828

On the Exchange Property for the Mehler-Fock Transform

Abhishek Singh

Department of Mathematics

Amity Institute of Applied Sciences

Amity University Uttar Pradesh, Noida

mathdras@gmail.com; asingh28@amity.edu

Received: December 15, 2015; Accepted: August 8, 2016

Abstract

The theory of Schwartz Distributions opened up a new area of mathematical research, which

in turn has provided an impetus in the development of a number of mathematical disciplines,

such as ordinary and partial differential equations, operational calculus, transformation theory

and functional analysis. The integral transforms and generalized functions have also shown

equivalent association of Boehmians and the integral transforms. The theory of Boehmians,

which is a generalization of Schwartz distributions are discussed in this paper. Further,

exchange property is defined to construct Mehler-Fock transform of tempered Boehmians.

We investigate exchange property for the Mehler-Fock transform by using the theory of

Mehler-Fock transform of distributions. Algebraic properties and convergence is also proved

for this relation on the tempered Boehmians which is a natural extension of tempered

distribution.

Keywords: Distribution spaces; tempered Boehmians; Fourier transform; Mehler-Fock

transform

MSC 2010 No.: 46F10, 46F99, 44A10

1. Introduction

The concept of Boehmians is motivated by the regular operator introduced by Boehme

(1973), which forms a subalgebra of the field of Mikusiński operators and thus they include

only such functions whose support is bounded from the left. The theory of Boehmians

(quotient of sequences), its properties and different classes of Boehmian spaces are studied by

Mikusiński et al. (1981), Mikusiński (1983, 1995).

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Appl. Appl. Math.

ISSN: 1932-9466

Vol. 11, Issue 2 (December 2016), pp. 828 - 839

Applications and Applied

Mathematics:

An International Journal

(AAM)

AAM: Intern. J., Vol. 11, Issue 2 (December 2016) 829

Tempered Boehmians is a natural extension of tempered distribution which, therefore, makes

it possible to define an extension of the Fourier transform for this class of Boehmians. The

Fourier transform of a tempered Boehmian is a distribution. An infinitely differentiable

function CRf n : is called rapidly decreasing if

, |)(|)...1(supsup 22

1||

xfDxx m

NRxm N

(1)

for every nonnegative integer m, where x = (x1, x2... xN), = (1,...N), N's are non-negative

integer, | | = 1+...+N , and

....1

1

||||

NNxxx

D

(2)

The space of rapidly decreasing functions is denoted by S(RN) or simply by S. If f J and

S, then the convolution

duuxufxfNR

)()())(( (3)

is well defined and f J. A sequence Sn is called a delta sequence if it satisfies the

following conditions

(i) 1)( dxxn

RN

, for all n N,

(ii) Mdxxn

R N

|)(| , for some constant M and for all n N,

(iii)

||||

,0|)(|||||lim

x

nk

n dxxx for every Nk and > 0.

If S and 1= , then the sequence of functions n is a delta sequence.

A continuous function CRf N : is called slowly increasing if there is a polynomial p on

𝑅𝑁 such that )(|)(| xp xf for all NRx . The space of slowly increasing function will be

denoted by J(RN) or simply by J. Let fn J. }{ n is a delta sequence under usual notation.

Then the space of equivalence classes of quotients of sequence will be denoted by J and its

elements will be called tempered Boehmians. For JnnfF ]/[ define

)].)/([(= nnnn DfFD

If F is a Boehmian corresponding to differentiable function, then J FD .

830 Abhishek Singh

If JnnfF ]/[

and Sfn , for all ,Nn then F is called a rapidly decreasing

Boehmian, the space of which is denoted by S . If JnnfF ]/[= and ,]/[= SnngG

then we define the convolution

.)])/([(= Jnnnn gfGF

In what follows, we will denote by S' the space of tempered distributions; that is, the space of

continuous linear functional on S. The Mehler-Fock transform of a tempered distribution f,

denoted by Mf, is the functional defined by Mf () = f (M), where M is the Mehler-Fock

transform of defined by Banerji et al. (2008). The Mehler-Fock transform on generalized

functions is studied by Pathak (1997). The Mehler-Fock transform of Boehmian spaces are

investigated by Loonker and Banerji (2008, 2009, 2009).

In Section 2 we study Mehler-Fock transform and its properties and investigate the exchange

property for the Mehler-Fock transform. In Section 3, algebraic properties and convergence is

proved for this relation on the tempered Boehmians.

