orthogonality measure for subsequences of hermite polynomials

DEMONSTRATIO MATHEMATICA Vol. XXXIV No 3 2001

Monika Maj

ORTHOGONALITY MEASURE FOR SUBSEQUENCES OF HERMITE POLYNOMIALS

Abstract. In this paper we prove the uniqueness of the orthogonality measure for special subsequences of Hermite polynomials.

1. Introduction and formulation of the result The aim of this paper is to prove the uniqueness of the orthogonality

measure for special subsequences of Hermite polynomials. All results formu-lated in this paper can be presented in both deterministic and probabilistic language.

Let (Wt, t > 0) be a Wiener process and Hn(x) be an Hermite polynomial (HP) of degree n defined by formula

(1.1) Hn(x) = (-1)« exp ( y ) ^ exp ( - ^ ) , n = 0,1, . . .

In this paper we consider (HP) defined by (1.1), because the form is more useful for our calculations and it is often used in the probability theory. Most authors define (HP) by formula

(1.1') Hn(t) = ( -1 )" exp (i2) ^ exp ( - t 2 ) , n = 0,1, . . .

By the definition of (HP) and formulas given in [5] and [6] we have another form of (HP)

W nU—"h7"

Hermite polynomials in the class of orthogonal polynomials are subject of many papers. They have fundamental importance in the probability theory. It is very well known that

{«••«MBs)'**}

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is a martingale and a Markov process (we use the typical denotation for cr-fields generated by a Wiener process

F<t = (r(Ws, s<t).

The proof is given, for example, in [1]. We introduce the following notation:

(#n)2 p + i = {#n+u>(2p+i) : t« = 0 , 1 , . . . } for fixed n and p,p € N, fc!! = 1 • 3 • . . . • k, where k = 2n + 1, n <E N,

1 p

P2p(x) = - = f l + ®oi + . . . + x2pa2p - £ ( 2 i - l ) ! ! a J , V ¿TT L

i = 1 J

where the sequence Oi : i = 1 , 2 , . . . , 2p is such that P2p(x) is nonnegative polynomial of degree 2p.

In the present paper we prove the uniqueness of the orthogonality mea-sure for special subsequences of Hermite polynomials. As we know (see [4]), the function

is the unique weight for Hermite polynomials, for which \ Hn (x) Hm (x) f(x)dx = VzirnlSnm, R

where Snm = 1 for n=m. Of course /i is the orthogonality measure when

J Hn (x) Hm (x) dfi = 0 for n ^ m. R

The main problem of this paper is to show that this equality characterizes orthogonality weight for (HP) subsequences. The above condition is not sufficient. We have to add an assumption about the form of 3p moments. We need to know the form of: (j) EX, E X 2 , E X 2 p , EX2p+2, EX2p+4,..., EX4p,p £ N.

It is worth pointing out that the basic idea considered in this paper can be formulated without using the probability language.

From now on, f2p (x) denotes the function given by formula

(1.3) f ( x ) = P2p(x)exp(~y) =

V̂ TT 1 - ¿(2i - l)!!a2t + xai + ... + x2pa2p

t=i exp

The following formulation shows our problem in the most general way.

Orthogonality measure 703

THEOREM 1. For fixed number p € N the following integrals

J xnf(x)dx, n < 2p or n = 2 (p + r ) , r < p R

are given

(1.4) EXn =

P - I E (n+2j)!!a 2 j+i for n = 2m + \

3=0 • .

(1 - E (2t - 1 ) ! !om) (n -1) ! ! + £ (n + 2 j -1 ) ! \a 2 j , n=2m (A «=l / j=i

If for every n € N \ {0}

J Hn (x) ffn+(2p+i)w(x)f(x)dx = j Hn (x) Hn+(2P+i)w(x)dn = 0, R R

Then n is the measure with density f{x).

We can formulate the above theorem in the probabilistic language. The only difference is based on taking advantage of the mean value.

