The Generalized Mehler-Fock Transform over Lebesgue Spaces

This paper focuses on establishing boundedness properties and Parseval-Goldstein type relations for the generalized Mehler-Fock transform initially introduced by B. L. J. Braaksma and B. M. Meulenbeld (Compositio Math.,{18}(3):235-287, 1967. Also, we derive an inversion formula for this transform over Lebesgue spaces.

Keywords:

Generalized Mehler-Fock transform

;

Lebesgue spaces

;

inversion formula

;

boundedness properties

;

Parseval-Goldstein type relations

Subject:

Computer Science and Mathematics - Mathematics

MSC: 44A15; 46E30; 47G1

1. Introduction and Preliminaries

The concept of the Mehler-Fock transform has its roots in the groundbreaking contributions of F. G. Mehler [5] and V. A. Fock [2]. Evolving from their seminal work, it emerged as a distinct integral transform, finding applications across a broad spectrum of mathematical physics problems. Extensive research by various scholars has led into its properties and applications. For an in-depth exploration, interested readers can consult the following references, among others [3,4,7,8,9,10,11,13].

We consider the generalized Mehler-Fock transform of a suitable complex-valued function f on

R_{+}

analyzed by B. L. J. Braaksma and B. M. Meulenbeld [11] and R. S. Pathak [13] (chapter 11) given by

\begin{matrix} (B_{m, n} f) (τ) = \int_{0}^{\infty} f (x) P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x d x, τ > 0, \end{matrix}

(1)

where

m, n

are complex numbers with

ℜ m < \frac{1}{2}

and

P_{- \frac{1}{2} + i τ}^{m, n} (cosh x)

is the associated Legendre function of the first kind [1] (Chapter 3) defined by

\begin{matrix} P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) & = & \frac{{(1 + cosh x)}^{\frac{n}{2}}}{Γ (1 - m) {(cosh x - 1)}^{\frac{m}{2}}} \\ \times & {}_{2}F_{1} (\frac{1}{2} + i τ + \frac{n - m}{2}, \frac{1}{2} - i τ + \frac{n - m}{2}; 1 - m; \frac{1 - cosh x}{2}), \end{matrix}

where

{}_{2}F_{1}

is the Gauss hypergeometric function [1] (p. 57).

A corresponding inversion formula of (1) for suitable f is given by

\begin{matrix} f (x) = \int_{0}^{\infty} χ (τ) P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) (B_{m, n} f) (τ) d τ, x > 0, \end{matrix}

(2)

where

\begin{matrix} χ (τ) & = & Γ (\frac{1 - m + n}{2} + i τ) Γ (\frac{1 - m + n}{2} - i τ) Γ (\frac{1 - m - n}{2} + i τ) \\ \times & Γ (\frac{1 - m - n}{2} - i τ) {[Γ (2 i τ) Γ (- 2 i τ) π 2^{n - m + 2}]}^{- 1} . \end{matrix}

(3)

The conditions of validity of (1) and (2) are provided by Braaksma and Meulenbeld [11].

Notice that for

m = n = 0

one obtains the Mehler-Fock transform of zero order [12] (Section 7-6, p. 390) and [11] (Theorem 7, p. 247)

\begin{matrix} F (τ) = \int_{0}^{\infty} f (x) P_{- \frac{1}{2} + i τ} (cosh x) sinh x d x, τ > 0 . \end{matrix}

(4)

and a corresponding inversion formula given by

\begin{matrix} f (x) = \int_{0}^{\infty} τ tanh (π τ) P_{- \frac{1}{2} + i τ} (cosh x) F (τ) d τ, x > 0 . \end{matrix}

(5)

The case

m = n

ℜ m < \frac{1}{2}

is contemplated in [11] (Theorem 7, p. 247)

\begin{matrix} F_{m} (τ) = \int_{0}^{\infty} f (x) P_{- \frac{1}{2} + i τ}^{m} (cosh x) sinh x d x, τ > 0 . \end{matrix}

(6)

and a corresponding inversion formula given by

\begin{matrix} f (x) & = & \frac{1}{π} \int_{0}^{\infty} τ sinh (π τ) Γ (\frac{1}{2} - m + i τ) Γ (\frac{1}{2} - m - i τ) \\ \times & P_{- \frac{1}{2} + i τ}^{m} (cosh x) F_{m} (τ) d τ, x > 0 . \end{matrix}

(7)

From [13] (formula (11.1.8), p. 345)

\begin{matrix} P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) = O (x^{- ℜ m}) a s x \to 0^{+} . \end{matrix}

(8)

