On the Exchange Property for the Mehler-Fock Transform · infinitely differentiable functions of...
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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
[email protected]; [email protected]
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).
Available at
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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)
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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
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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
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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