2. The Mehler-Fock Transform and the Exchange Property

The Mehler-Fock transformation is defined as [cf. Yakubovich and Luchko (1994, p. 149)]:

1)()()()]([

2

1 dxxfxPrFxfMir

, 0r , (4)

and its inversion is given by

1

20( ) tanh( ) ( ) ( ) , 1

irf x r r P x F r dr x

. (5)

The generalization of the Mehler-Fock transformation is given by [cf. Pathak (1997, p. 343)]

xdxxPxfrF nm

irsinh)(cosh)()(

0

,

2

1

, (6)

where )(cosh,

2

1 xP nm

ir is the generalized Legendre function, defined for complex values of the

parameters mk, and n by

2

1;1;

2;1

2)1)(1(

)1()( 122/

2/, z

mmn

kmn

kFzm

zzP

m

nnm

k , (7)

for complex z not lying on the cross-cut along the real x-axis from 1 to .

The inversion formula of (6) is

0

, )()(cosh)()(2

1 drrFxPrxf nm

ir , (8)

AAM: Intern. J., Vol. 11, Issue 2 (December 2016) 831

Where

ir

nmir

nmir

nmir

nmr

2

1

2

1

2

1

2

1)(

1

22)2()2(

mnirir . (9)

When nm , (6) and (8) can be written as

0)(sinh)(cosh)(

2

1 dfPrFir

, (10)

and

0)()(cosh)tanh()(

21 drrFPrrf

ir , (11)

whereas for 0 nm , (6) and (8) reduce to (4) and (5), respectively.

The Parseval relation for the Mehler-Fock transformation is defined as [cf. Sneddon (1974,

pp. 393-94)]

0)()()tanh( drrGrFrr

1

)()( dxxgxf , (12)

whose convolution is

][][][ gMfMgfM . (13)

The asymptotic behavior for (7) is defined by Pathak (1997, p. 345) as

Re

,

1/2 (1/2)

( ) , 0 ,(cosh )

( ), ,

m

m n

ir x

O x xP x

O e x

(14)

and

,

1/2 1/2( 1) 1/2 1/2 1/2 ( ) 1

(1) , 0 ,(cosh )

2 (sinh ) ( ) { ( )}, .

m n

ir n m m irx i m xt

O rP x

x ir e ie O r r

(15)

Similarly, the function )(r , defined by (9), possesses the following asymptotic behavior [cf.

Pathak 1997, p. 345)]

2

1 21

2

( ), 0 Re 1 Re ,

( ) ( )[1 ( )], .

2

m

n m

O r r n m

r irO r r

(16)

832 Abhishek Singh

The distributional generalized Mehler-Fock transform )( Rf M , where R denotes the set

of positive real numbers and )Re(m , 2/1 , is defined as [cf. Pathak (1997, p. 346)]

12

,( ) : ( ), (cosh ) , 0,m n

irF r f x P x r

(17)

where the space )( RM is the dual of the space )( R

M , which is the collection of all

infinitely differentiable complex valued function defined on open interval (0, ∞) denoted

by 𝑅+ such that for every non-negative integer q,

)()(sup)(0

xx q

xx

q , (18)

where

)cosh1(2)cosh1(2)(coth

222

x

n

x

mDxD xxx

, (19)

and

,

( ), 0,( ) ( )

( ), .

O x xx x

O x x

(20)

The topology over )( RM is generated by separating collection of seminorms

0}{ qq and

is a sequentially complete locally convex topological vector space. 𝐷(𝑅+), the space of

infinitely differentiable functions of compact support with the usual topology, is a linear

subspace of )( RM .

From the properties of the hypergeometric functions, the generalized Legendre function [cf.

Pathak (1997, p. 346)], satisfies the following differential equation

04

1

)cosh1(2)cosh1(2)(coth 2

222

yr

x

n

x

mDyxyD . (21)

Therefore,

)(cosh4

1)(cosh ,

2/1

2,

2/1 xPrxP nm

ir

nm

irx

.

(22)

Relations (14) and (15) prove the boundedness for the Legendre function [cf. Pathak (1997,

pp. 346-347, Lemma 11.3.1, Equation (11.3.2))]

)(cosh2

1

2)()(cosh)(

0,

2/1

2/1

,

2/1 xPmxxCxPr

x rm

r

qnm

ir

q

, (23)

where C is a constant independent of x and r and, rm is Re( ).m

AAM: Intern. J., Vol. 11, Issue 2 (December 2016) 833

The differentiability of the Mehler-Fock transform is defined by [cf. Pathak (1997, p. 347)]

)(cosh),(:)( ,

21 xP

rxfrF nm

ir

, (24)

where )( Rf M , ,2/1)Re(),Re( mm 0,2/1 r .