Let X be a random value with density (1.3). Then

j xnf(x)dx = EXn

R

and our theorem has the following form:

THEOREM 2. For fixed number p e N the moments

EXn, n<2p or n = 2(p + r), r <p

are given by (1.4) and

(1.5) EHn(X)Hn+{2p+1)w(X) = 0. Then fj, is the measure with density f(x).

In the first example we show that orthogonality conditions for subse-quences do not uniquely determine the weight function. We have to add some assumptions connected with moments.

The opposite theorem to the above is proved by Plucinska. She shows in [3] that if f(x) is given by (1.3) then

EHn (X) Hn+(2P+i)w{X) = 0.

2. P roof of T h e o r e m 1 Suppose we have an (HP) of degree n. We are going to prove the following

lemmas, which show us some properties of (HP) used in the main proof.

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LEMMA 3. Letf(x) be a function given by formula (1.3). Then the following equation holds

{ p-1

£ (n + 2j)Ua,2j+i for n = 2m + 1 j=o ( l - J ) (2i - l) ! !a2i) (n - 1)!! + £ (n + 2j - l)!!a2 j /or n = 2m

LEMMA 4. For real coefficients ai, 0 2 , . . . , a2P i/iere is a special relationship between consecutive even moments given by formula pv2m+2 py2m P K-1

<2 1> (2m+~ï)ÏÏ " ( ^ T î j ï ï = £ ( » * » H + « + 1 ) ) .

where n°=i(2Tn + 2j + 1) = 1.

jet us us

1 = 0 = (2fc - l)!!<T2fc, k € N,

which describe central moments of Gaussian distribution. Then for odd n we get

lJ=

P r o o f of Lemma 3. Let us use formulas

EXn = \ xnP2p(x) exp j d x

= n!!oi + (n + 2)!!o3 + . . . + (n + 2p - 2)!!a2p_i,

which means that our equation is proved. In the same way we show that formula (1.4) holds for even n. •

P r o o f of Lemma 4. We must only to show forms of two next even moments. For given function f(x) we have

v EX2m = (2m - l ) ! ! ( l - £ ( 2 i - l)! !a2 i)

i=i

+ (2m + l)!!a2 + . . . + (2m + 2p - l)!!a2p, P

EX2m+2 = (2m + ^„^ _ _ j j , , ^

i=l + (2m + 3)!!a2 + ... + {2m + 2p+ 1 )!!a2p.

Formula (2.1) follows from above equations. •

The remainder of this section will be devoted to the uniqueness of the orthogonality measure for subsequences of (HP).


It is clear that we can calculate coefficients a i ,a 2 , . . • ,a2P if we know the form of moments. Of course our moments are represented by linear combinations of those coefficients. Note that we can really use moments given by equation (1.4) in our calculations. Let us first prove that formula

We must only show that the moment problem considered by Hamburger and Carleman (see for example Shirayew) has the unique solution. Showing that (1.4) is the unique orthogonality measure for our special subsequences of Hermite polynomials is equivalent to moment problem.

We shall consider two cases:

(i) n = 2m (even), (ii) n = 2m + 1 (odd).

In both of them the proof is by induction.

2.1. Case for even n. Let us assume that n=2m. The main idea of the proof is to take the following subsequences of (HP)

(2.2) (1.5) = • (1.4).

{h2z(2P+I){X) : 2 = 0,1, . . . j .

1) We will first show that (2.2) is correct for z=l. Taking (1.2), n

Hn(x)Hm(x) = Y, (k) (k) k\Hm+n-2k(x) for n < m fc=0

EHq ( X ) H2(<ip+i){X) = EH2(2p+\){X)

Thus 2p+l [2(2p + l ) ] ! ( - l ) r „^ep+p-ar E > f y

[2(2p+ 1) — 2r]!r!2r EX = 0 r=0

[2(2p+l)]!(- l ) r

¿ J [2(2p + 1) - 2r]!r!2r EX , 2 (2p+l ) -2r

706 M. Maj

According to the above assumption, we have

EX2(2p+1) = ( l - ^ ( 2 z - l ) ! ! a 2 i ) ( 2 ( 2 p + l ) - l ) ! ! + f ^ (2 (2p+ l ) + 2 j - l ) ! ! a 2 j ) i = l j=1

which means that for z= l our implication is true. 2(2p+l) 2r

2) Assuming that all moments, including EX , are determined and represented by (1.4), we have (2-3) EH0(X)H2z{2p+1)(X) and we can make inductional step.