From [13] (formula (11.1.9), p. 345)

\begin{matrix} P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) = O (e^{- \frac{x}{2}}) a s x \to + \infty . \end{matrix}

(9)

Thus one has

\begin{matrix} P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x = O (x^{1 - ℜ m}) a s x \to 0^{+}, \end{matrix}

(10)

\begin{matrix} P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x = O (e^{\frac{x}{2}}) a s x \to + \infty . \end{matrix}

(11)

From [13] (formula (11.1.9), p. 345)

\begin{matrix} |P_{- \frac{1}{2} + i τ}^{m, n} (cosh x)| \leq C \sqrt{\frac{π}{2}} Γ (\frac{1}{2} - ℜ m) P_{- \frac{1}{2}}^{ℜ m, 0} (cosh x), \end{matrix}

(12)

where C is a constant independent of x and

τ

ℜ m < \frac{1}{2}

x > 0

τ > 0

Now, observe that from () the convergence of the integral

\begin{matrix} \int_{0}^{\infty} P_{- \frac{1}{2}}^{ℜ m, 0} (cosh x) sinh x d x \end{matrix}

is not assured in ∞.

For

γ \in R

we consider the vector space

E_{γ}

consisting of all complex-valued measurable functions f on

R_{+}

such that

e^{γ x} f (x) \in L^{\infty} (R_{+})

. In their recent study [6], Maan and Negrín made extensive use of spaces of type

E_{γ}

A norm

{∥ \cdot ∥}_{γ}

E_{γ}

is given by

\begin{matrix} {∥ f ∥}_{γ} = {∥ e^{γ x} f (x) ∥}_{L^{\infty} (R_{+})} . \end{matrix}

With this norm, the map

\begin{matrix} T_{γ} : E_{γ} \to L^{\infty} (R_{+}) \end{matrix}

where for any

f \in E_{γ}

\begin{matrix} (T_{γ} f) (x) = e^{γ x} f (x), x \in R_{+}, \end{matrix}

is an isometric isomorphism from

E_{γ}

L^{\infty} (R_{+})

. Thus, since

L^{\infty} (R_{+})

is complete, then the space

E_{γ}

becomes a Banach space.

The

C_{c}^{k} (R_{+}), k \in N

, denotes as it is usual the space of compactly supported functions on

R_{+}

which are k-times differentiable with continuity.

The content of this paper is as follows: Section 1 is concerned with the definitions and useful results which are used in the entire sequel. Section 2 provides an inversion formula for the generalized Mehler-Fock transform given by (1). Section 3 deals with the continuity features over Lebesgue spaces and Parseval-Goldstein type relations for the generalized Mehler-Fock transform given by (1). Section 4 gives concluding remarks.

2. An Inversion Formula over the Spaces $E_{γ}$

By means of [11] (Theorem 5, p. 245) one obtains the next inversion formula for the transform (1) over the spaces

E_{γ}

γ > \frac{1}{2}

Theorem 1

(Inversion formula). Assume

f \in E_{γ}

γ > \frac{1}{2}

, and let

m, n

be complex numbers with

| ℜ n | < 1 - ℜ m

and

- \frac{3}{2} < ℜ m < \frac{1}{2}

. Suppose that the function f is continuous at the point x and of bounded variation in the neighbourhood of the point x. Then

\begin{matrix} \int_{0}^{\infty} χ (τ) P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) (B_{m, n} f) (τ) d τ = f (x), \end{matrix}

where

χ (τ)

is given by (3).

Proof.

Observe that being

f \in E_{γ}

γ > \frac{1}{2}

, the function

g (t) = f (log (t + \sqrt{t^{2} - 1})), t > 1

, satisfies the condition (0.6) of p. 236 of [11].

Also the function g is continuous at the point

cosh x

and of bounded variation in the neighbourhood of the point

cosh x

Thus from Theorem 5 of [11] one has

\begin{matrix} \int_{0}^{\infty} χ (τ) P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) \int_{1}^{\infty} g (t) P_{- \frac{1}{2} + i τ}^{m, n} (t) d t d τ = g (cosh x) . \end{matrix}

(13)

Now, taking in (13)

t = cosh u, u > 0

, and taking into account that

g (cosh u) = f (u)

, one obtains

\begin{matrix} \int_{0}^{\infty} χ (τ) P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) \int_{0}^{\infty} f (u) P_{- \frac{1}{2} + i τ}^{m, n} (cosh u) sinh u d u d τ = f (x) . \end{matrix}

□

As a consequence of Theorem 1 one obtains the next result.