When q is a non-negative integer depending on f, the asymptotic behavior of the Mehler-Fock

transform is

2

(1), 0,( )

( ), ,q

O rF r

O r r

(25)

where

q

qrCrF

4

1max)( 2 . (26)

For the operator )()(:*

RRx

MM under already stated symbols and for )(

Rf M ,

)( R M , we define the operator transformation formula by

,)(),()(),()( * xxfxxf q

x

q

x (27)

and for f being the generalized Mehler-Fock transformation,

)(4

1)1()()( 2* xfMrxfM

q

qq

t

.

If )( Rf M and )( R

M we have by transposition by Banerji et al. (2008)

( ), = , ( ) ,M f f M (28)

where function f is absolutely integrable and is a testing function of rapid descent.

For a family {𝜑𝑖}𝑖∈𝐼 = {𝜑𝑖}𝐼, where I is an index set and 𝜑𝑖 ∈ )( RM ⊂ 𝑆, ∀𝑖 ∈ 𝐼, we

define [Atanasiu and Mikusiński (2005)]:

Ψ({𝜑𝑖}𝐼) = {𝑥 ∈ 𝑅+𝑁 : 𝑀𝜑𝑖(𝑥) = 0, ∀ 𝑖 ∈ 𝐼}. (29)

A family of pairs {(fi, i)}I, where fi )( RM ⊂ 𝑆′ and 𝜑𝑖 ∈ )( R

M ⊂ 𝑆, ∀𝑖 ∈ 𝐼, is said

to have the exchange property if

834 Abhishek Singh

ikki ff , Iki, . (30)

We will denote by A the collection of all families of pairs {(fi, i )}I, where I is an index set,

fi )( RM ⊂ 𝑆′ and 𝜑𝑖 ∈ )( R

M ⊂ 𝑆, ∀𝑖 ∈ 𝐼, satisfying the exchange property such that

)}({ Ii . If (i) is a delta sequence, then )}({ Ni .

Definition 1.

If {(fi , i)}I A, then the unique F D' (R) such that Mfi = Mi F for all i I will be

denoted by M{(fi , i)}I .

Let Agf kkkIii )},{(,)},{( . If ikki gf for all i I and k K, then we write

KkkIii gf )},{(~)},{( . This relation is clearly symmetric and reflexive. We will show that

it is also transitive.

Let

.)},{(,)},{(,)},{( Ahgf LllKkkIii

If

KkkIii gf )},{(~)},{( and LllKkk hg )}{(~)}{( ,

then

kllkikki hggf , ,

(31)

for all i I, k K, l L. Therefore,

iklilkliklki hggf , , (32)

for all i I, k K, l L. Since is commutative, we have

kilkli hf . (33)

Now fix i I and l L. Since )}({ Kk and (32) holds for every k K, we conclude

that illi hf for all i I and l L, which means that .)},{(~)},{( LllIii hf

Theorem 1.

If a family of pair Iiif )},{( has the exchange property and {( ) }c

i I (the complement

of {( ) }i I in 𝑅+), then there exists a unique F D' () such that

M[𝑓𝑖] = 𝐹𝑀[𝜑𝑖], ∀ 𝑖 ∈ 𝐼. (34)

AAM: Intern. J., Vol. 11, Issue 2 (December 2016) 835

Proof:

For every x there exists i I and > 0 such that |)(| xM i in an open neighborhood

of x. Then we can define ii MMfF / in that neighborhood. Let for some > 0, we have

|)(| xM i for all x U and |)(| xM k for all x V, where U and V are open sets.

Since ikki ff , we have

ikki MMfMMf and k

k

i

i

M

Mf

M

Mf

, (35)

on VU . This shows that F is a unique function.

Theorem 2.

There exists Iiif )},{( A, for every 𝐹 ∈ 𝐷′(𝑅) and such that ).)},({( IiifMF

Proof:

Since 𝐷 (𝑅) denotes the space of smooth function with compact support, there exists a total

sequence Ni}{ such that Mi ∈ 𝐷 (𝑅) for all I N. Then for every I N, there is fi

)( RM ⊂ 𝑆′ such that Mfi = Mi F. Clearly Niif )},{( A and ))},({( NiifMF . This

completes the proof of the theorem.

Definition 2. [Atanasiu and Mikusiński (2005)]

Let {Ui}I be an open covering of 𝑅+ and let Ii}{ be such that 0 |)(| xM i for x Ui. A

family Ii}{ such that )}({ Ii will be called total.

Lemma 1. [Atanasiu and Mikusiński (2005)]

If Ii}{ and Kk }{ are total, then KIki }*{ is total.

Theorem 3.

Let

Iiif )},{( , Ag Kkk )},{( .

