Continuity of the Bessel wavelet transform on certain Beurling type function spaces

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R E S E A R C H

Open Access

Continuity of the Bessel wavelet transform on

certain Beurling-type function spaces

Akhilesh Prasad

1

_{, Ashutosh Mahato}

1*

_{and MM Dixit}

2

*_{Correspondence:}

[email protected] 1_{Department of Applied}

Mathematics, Indian School of Mines, Dhanbad, 826004, India Full list of author information is available at the end of the article

Abstract

In this paper, continuity of the Bessel wavelet transform of a suitable function

φ

in terms of an appropriate mother wavelet

ψ

is investigated on certain Beurling-type function spaces.

MSC: 46F12; 26A33

Keywords: Bessel wavelet transform; Hankel transform; Hankel translation; Zemanian space

1 Introduction

Bessel wavelet transforms have applications in the study of boundary value problems on the half-line. The Hankel transform and pseudo-diﬀerential operators associated with the Bessel operator have been studied on some Beurling-type function spaces by [, ] and [] respectively. The aim of the present paper is to study the continuity of the Bessel wavelet transform on certain Beurling-type function spaces.

Let us deﬁneLμ,p(R+), ≤p<∞, as the space of those real measurable functionsφon R+= (,∞) for which

φLμ,p=

∞



φ(x)pdμ(x)

/p

<∞, ≤p<∞, ()

φLμ,∞=ess sup <x<∞

φ(x)<∞, ()

wheredμ(x) = /μ₍_μ_{+ )}_xμ+_dx_.

From [, ], the Hankel translationφ∈Lμ,p(R+) is deﬁned by

τyφ(x) :=φ(x,y) :=

∞



φ(z)D(x,y,z)dμ(z),  <x,y<∞, ()

where

D(x,y,z) :=

∞



j(xt)j(yt)j(zt)dμ(t) ()

withj(x) = μ₍_μ_{+ )}_x–μ_J μ(x).

From [], in terms of the Hankel translationτyand dilationDa deﬁned byDaφ(x,y) := φ(x/a,y/a), we deﬁne the Bessel waveletθb,aby

θb,a(x) :=Daτbθ(x) =Daθ(b,x) =θ(b/a,x/a) :=

∞



D(b/a,x/a,z)θ(z)dμ(z). ()

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Letθ ∈Lμ,p(R+), then the Bessel wavelet transform ofg∈Lμ,q(R+), (/p+ /q= ), is

de-ﬁned by

(Bθg)(b,a) =

g(t),θb,a(t)

=

∞



g(t)θb,a(t)dμ(t) ()

=

∞



∞



g(t)θ(z)D(b/a,t/a,z)dμ(z)dμ(t)

=

∞



(bx/a)/Jμ(bx/a)x–μ–/b–μ–/aμ+

×

∞



tμ+/_g₍_t₎₍_tx_/_a₎/_J

μ(tx/a)dt

×

∞



zμ+/θ(z)(zx)/Jμ(zx)dz dx. ()

Now, using the Hankel transformation,

(hμφ)(u) =

∞



(xu)/Jμ(xu)φ(x)dx, μ≥–/ ()

which is known to be an automorphism on the Zemanian spaceHμ(R+), R+= (,∞),

consisting of all complex-valued inﬁnitely diﬀerentiable functionsφonR+which satisfy

γ_m,kμ (φ) = sup

x∈R+

xmx–Dkx–μ–/φ(x)<∞, ∀m,k∈N. ()

From (), we have

(Bθg)(b,a)

=aμ+b–μ–/

∞



x–μ–/(bx/a)/Jμ(bx/a)hμ

tμ+/g(t)(x/a)hμ

zμ+/θ(z)(x)dx

=aμ+/b–μ–/

∞



(u)–μ–/(bu)/Jμ(bu)hμ

tμ+/g(t)(u)hμ

zμ+/θ(z)(au)du.

