k AOA093 612 WISCONSIN UNIV-ADISON MATHEMATICS RESEARCH CENTER F/6 12/1HAUSOORFF S MOMENT PROBLEM AND EXPANSIONS IN LEGENDRE POLYNOMIA-ETC(U)OCT 80 R ASKEY, I J SCHOENBERG, A SHARMA DAA62"8O-C-001

UNCLASSIFIED MRC-TSR-2130 NL.EEEEIIIIEEEEE.iiflffl7

I"

MRC Technical Summary Report #2130

HAUSDORFF'S MOMENT PROBLEM AND

EXPANSIONS IN LEGENDRE POLYNOMIALS

R. Askey, I. J. Schoenberg,Wand A. Sharma

I)

Mathematics Research Center

University of Wisconsin-Madison

610 Walnut Street \ AMadison, Wisconsin 53706 j u

October 1980

(Received August '20, 1980)

Approved for public release

Distribution unlimited

' Sponsored by

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LL. Research Triangle ParkNorth Carolina 27709 80~S zzu

'- ), - - 'I'-;...di

UNIVERSITY OF WISCONSI\ -MADI~qn!Z

MATHEMATICS RESEARCh 'ENTEP

-- USDORF~ MOMENT -ROBLEM AND _XPANSIONS I' 7E'.DPE P'L," .,Sg

R./Askeyl I. Jo/Schoenber 9 and A. /S a-m,..

.9Technical /ummary eporti42130

October 98O.-

ABSTRACT

A new proof is given for Fausdorff's conlition on a set riF -omc.r ts Ab

determines when the function generating these moments is in L 2 . 11- pr-nrW

uses Legendre polynomials and their discrete -xtensions found by chobvcr"o.

Then an extension is given to a weighted L2 space usina Jacobi rolvnonria1c

and their discrete extensions.

AMS (MOS) Subject Classifications: 42A70, 33A5

Key Words: Moment problem, Orthogonal polynomials

Work Unit Number 3 (Numerical Analysis and Computer Science)

Sponsored by the United States Army under Contract No. PAAG2O-n0-C-e.04i.

i z.

SIGNIFICANCE ANID EXPLANATION

The paper describes a method of obtaining in terms of the moments

approximations to the solutions of the finite moment problem

(1) f f(x)x dx = , (v = 0,1,2 ... )0 Y

In his paper [21 Hausdorff gave conditions on the moments Pi for the

problems (1) to have a solution f(x) which is squares integrable. However,

our approximations are constructed in terms of the coefficients c of theV

Legendre series expansion

f(x) - c P (2x- 1), (0 < x < 1)V - -V=0

where P (x) are the Legendre polynomials. The main result is that

Hausdorff's condition for a square integrable f(x) are here expressed in

terms of the c • This transition from the pj to the c is done by usinq

a set of orthogonal polynomials on the discrete set x 0, x = 1,.... x = n

due to Tchebvchef. [Accession ror

I-. .- - - - ;;

: t i f , i 0!

B y ,. . . . .. . . . . .

The responsibility for the wordinq and views expressed in this descriptivesummary lies with MPC, and not with the authors of this report.

HAUSDORFF'S MOMENT PROBLEM AND EXPANSIONS IN LFGENDRE POLY)70MIALS

R. Askey, I. J. Schoenber.?, and A. Sharma

1. Introduction. We refer to (3] for a description of the problem of rlplfan,

Kalaba and Lockett (13 of obtaining approximation to the inverse Laplace

transform. They reduce the problem to the solution of the finite moment probI'h

(1.1f f(x)x dx = (v = 0,1,...,n - 1)

0

and obtain approximations for f(x) by applying Gauss' n-point quadrat,r, _r-

to the integrals (1) and use numerical approximations to the inverse of the r-tr,"

of the system so obtained.