Then,

KkkIii gf )},{(~)},{( if and only if ))},({())},({( KkkIii gMfM .

Proof:

836 Abhishek Singh

Here,

})),({( IiifMF and ))},({( KkkgMG .

If

KkkIii gf )},{~)},{( ,

then

kiki MMfMMF

ik MMg

., , KkIiMGMMFM ikki (37)

Hence, F = G, by Lemma 1. Now assume F = G. Then,

., , KkIiMMgMMGMMFMMf ikikkiki (38)

Hence,

.)},{(~)},{( KkkIii gf

This completes the proof of the theorem.

Theorem 4.

There exists a delta sequence (n) such that for every T J, ])},[{( NnnfT for some

fn J.

Proof:

Let )( n be a delta sequence such that Mn D(𝑅). Then for any TJ, we have 𝑇𝑀𝜓𝑛 ∈

)( RM ⊂ 𝑆′, since MT D'(𝑅). Consequently, n nMTM Mg for some gn )(

RM

⊂ 𝑆′. It is easy to check that ].)},[{( NnnnngT Since Jgf nnn and

)()( nnn is a delta sequence, where (n) does not depend on T, hence the theorem is

proved.

3. Algebraic Properties and Convergence

J becomes a vector space with the addition operation, defined by

])},[{(])},[{(])},[{( KIkiikkiKkkIii gfgf . (39)

Moreover, multiplication by a scalar and the operation are defined by

AAM: Intern. J., Vol. 11, Issue 2 (December 2016) 837

C ff IiiIii ],)},[{(])},[{( . (40)

If

JKkkIii gf ])},[{( ],)},[{( and gk )( RM for all k K,

then for the operation we can define

].)},[{(])},[{(])},[{( KIkikikkkIii gfgf (41)

Definition 3. [Atanasiu and Mikusiński (2005)]

Let T0, T1, T2, ... J. Then the sequence (Tn) is said to converge to T0, which is written as

TnT0 if there exists a total family {i }I such that

(a) there exists tempered distribution fi,n, where i I and n N such that

]},[{ , Iinin fT for all n = 0,1,2,...,

(b) 0,, ini ff in )( RM as n for every i I.

Theorem 5.

The Mehler-Fock transform is an isomorphism from J to D'(𝑅).

Proof:

Since TnT0 in J if and only if nT T0 0 , it suffices to prove the continuity at 0. Let

Tn0 is in J . Then there exists tempered distribution fi,n where i I and n N such that

]},[{ , Iinin fT for all n = 1,2,... and fi,n 0 in )( RM ⊂ 𝑆′ as n for every i I. If

D(𝑅), then there are i1,...,ik such that

mi

k

m

M suppsupp1

.

Then,

2

1

1 ||

)(limlim

m

m

m

i

k

m

i

in

k

mn

nn

M

MMMTMT

0

||

lim2

1

,

1

m

m

m

i

k

m

i

nin

k

m M

MMf

, (42)

838 Abhishek Singh

because 0lim ,

nin

Mf for i I, due to the continuity of the Mehler-Fock transform in

)( RM . This proves the continuity of M : J D′(𝑅) , because 0lim

n

nMT in )(

RM

for every D(𝑅), implies 0lim

nn

MT in D' (𝑅).

Now, assume 0lim

nn

MT in D′(𝑅). By Theorem 4, there exists a delta sequence Ni i ),(

such that for every n N, we have ])},[{( , Ninin fT for some ., Jf ni Let Nkk ),( be

a delta sequence such that kM D(𝑅) for every k N. Then,

0lim

kinn

MMMT in )( RM for every i, k N .

Since

niin fMMT , i, k N,

0lim ,

knin

MMf in )( RM ⊂ 𝑆′,

which implies

0lim ,

knin

f , in )( RM ⊂ 𝑆′.

But,

])},[{(])},[{( ,, KIkkkniIinin ffT , (43)

for all n = 0,1,2,... . Thus, we have Tn 0 in J . This proves the theorem.

3. Conclusions

The present paper focused on the exchange property for the Mehler-Fock transform via

tempered Boehmians which is the natural extension of tempered distributions. Algebraic

properties and convergence proved for this relation are useful in this area for development of

the convolution properties and other operations of Mehler-Fock transform of distributions

and Boehmians [Pathak et al. (2016)]. The formula and the property established in this paper

may also be suitable for an ultraBoehmians. The aforesaid analysis can be used to develop the

Calderon’s formula for Mehler-Fock transform [Pathak et al. (2016)].

Acknowledgement:

The author is thankful to the referees for fruitful criticism and comments for the

improvement.

AAM: Intern. J., Vol. 11, Issue 2 (December 2016) 839

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