Corollary 2 (Injectivity)

Assume that the functions

f, g \in E_{γ}

γ > \frac{1}{2}

, and let

m, n

be complex numbers with

| ℜ n | < 1 - ℜ m

and

- \frac{3}{2} < ℜ m < \frac{1}{2}

. Suppose that the function f and g are continuous on

R_{+}

and of bounded variation in the neighbourhood of each point of

R_{+}

. Then, if

B_{m, n} f = B_{m, n} g

R_{+}

it follows that

f = g

R_{+}

3. Boundedness Properties and Parseval-Goldstein Type Relations Over the Spaces $E_{γ}$

Observe that for

m, n

being complex numbers with

ℜ m < \frac{1}{2}

f \in E_{γ}

γ > \frac{1}{2}

τ > 0

, one obtains from (12)

\begin{matrix} | (B_{m, n} f) (τ) | \\ \leq & {∥ f ∥}_{γ} \int_{0}^{\infty} e^{- γ x} f (x) | P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) | sinh x d x \\ \leq & {∥ f ∥}_{γ} C \sqrt{\frac{π}{2}} Γ (\frac{1}{2} - ℜ m) \int_{0}^{\infty} e^{- γ x} P_{- \frac{1}{2}}^{ℜ m, 0} (cosh x) sinh x d x . \end{matrix}

(14)

Now, from (10) and () the integral

\begin{matrix} \int_{0}^{\infty} e^{- γ x} P_{- \frac{1}{2}}^{ℜ m, 0} (cosh x) sinh x d x \end{matrix}

converges for

γ > \frac{1}{2}

Thus the next result holds.

Proposition 3.

Set

m, n

be complex numbers with

ℜ m < \frac{1}{2}

. The generalized Mehler-Fock transform

B_{m, n}

given by (1) is a bounded linear operator from

E_{γ}

γ > \frac{1}{2}

, into

L^{\infty} (R_{+})

. If

f \in E_{γ}

γ > \frac{1}{2}

, then

\begin{matrix} ∥ B_{m, n} {f ∥}_{L^{\infty} (R_{+})} \leq M {∥ f ∥}_{γ}, f o r s o m e M > 0, \end{matrix}

and

B_{m, n} f

is a continuous function on

R_{+}

. Moreover, the generalized Mehler-Fock transform

B_{m, n}

is a continuous map from

E_{γ}

γ > \frac{1}{2}

, to the Banach space of bounded continuous functions on

R_{+}

Proof.

Let

τ_{0} > 0

be arbitrary. Since the map

\begin{matrix} τ \to P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x \end{matrix}

is continuous for each fixed

x > 0

, we have

\begin{matrix} P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x \to P_{- \frac{1}{2} + i τ_{0}}^{m, n} (cosh x) sinh x a s τ \to τ_{0} . \end{matrix}

Further, we have that

\begin{matrix} |P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x - P_{- \frac{1}{2} + i τ_{0}}^{m, n} (cosh x) sinh x| | f (x) | \end{matrix}

is dominated by the integrable function

\begin{matrix} 2 C \sqrt{\frac{π}{2}} Γ (\frac{1}{2} - ℜ m) P_{- \frac{1}{2}}^{ℜ m, 0} (cosh x) sinh x | f (x) | . \end{matrix}

Therefore, by using dominated convergence theorem, we get

\begin{matrix} |(B_{m, n} f) (τ) - (B_{m, n} f) (τ_{0})| \\ \leq \int_{0}^{\infty} |P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x - P_{- \frac{1}{2} + i τ_{0}}^{m, n} (cosh x) sinh x| | f (x) | d x, \end{matrix}

which tends to 0 as

τ \to τ_{0}

Thus,

B_{m, n} f

is a continuous function on

R_{+}

Since for each

τ > 0

and from (14) one has

\begin{matrix} | (B_{m, n} f) (τ) | \\ \leq & {∥ f ∥}_{γ} C \sqrt{\frac{π}{2}} Γ (\frac{1}{2} - ℜ m) \int_{0}^{\infty} e^{- γ x} P_{- \frac{1}{2}}^{ℜ m, 0} (cosh x) sinh x d x \\ = & {M ∥ f ∥}_{γ}, for some M > 0, \end{matrix}

(15)

then

B_{m, n} f

is a bounded function.

The linearity of the integral operator implies that the

B_{m, n}

transform is linear. Also from (15) we get that

\begin{matrix} ∥ B_{m, n} {f ∥}_{L^{\infty} (R_{+})} \leq M {∥ f ∥}_{γ} \end{matrix}

and hence

\begin{matrix} B_{m, n} : E_{γ} \to L^{\infty} (R_{+}), γ > \frac{1}{2}, \end{matrix}

is a continuous linear map. □

Also, the next result holds.

