o l y n o m i a l r a t i o n a l

18 °/o 5.1 0 . 8 % 8 . 2 °i° k . 9 °/o 1 8 # « G £ . 0%

W n 2 - r \ ■ i g » * f " _h _i

Ц n

Я

T a b l e 1.

G i b b s c o n s t a n t a n d the s t e e p n e s s

A c k n o w l e d g e m e n t

The a u t h o r s are i n d e b t e d to Dr. M. H u s z á r for his c o n  t i n u e d i n t e r e s t a n d for h e l p f u l d i s c u s s i o n s on h y p e r g e o m e t r i e i d e n t i t i e s .

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R e f e r e n c e s

£]~j Z y g m u n d , A.: T r i g o n o m e t r i c series, C a m b r i d g e / 1 9 5 9 / И A r f k e n , G . : M a t h e m a t i c a l M e t h o d s for P h y s i c i s t s ,

A c a d e m i c Press, N e w Y o r k / 1 9 6 8 /

И Cheney, E . W . : I n t r o d u c t i o n to A p p r o x i m a t i o n T h e o r y M c G r a w - H i l l , N e w Y o r k /1 9 6 6/

И

C l e n s h a w , C . W . , Lord, K. : R a t i o n a l A p p r o x i m a t i o n s from C h e b y h e v series, in " S t u d i e s in N u m e r i c a l A n a l y s i s /В.К.Р. Scaife, e d /

A c a d e m i c Press, Lon d o n , pp. 9 5 - 1 1 3 /197**/

LG

S e m e r d j i e v , C . , N e d e l c h e v , C. : D i a g o n a l Pádé A p p r o x i m a n t s for a d i s c o n t i n u o u s f u n c t i o n g i v e n b y its F o u r i e r series.

/Plovdiv U n i v e r s i t y P r e s s / V. 15. / 1 9 7 7 / p. **27 /in B u l g a r i a n /

DO

E r d é l y i , A., M a g n u s , W . , O b e r h e t t i n g e r , F . , T r i c o m i , F . G . : H i g h e r T r a n s c e n d e n t a l F u n c t i o n s Vol. 1.

M c G r a w - H i l l , N e w Y o r k / 1 9 5 3 /

w D o u g a l l , J . : Proc. E d i n b u r g h Math. Soc. Vol. 25 / 1 9 0 7 / pp. 114-132.

LG

G r o n w a l l , T . H . : Ann. M a th. V. 31. /1 9 3О / pp. 232-240.

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Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Lőcs Gyula

Szakmai lektor: Pócs Lajos Nyelvi lektor: Harvey Shenker

Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1984. március hó

Példányszám: 65 Törzsszám: 84-244