Appell
transforms associated with
L^{2} 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 outside 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
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