Proposition 4.

Set

m, n

be complex numbers with

ℜ m < \frac{1}{2}

γ > \frac{1}{2}

, w be a measurable function on

R_{+}

such that

w > 0

a.e. on

R_{+}

and

\int_{0}^{\infty} w (x) d x < \infty

. Then, for

0 < q < \infty

\begin{matrix} B_{m, n} : E_{γ} \to L^{q} (R_{+}, w (x) d x) \end{matrix}

is a bounded linear operator.

Example 1.

Examples of weights w for Proposition 4 are:

\begin{matrix} (i) w (x) = {(1 + x)}^{r}, f o r r < - 1 . \\ (ii) w (x) = e^{r x}, f o r r < 0 . \end{matrix}

For a suitable function g we denote

\begin{matrix} (B_{m, n}^{*} g) (x) = sinh x \int_{0}^{\infty} g (τ) P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) d τ, x > 0, \end{matrix}

(16)

where m and n are complex numbers with

ℜ m < \frac{1}{2}

The

B_{m, n}^{*} g

exists for

g \in L^{1} (R_{+})

In fact, from (12)

\begin{matrix} | (B_{m, n}^{*} g) (x) | \\ \leq & sinh x \int_{0}^{\infty} | g (τ) | | P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) | d τ \\ \leq & sinh x C \sqrt{\frac{π}{2}} Γ (\frac{1}{2} - ℜ m) P_{- \frac{1}{2}}^{ℜ m, 0} (cosh x) \int_{0}^{\infty} | g (τ) | d τ < \infty, \end{matrix}

(17)

for each

x \in R_{+}

Now, observe that for

γ \leq - \frac{1}{2}

and having into account (10) and () the expression

\begin{matrix} e^{γ x} P_{- \frac{1}{2}}^{ℜ m, 0} (cosh x) sinh x \leq A, f o r a l l x > 0 a n d s o m e A > 0 . \end{matrix}

Thus, for

g \in L^{1} (R_{+})

one has

\begin{matrix} {∥ B_{m, n}^{*} g ∥}_{γ} \leq A \cdot {∥ g ∥}_{L^{1} (R_{+})}, γ \leq - \frac{1}{2} . \end{matrix}

So one obtains the next result.

Proposition 5.

Set

m, n

be complex numbers with

ℜ m < \frac{1}{2}

. The linear operator

B_{m, n}^{*}

given by (16) is a bounded linear operator from

L^{1} (R_{+})

into

E_{γ}

γ \leq - \frac{1}{2}

The next result holds.

Proposition 6.

Set

m, n

be complex numbers with

ℜ m < \frac{1}{2}

and w be a measurable function on

R_{+}

such that

w > 0

a.e. on

R_{+}

and

0 < q < \infty

. If

\int_{0}^{\infty} {(P_{- \frac{1}{2}}^{ℜ m, 0} (cosh x) sinh x)}^{q} w (x) d x < \infty

. Then

\begin{matrix} B_{m, n}^{*} : L^{1} (R_{+}) \to L^{q} (R_{+}, w (x) d x) \end{matrix}

is a bounded linear operator.

Proof.

From (17) one has

\begin{matrix} {(\int_{0}^{\infty} {| (B_{m, n}^{*} g) (x) |}^{q} w (x) d x)}^{\frac{1}{q}} \\ \leq M \cdot {∥ g ∥}_{L^{1} (R_{+})} \cdot {(\int_{0}^{\infty} {(P_{- \frac{1}{2}}^{ℜ m, 0} (cosh x) sinh x)}^{q} w (x) d x)}^{\frac{1}{q}}, \end{matrix}

for some

M > 0

, and so the result holds. □

Example 2.

Examples of weights w for Proposition 6 are:

\begin{matrix} (i) w (x) = e^{r x}, f o r r < - \frac{q}{2} . \\ (ii) w (x) = e^{r x^{2}}, f o r r < 0 . \end{matrix}

The next Parseval-Goldstein type relation holds.

Theorem 7.

Set

m, n

be complex numbers with

ℜ m < \frac{1}{2}

f \in E_{γ}

γ > \frac{1}{2}

and

g \in L^{1} (R_{+})

, then the following Parseval-Goldstein type relation relation holds

\begin{matrix} \int_{0}^{\infty} (B_{m, n} f) (x) g (x) d x = \int_{0}^{\infty} f (x) (B_{m, n}^{*} g) (x) d x . \end{matrix}

(18)

Proof.