In (3) it is shown that the inverse of the Gauss matrix is not needed.

approximations to f(x) are obtained if we determine the polynomial

n-1(1.2) R n 1 (x) = c P v (1 - 2x)

0

of degree n - 1 which is the least square approximation to f(x) in j,1',

having moments p0, 1,..., In 1 The coefficients cv in (1.2) are ai , n 1- -

lower triangular transformation

(1.3) cv = (2v + 1) V (-)1i(V + i)Yi 14i' (v = 0,1,... n -1V V 1']i=O

The numerical problem of Bellman, Kalaba and Lockett is thereby nov,.. H-w.".,

this approach shows that the infinite problem in [0,1)

Sponsored by the United States Army under Contract No. DAAG2O-0-'-0041.

(1.4) f f(x)x dx (v = 0,1,2 ... to infinity)0

might be attacked in terms of the Legendre series expansion

(1.5) f(x) - I cVP V(1 - 2x)v=0

Hausdorff devoted to the problem (1.4) his famous paper [2) in which he showed the

following:

A. The system

1

0

has a non-decreasing solution (x) if and only if

Anl -(fl),,++ 0 + (-1)n,, 0m m - + m+n

for m,n > 0.

B. The system (1.6) has a solution f(x) of bounded variations in [0,1) if

and only if

n' J V IV = 0(1) as n +

V=D

For a direct derivation of Hausdorff's conditions for A and B see [4].

C. The system (1.6) has a solution

x

(x) f o(x)dx0

where ;(x) E LP(0,1) with 1 < p < , if and only if

-2-

L ,

n

(n 1 PI J { nI n-v 0(1)V=O

A particular case of C is this (p = 2):

The moment problem

1

(1.7) f f(x)x Vdx = p , (u = 0,1,2,...)0

has a solution f(x) e L2 (0,1) if and only if

n 2 2S = (n + 1) 7 (n)(.n-)l -0(1)

n v=V

While Hausdorff's results A and I are most apt, it seems that the result (1.8) miqht

be profitably deduced from the expansion (1.5). Since ' + P (1 - 2x) aren

orthonormal, we derive from the Riesz-Fisher theorem and

0 cf(x) - /2v+ I P (1 - 2x)

0OY2 V+ 1

the following: The moment problem (1.7) has a solution f(x) f L2 (0,1) if ant only

if

2n c

(1.9) v=O 2V = 0(1) for all nV=O 2

Formula (1.3) can he inverted and assumes the form

1 P2v 1 _ (2v+1 v(2v+1(1.10) K = (v 2, 1~~ 1, )c0 ,-I)c 1 + ' ' '+(-1) 0 )c}

Substituting (1.10) in fl.,) leads to

-- -3-

2 1 2S 1 C co + Cl

2 1 2 +1 22 '0 + c1 50 c2

2 1 2 1 2 1 23 0 Cl + 5 c2 + 245 3

These simple expressions were a surprise and sugqested that

n2

S = a cn 4. n,V v

with a given by a reasonably simple expression. This will be shown in the n' x:

section

2. Hausdorff Theorem via Orthogonal Polynomials

Using the notation introduced in the first section, we have

1Sn-vv =I f x V( - X)n-Vf(x)dx

0

n 1j c k P k (1 - 2x)xV (1 - x) nvdx

k=0 0

n k (-k).(k + 1 ) 1ck [ J J V+

= ' k (1).j! x (- ~k=0 J= i 0

n(2.1)= r(v + 1)r(n - v + 1) n FC- ' l , -k, 1

r(n + 2) k 3 , n+7

The shifted factorial (a)n is defined by

( (n + a)* ~(a) = ran r(a)

Murphy's formula for Leqendre polynomials was used

_n" n+1 1- X(2.2) P (x) 1n 2

-4-

7,

and the ceneralized hypergeometric function is defined by

d ...,a (a ) * (a p n

(2.3) pFb...,b n (b1) n ... (b Y n n!