In virtue of (1.1')) n

Hn(x)Hm(x) = £ (?) (2) k\Hm+n_2k(x) for n < m fe=o

and N

r = l

equation (2.3) has the following form EHq (X) H2z(2P+i)(X:) = EH2z(2p+\){X) =

/ ( 2 p f 1 ) Z [2z (2p+l ) ] l ( - l ) r y2z(2p+1)—2r\ n V [2z(2p+1) — 2r]!r!2r J '

(2pt1}* [2z(2p + l ) ] l ( - l ) r =

¿ J [2z(2p + 1) - 2r]!r!2r

Thus

2,(2P+D (2p+1} ' [2z(2p + 1)]! (—l)r W ) - * ¿ J [2z(2p + 1) — 2r]!r!2r

and taking (1.4) we get

M2P+1) = _ [2z(2p + 1)]! (—l)r

¿ J [2z(2p + 1) — 2r]!r!2r

p p

• [ ( l - ¿ ( 2 » - l)!!a2i) (2*(2p+1) - 2r - 1)!! + £ ( 2 z ( 2 p + 1 ) + 2 j - l)!!a2j]. ¿=1 j=l

Easy calculations yield

p p = ( l - 5^(2» - l)!!flw)(2z(2p+ 1) - 1)!! + £(2.z(2p + 1) + 2j - l)!!o2i.

¿=1 j=l


For the following subsequences of (HP):

Hl(x),Hi+2(2p+l)(x), Hl+4(2p+l)(x), • • • ( H 1 + 2 z ( 2 p + l ) ( x ) , 2 = 0, 1, . .

H2(X), H2+2(2p+l)(X),H2+4(2p+l)(X), • • • (•Hr2+2z(2p+l)(:c)) 2 = 0, 1, . .

H2P(X), H2p+2(2p+l){x), H2p+4(2p+l)(x), • • • (-ff2p+2z(2p+l)(®). * = 0, 1,. . ,

the arguing is analogical and we shall not repeat it. Similar transformations give (1.4) for even n. Theorem has just been proved for n=2m.

2.2. Case for odd n For odd n, assuming that all moments, including the moment of degree

(z(2p + 1) — 2r), are known, we have

EH0 ( X ) HZ(2P+I)(X) = EHZ(2P+I)(X) = 0.

Of course (2.2) is correct for z=2, because we can determine moments from

EH0(X)H2{2P+1)(X) = 0.

The inductional step is following r*(2p+in 1 ' J [ z ( 2 p + l ) ] ! ( - i y

¿ J [z(2p+ 1) — 2r]!r!2r

L 2 J [z(2p +1)]! (—l)r ^ = 0. S + 1 ) - » +

It is easy to check that the main calculations made above give r«»p+in

-(ip+l) _ 1 A J [Z(2p+ 1)]! (—l)r x(2P+l)-2r ¿ i [z(2p+ 1) — 2r]!r!2r

Thus

EX'{2p+1) = X > ( 2 p + 1) + 2 j ) \ \ a 2 j + i . j=0

The same calculations for subsequences

Hi(x), H1+(2P+I)(X), H1+^2P+I)(x), •.. (HI+2Z(2P+I)(X), Z = 0 , 1 , . . . )

#2(x), H2+(2p+l)(x), H2+3(2p+l)(®)» • • • (#2+2i(2p+l)0»0> Z = 0,1,. .

708 M. Maj

H2p(x), H2p+(2p+i)(x), #2p+3(2p+l)(®)> • • • (#2p+2z(2p+l)(®)> 2 = 0,1,. .

end the proof for odd n. The opposite implication to (2.2) follows immediately from orthogonality

condition for (HP). Now we shall check if the moments which we determine really satisfy the

conditions required for unique solution of the moment problem. Of course our moments satisfy the following formula

lim < oo, where mn — EXn. n—foo n

Plucinska has shown in [3] that the moments are bounded by a constant. Thus /i is the unique orthogonality measure for the set of (HP) subse-

quences.