Applying Fubini’s theorem in the following we get

\begin{matrix} \int_{0}^{\infty} (B_{m, n} f) (τ) g (τ) d τ & = \int_{0}^{\infty} \int_{0}^{\infty} f (x) P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x d x g (τ) d τ \\ = \int_{0}^{\infty} sinh x \int_{0}^{\infty} g (τ) P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) d τ f (x) d x \\ = \int_{0}^{\infty} f (x) (B_{m, n}^{*} g) (x) d x . \end{matrix}

Observe that from (17) the

B_{m, n}^{*} g

exists for

g \in L^{1} (R_{+})

. □

Consider the differential operator

\begin{matrix} L_{m, n, x} = D_{x} sinh x D_{x} {(sinh x)}^{- 1} + [\frac{m^{2}}{2 (1 - cosh x)} + \frac{n^{2}}{2 (1 + cosh x)}] . \end{matrix}

From [13]

\begin{matrix} L_{m, n, x} (P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x) = - (τ^{2} + \frac{1}{4}) P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x, \end{matrix}

(19)

and then for

k \in N

\begin{matrix} L_{m, n, x}^{k} (P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x) = {(- 1)}^{k} {(τ^{2} + \frac{1}{4})}^{k} P_{- \frac{1}{2} + i τ}^{m, n} (cosh x) sinh x . \end{matrix}

(20)

Denote

\begin{matrix} L_{m, n, x}^{*} = {(sinh x)}^{- 1} D_{x} sinh x D_{x} + [\frac{m^{2}}{2 (1 - cosh x)} + \frac{n^{2}}{2 (1 + cosh x)}] . \end{matrix}

From (20) and

f \in C_{c}^{2 k} (R_{+})

k \in N

, it follows

\begin{matrix} (B_{m, n} (L_{m, n, x}^{*^{k}} f)) (τ) = {(- 1)}^{k} {(τ^{2} + \frac{1}{4})}^{k} (B_{m, n} f) (τ), τ > 0 . \end{matrix}

(21)

Then from Theorem 7 one has the next result.

Theorem 8.

f \in C_{c}^{2 k} (R_{+})

k \in N

m, n

be complex numbers with

ℜ m < \frac{1}{2}

, and

g \in L^{1} (R_{+})

, then the following Parseval-Goldstein type relation holds

\begin{matrix} {(- 1)}^{k} \int_{0}^{\infty} (B_{m, n} f) (x) g (x) {(x^{2} + \frac{1}{4})}^{k} d x = \int_{0}^{\infty} (L_{m, n, x}^{*^{k}} f) (x) (B_{m, n}^{*} g) (x) d x . \end{matrix}

(22)

4. Conclusions

In conclusion, this paper has been dedicated to the establishment of boundedness properties and Parseval-Goldstein type relations for the generalized Mehler-Fock transform, as first introduced by B. L. J. Braaksma and B. Meulenbeld in 1967 and subsequently by R. S. Pathak in 1997. Moreover, we have derived an inversion formula for this transform across Lebesgue spaces. These results not only contribute to a deeper understanding of the generalized Mehler-Fock transform but also pave the way for future investigations into analogous properties within the spaces of type

E_{γ}

for diverse integral transforms.

Declarations Note

The manuscript has no associated data.

Conflicts of Interest

No potential conflict of interest was reported by the authors.

References

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González B., J. and Negrín E. R., Operational calculi for Kontorovich-Lebedev and Mehler-Fock transforms on distributions with compact support, Rev. Colombiana Mat., 32(1):81-92, 1998.
González B., J. and Negrín E. R., Lp-inequalities and Parseval-type relations for the Mehler-Fock transforms of general order, Ann. Funct. Anal., 8(2): 231-239, 2017.
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Srivastava H., M., González B. J. and Negrín E. R., New Lp-boundedness properties for the Kontorovich-Lebedev and Mehler-Fock transforms, Integral Transforms Spec. Funct., 27(10):835-845, 2016.
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The Generalized Mehler-Fock Transform over Lebesgue Spaces

Abstract

Keywords:

Subject:

1. Introduction and Preliminaries

2. An Inversion Formula over the Spaces E γ

3. Boundedness Properties and Parseval-Goldstein Type Relations Over the Spaces E γ

4. Conclusions

Declarations Note

Conflicts of Interest

References

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2. An Inversion Formula over the Spaces $E_{γ}$

3. Boundedness Properties and Parseval-Goldstein Type Relations Over the Spaces $E_{γ}$