Using (2.1) in Sn gives

(2.4)I n n n -k, k+1, v+1 F L+I, v+1

(.n k= 0 kc v 372( 1, n+2 1 3 2 1, n+2k=0 -M '=

If this quadratic form is to be diagonal, then the followina orthogonaljty re'atic

must hold:

n _k k k+1, x+1 1 F F Z+ ; 1 = 0, (0 < k*0 1, n+2 32 1

Now

R (~~k, k+1, X+1.1Rk(X)= 3F2 ( - 1, n+2 1)

is a polynomial of degree n in x and it is relatively well-known that Tc'er-

found a set of polynomials which are orthogonal on x = 0,1,..., n wjth respec ,

the uniform distribution (see 15], 2.8). This is what we want, but at first

glance, it seems we do not have it, since Tchebychef's polynomials are unallv ,'

as

(,n ) = 3F (k, k+1, -x =k 3 2 1, -n ,,

and this does not seem to be the same as Rk (x). However, there is a tran,'rT-rn,

formula which reconciles this difference,

(2.5) F-k, a, b (c - a)k F-k, a, d-b32 c, d = (c)k 3 2 d, a+1-k-c

To obtain (2.5), write ([4), (4.1.3)) as an identity between hyperqnemetric rprio-,

-5-

that is

1F(-k, a (c a)k 1-, a 1 _ -)2P 0 (a k 2c) a+1-k-c'

and integrate with respect to a beta distribution. Take a = k + 1, b = x + 1,

c = n + 2 and d = 1 in (2.5) to get

( 2 6 k) k F-k, k+1, -x(2.6) Rk(x) = (n + 2)k 3 2 1, -n

Using (2.6) above gives

n (n + 1 - ) nS 1 2 2 k * Q ~ ) 2

n+1 kO (n+) x0k

The orthogonality relation for Q k(X) is

n (n + 1)(n + 2)

Qk(x'n)Qz(x 'n) kk (n + 1 - k)k (2k + 1)x=O k

so

n c2 (n+ 1I k)

k=0 2k + I (n + 2)

nSince (n + 1 - I)k/(n + 2)k 1, S c 2 /(2 + 1), which proves one of the

k k n ~ k/(k=O

required inequalities.

Conversely, if Sn = 0(1), then

2 222

n c n ck (n2 + 1- k) (n2 + 2)k -k *k k

L 2k + 1 2k+1) 2 2k=0 k=O (n + 2) (n 2 + I - k) k

22 n2

(n 2 + 2) ( +1- k)n ck ( * -k k

2 * -2k +1 2+2(n 2 + 1 - n) k=O (n 2 2)

n

(1 + 2n n

n +1I S = 0(1)

- n n

n +1

3. A Weighted Hausdorff Moment Problem

Extensions of Legendre polynomials and the discrete Tchebvchef polynomials

exist, so it is natural to see if they can be used to obtain an extension of

Hausdorff's theorem. To this end, set

(3.1) = f(x)xV+(1 - x) Pdx, v 0,1,2.... (a;6 > -1)0

Polynomials orthooonal with respect to x (1 - x) on (0,11 are known. They are

called Jacobi polynomials and are qven by

(a+ 1)(3.2) P 2x) n -n, n+X+1+1

n n! 2 F1 a+1 x•

Set

- ( , 3)t(xj - c p (1 - 2x)0

-7-

where c is determined by

(3.3) c f(x)P (a,) 2x)X(1 x)dxV h(a 8 ) 0

V

and

I(a,8) (a,S) a 8 ~ a

(3.4) J P (I - 2x)P (1 - 2x)xa(1 - x) dx =

n k kn n

with

h(aB) r(n + a + 1)r(n + 6 + 1)n (2n + a + 6 + 1)F(n + a + $ + ) •

As in the last section

1

An-vU = f f(x)xv (1 - x)n-\) dx

0

n (a )aI- +

=kO 0 ck f P(08)ll - 2x)xv+(0 - x)lVdx

r(v+a+1)r(n-v+8+1) k (a + -k, k+a+ +I, v+a+1r(n+a+8+2) k! Ck 3 2( a+1, n-a+6+2

k=0

Using (2.5) gives

r(n + a + $ + 2)An-vvV

f(v + a + 1)r(n - v + 0 + 1)

(3.5) n (a + 1)k (n + k- k) k

k=O k! Ck (n + a + + )k 2 3 2 ( - +1, - n

The general discrete Tchebychef polynomials (6] (or to use their common nam,-,

Hahn polynomials) are given by

(3.6) Q (X;a ,6, 3 F 2' + - I , (k,x 0,1,...,n)

