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Appell transforms associated with L²- expansions in terms of generalized heat polynomials

Abstract: The main purpose of this paper is to characterize functions which have expansions in terms of polynomial solutions  of the generalized heat equation                                        (A)

and in terms of the Appell transforms  of the . denotes the class of functions  which, for  satisfy (A) and for which

for all ,  the integral converging absolutely; where  is the source solution of (A).

Keywords: Appell transforms, heat polynomial, Huygens property.

2000 Mathematics subject classification: 44A15, 46F12.

1. Introduction and preliminary results: The generalized heat polynomial is a polynomial defined by

                                                       (1.1)

 a fixed positive number. Note that when    the ordinary heat polynomials defined in [7, p.222]. 

For  has the following integral representation.

                                        ,                                    (1.2)

We may readily verify for, that satisfy the generalized heat equation

                                                                                                  (1.3)

where

We denote by the class of all functions which satisfy (1.3). The source solution of (1.3) is given by , where

                                                                          (1.4)

with

being the Bessel function of imaginary argument of order and where

 (Sec [1] for details).

Corresponding to the generalized heat polynomial its Appell transform  is defined by

                           �..,                    (1.5)

which is also a solution of (1.3). It follows from the definition of  that

                          �..                   (1.6)

Interesting fact about and  is that they form a bioorthogonal system on.For, we have

                                                                  (1.7)

where

                                      .                                (1.8)

A consequence of (1.7) is a fundamental generating function for the bioorthogonal set . We have, for    

                                  .                              (1.9)

2. Inversion: For  Set

where is defined by (1.8). Then as a consequence of the definitions and of (1.9), we have

           (2.2)

Following [1; � 4], we have the following results:

Lemma 2.1:

(i)                                                                           (2.3)

(ii)                                                                      (2.4)

(iii) uniformly                                                           (2.5)   

             any fixed positive number.

(iv) For fixed  ,  ,

                                                             (2.6)

It is now easy to establish the following fundamental inversion theorem.

Theorem 2.2: If  and is continuous at , then

                                                                       (2.7)

3. The Huygens property: A function is said to have the Huygens property for  if and only if and for every

                                                                         (3.1)

the integral converging absolutely. We denote the class of all functions with the Huygens property by . Functions of class have a complex integral representation as given in the following result.

Lemma 3.1: If  then for

                                                                       (3.2)

The fact that  for   and  for enables us to conclude that certain integrals involving functions of are constant. A general result we can find in [4], but we state here the specific forms required in this paper.

Theorem 3.2: If  for  then

                                                                                         (3.3)

is a constant.

Theorem 3.3: If  for  then

                                                                                         (3.4)

is a constant.

Theorem 3.4: If  for  then

                                                                                         (3.5)

is a  constant.

4. Criteria for - expansions: In this section we establish criteria for a function  so that the series  converges in mean, with weight functions, to.

Theorem 4.1: Let  for   and

for  Then for

                                             (4.1)

and

                                                                    (4.2)

where is given by (1.8) and

                                          .                                   (4.3)

Proof: For fixed , let  be a continuous function vanishing out-side a finite interval and such that, for

                                               (4.4)

Now set

                                      (4.5)

Then by (2.1), we have

                                                                    (4.6)                            

where is defined by (1.8). Hence

                     .

If we set

                                                                       (4.7)

and apply Theorem 2.2, we find that

                                       (4.8)

If we multiply both sides of (4.8) by  and integrate between  and , we obtain

or by (4.7),

                                                                                   (4.9)

Now, let

                                                (4.10)

Consider

                                             (4.11)

Since by (1.7), we have

                                                                          (4.12)

with given in (1.8), it follows that

                               .

By (4.4), we have

It follows, therefore, by (4.9), that if is sufficiently large,

Thus      

                            (4.13)

or by (4.5), we have (4.1) with  Theorem 3.4 establishes the fact that  is independent of Parseval�s equation (4.2) follows since

with the last equality a result of (4.12).Thus proof is completed.

            An example illustrating the theorem is given by

This function satisfies the hypothesis for  and we find that

                                   (4.14)

where as

                                                                               (4.15)

Since

                                                 (4.16)

Although, in this example,  for  the expansion (4.1) does not hold in the extended strip. Note that in this case the requirement that be in fails for  A modification of Theorem 4.1 when  for  is given by the following result.

Theorem 4.2: If   for  and if

.

For each fixed ,  ,    then for

                                               (4.17) 

and

                                                                    (4.18)

where  is given by (1.8) and

                                                                 (4.19)

Proof: As in the preceding proof, we have

with

Thus (4.17) holds with  which by Theorem 3.5, is independent of . Further,

which is the Parseval equation (4.18). This completes the proof.

            The example of the preceding theorem satisfies these hypotheses for  and we have for

where as

so that

Criteria for expansions in terms of  are given in the following result.

Theorem 4.3: If  for   and if

for each fixed ,  then for

                                               (4.20)

where  is given by (1.8) and

                                                         (4.22)

Proof: Again, as in Theorem 4.1 , since we have

                                        (4.23)

with

                                                    (4.24)

Now, (4.23) can be written in the form

with (4.24) becoming

Hence if we set  is independent of , by Theorem 3.6,  (4.20) is established. Moreover Parseval formula is

Note that the function  satisfies the conditions of the theorem for  In this case we have

and hence

where as

References:

[1]     F.M. Cholewinski and D.T. Haimo, the Weierstrass-Hankel convolution transform, J. d Analyse Math. Vol. 17, No. 1 (1966), 1-58.

[2]     A. Erdelyi et al., Higher transcendental functions, Vol. II, 1953.

[3]     D.T. Haimo, Integral equations associated with Hankel convolutions, Trans. Amer Math. Soc. 116(1965), 330-375.

[4]     D.T. Haimo, Generalized temperature functions, Duke Math. J. (to appear).

[5]     D.T. Haimo, Functions with the Huygens property, Bull Amer. Math. Soc. 71 (1965), 528-532.

[6]     I.I. Hirschman, Jr, Variation diminishing Hankel transforms, J.  Analyse Math. 8 (1960-61), 307-336.

[7]     P.C. Rosenbloom and D.V. Widder, Expansions in terms of heat polynomials and associated functions, Trans. Amer. Math. Soc. 92 (1959), 220-266.

[8]     E.C. Titchmarch, The theory of Fourier integrals, 1937.

[9]     B.B.Waphare, Appell transforms associated with expansions in terms of generalized heat polynomials (communicated).