3. Examples 3.1. Example 1

In the present paper we show that f(x) given by formula (1.3) is the unique weight for special subsequences of (HP). Now we are going to show an example which demonstrates that a function different from f(x) cannot satisfy all assumptions.

Let us consider

(3.1) f(x) = ci/i(x) + c2/2(z) + 03/3(1),

where

1 / x2 \ fi(x) = -^==[1 - A2 + xAi + X2A2] exp ^ — J >

MX) = ̂ I1 ~B2 + XB1+ X2B2\ GXP ( " R)' M X ) = ̂ I1 ~D2 + X D l + GXP ( '~ ¿i) '

fi(x) >0 , i = 1,2,3 and ci + c2 + c3 = 1, Cj > 0 for i = 1,2,3

<Ti, o"2 , <73 are different. Of course our function f(x) is nonnegative and \f{x)dx = 1. Then f(x)

is a density function. Let us consider the case when

p = 1, ci = c2 = c3 = J and a\ = 1, a\ = 2, = 4.


Then our moments are given by formula 1 2 4

EX = -A1 + -B1 + -DL

2 2 10 „ 44 7 (3.2) = 3 ^ 2 + T 5 2 + T D 2 + 3

EX4 = 4^2 + 365 2 + 304£>2 + 21 EX6 = 30A2 + 390S2 + 3900£>2 + 330.

On the other hand, from the assumption

EHN(X)HN+3(X) = 0

we get EX6 = 1 5 E X 4 - 4 5 E X 2 + 15.

Let us put EX2 and EX4 given by (3.2) to the above equation. Then we have

EX6 = 30A2 + 520^2 + 864D2 + 365.

Thus the linear combination of the function f(x) is not unique orthogo-nality weight for subsequences of (HP). We can see that the sixth moment calculated from the assumption

EHO(X)H6(X) = 0

is different from the one we obtained from the moments' definition. We can obtain a lot of such examples talcing another values of coefficients

in (3.1) satisfying given conditions.

3.2. Example 2 In the second example we are going to show that the assumption about

(j) is very important. Let us consider the same density function as we did in the first example.

Suppose that EX = 3 and EX2 = 25.

Then the general assumption

EHN(X)HN+3(X) = 0

yields EX4 = 147.

For

A2 = 12, B2 = 0, D2 = 1, AI =3, BI = DI = 1,

some of equations in (3.2) have the following form

EX = 3, EX2 = 25, EX4 = 373.

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In this case, we can see that we cannot omit the assumption about (j). If we assume that we know only 2p moments, then f(x) will not be a unique weight for (HP).

Acknowledgment. The author wishes to express her thanks to A. Plucinska for suggesting the problem and for many stimulating conversa-tions.

References

[1] S. K w a p i e n , A. W o y c z y n s k i , Random Series and Stochastic Integrals: Single and Multiple, Boston 1992.

[2] A. P l u c i n s k a , A stochastic characterization of Hermite polynomials, J . Math. Sei., Vol. 89, No 5, (1998), 1541-1544.

[3] A. P l u c i n s k a , Some properties of polynomial-normal distributions associated with Hermite polynomials, Demonstratio Math., Vol. 32, No 1, (1999), 195-206.

[4] G. Szego, Orthogonal Polynomials, American Mathematical Society Providence, Rhode Island 1975.

[5] A. N. S h i r y a e w , Probability, Springer Verlag, New York 1989. [6] A. P. P r u d n i k o v , J. A. B r y c z k o v and O. I. M a r i c z e v , Integrals and Special Func-

tions, Nauka, Moscow 1983 (in Russian).

FACULTY OF MATHEMATICS AND INFORMATION SCIENCE WARSAW UNIVERSITY OF TECHNOLOGY PI. Politechniki 1 00-661 WARSAW, POLAND

Received June 28, 1999; revised version February 1st, 2000.

orthogonality measure for subsequences of hermite polynomials

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