Their orthogonality relation i,

nx ' N-x+

x0 K

(3.7) (a + 6 + 2) • k(n + a + + 2)k (, + 1) k a - 1)

jk n!(n + 1 - k) (x + 1)k ( + " + 1 )2k + - - 1)

0 '. 7,k < n

9+a1rn-V+B) n u, fe ipiiain hSquare (3.5), multiply by n and sur. After sirpification, te

resulting identity is

n n (n + 1 kFn-v 2(n) (n + ax + + 2) . - 2 (,£) B)

v=0 v v r(v + a +1)r(n - v + 1) k- kk (n + a + :+ 2).

The Riesz-Fisher theorem for Jacobi series is

1r(f(x)) 2x X) = 2 (~: F x x(1 x)dx c

0 kr

so an argument similar to the one in 2 gives the folinwIna:

Theorem 1. Define wV b (3.1). Then for a, > -1,

12 af If(x)l x (1 - xI' ' <

0

if and only if

nA ,n, (n + + 2)0(1)

v kv" f(v + a + 1)1(n - v + 5 +1)Th=0

This can Ibe rephrased as

0 x (1 x)

if and only if

(n + I JA a -01

V=0

when a,B 1

REFERENCES

(11 R. E. Bellman, R. E. Kalaba and J. A. Lockett, "Numerical Inversion of the

Laplace Transform", Amer. Elsevier Publ. Co., Inc., New York, 1966.

(2] Felix Hausdorff, Moment probleme fur ein endliches Intervall, Math. Z. 16

(1923), 220-248.

[3] I. J. Schoenberq, Remarks concerning a numerical inversion of the Laplace

transform due to Bellman, Kalaba and Lockett, J. Math. Anal. Appl. 43 (1973),

823-828.

(41 I. J. Schoenberg, On finite and infinite completely monotonic sequences, Bull.

Amer. Math. Soc., Feb. 1932, 72-76.

(5] G. Szego, "Orthogonal Polynomials", Fourth edition, Amer. Math. Soc.,

Providence, RI, 1975.

(6] P. L. Tchebychef, Sur l'interpolation de valeurs ecuidistants, Oeuvees Me P. L.

Tchebychef, Vol. II, reprinted by Chelsea, New York, 1961, 217-242.

PA/I JI/AS/scr

REFORT [),)CU. ENIAI PAGLE .~i.. ,. , _., ____. _

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4. TITLE (im.J 5btirle) TYPE CI %;j'"

F. ItcuI~~mr 1 :,Oe, - r.O :~

HAUSDORFF'S MOMENT PROBLEM ANC 2 XANSIONS S . n[cr~ i r - :C':

IN LEGENDRE . YNOMIALS C P RFC.k F .km. I ,. ,.--Z

7. AUTHOR(s) 6. ~CotiTAC7 CR G4

R. Askey, I. J. Schoenberg, and A. Sharma "DIaG29-80-C-041

9. PER(=OrldtNO ORGA'.-'-ATION KlA'ME AN4 A SS 12 ." , _ '''" '7 [-" z ' -

--

A F, E A 01 RK ,Mathematics Research Center, University of ork Unit !umhor 3 r..

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UNCLUSSI FIEDISa. DECLASSIFICATICN . ZdNGRA,1-

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18, SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side if necessary and iden!Ify by block r'nuber)

Moment problem

orthogonal polynomials

20. ABSTRACT (Continue on -averse side If necessary and identify by block ntrmbor)

\A new proof is given for Hausdorff's condition on a s,'t of men('tts wh,h

determines when the function generating these moments is in L2. Th,, iroolfuses Legendre polynomials and their discrete extensions found by Tchebvcht.f.Then an extension is given to a weighted L2 space usinq Jacobi Pol\'omia]1

and their discrete extensions.,

n a, ii-I II I i I -- - -

DD 1' 147.3 EOll ION or 1 NOV611 IS OUSOLET E .T I- S I -)

SECURITY CL Ass IC AI ION CF NHIS A,;L (0 i ,;,. t .