Now, let us set

u–μ–/_h

μ

tμ+/_g₍_t₎_{= (}_h

μf)(u),

zμ+/_θ₍_z_{) =}_ψ₍_z₎

and

(Bψf)(b,a) =bμ+/a–μ–/(Bθg)(b,a)

to get the following convenient form of the Bessel wavelet transform:

(Bψf)(b,a) :=

∞



(bu)/Jμ(bu)f˜(u)ψ˜(au)du:=hμ

_˜

f(u)ψ˜(au)(b), ()

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From [, p.], we shall use the following Leibnitz-type formula:

x–Dkx–μ–/_{ψ φ}₌ k

s=

k s

x–Dsx–μ–/_φ_x–_Dk–s

ψ. ()

2 The spaceHω_μ

Letωbe a continuous real-valued function deﬁned onR+= (,∞) such thatω() =  and

(a) ≤ω(u+v)≤ω(u) +ω(v), ∀u,v∈R+, ()

(b)

∞



ω(u)du

 +u <∞, ()

(c) ω(u)≥l+dlog( +u) ()

for some reallandd> .

The class of all suchωfunctions is denoted byM.

Now, assume thatωis a function inM. A functionfis said to be in the spaceHω μ, where

f andhμf are smooth functions, and for everyμ∈R+,n∈N and m is a positive real

number,

αμ_m,n(f) = sup

u∈R+e

mω(u)_u–_Dn

u–μ–/f(u)<∞. ()

OnHω

μ we consider the topology generated by the family{α μ

m,n}m∈R+,n∈N of seminorms.

From [, ], we have

β_m,nμ (f) = sup

u∈R+e

mω(u)_u–_Dn_u–μ–/₍_h

μf)(u)<∞, ∀m∈R+,n∈N. ()

In what follows, we study the Bessel wavelet transformBψof inﬁnite order onHμω. For this

purpose, we deﬁne the symbol classSρ,ω_.

Deﬁnition . The function ψ˜(x) :C∞(R+)→C belongs to class Sρ,ω if and only if

∀m∈R+

emω(x)x–∂/∂xpψ˜(x)≤cp,ρ +xρ–p, ∀p∈N,ρ∈R, ()

wherecis a constant andψ˜ denotes the Hankel transform of the basic waveletψ.

In this section, certain spaces of functions of Beurling-type are introduced on which Bessel wavelet transforms can be deﬁned. First, we recall the deﬁnition of the Zemanian spaceHμ(R+).

Definition . The set of all infinitely differentiable functions (Bψφ)(b,a) onR+satisfying

the condition

γ_n,lμ,m,m(Bψφ) =sup

a,b

emω(b)–mω(a)b–Dna–Dlb–μ–/(Bψφ)(b,a)<∞, ()

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Theorem . The Bessel wavelet transform Bψφ is a continuous linear map of Hμω(R+)

into Hω

μ(R+)forμ≥–/.

Proof To complete the proof of the theorem, we need to show that (Bψφ)(b,a)

satis-ﬁes (). From the property (b) of the functionω(u), it follows that to every> , there exists a constantc() such that

ω(u)≤u+c(); ()

so that

emω(u)≤emc()

∞

ν=

(m)ν

ν! u

ν

. ()

Using equation () and the technique of Zemanian [, p.], we can write

emω(b)_b–_Dn

b–μ–/₍_B

ψφ)(b,a)

≤emc()

∞

ν=

(m)ν

ν! b

ν

b–Dnb–μ–/(Bψφ)(b,a)

=emc()

∞

ν=

(m)ν

ν! b

ν

b–Dnb–μ–/hμ

_˜

φ(u)ψ˜(au)

=emc() ∞

ν=

(m)ν

ν!

×

_∞



uμ+n+ν+u–Dνu–μ–/ψ˜(au)φ˜(u)(bu)–μ–nJμ+ν+n(bu)du

. ()

Since forμ≥–/,|(bu)–μ–n_J

μ+ν+n(bu)|is bounded on  <b,u<∞byQμ, the right-hand

side of equation () can be estimated by

≤emc()_Q

μ

∞

ν=

(m)ν

ν!

∞



uμ+n+ν+_u–_Dν

u–μ–/_ψ˜₍_au₎_φ˜₍_u₎_du

=emc()Qμ

∞

ν=

(m)ν

ν!

∞



uμ+n+ν+

×

ν

r=

ν

r

u–Dru–μ–/φ˜(u)u–Dν–rψ˜(au)

du.

Therefore,

emω(b)b–Dna–Dlb–μ–/(Bψφ)(b,a)

≤emc()_Q

μ

∞

ν=

(m)ν

ν!

∞



uμ+n+ν+

ν

r=

ν

r

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In view of estimate (), we have fort=au

u–Dra–Dlψ˜(au)

=arult–d/dtr+lψ˜(t)

≤cr+l,ρ +tρ–r–larul

=cr+l,ρ +auρ–r–larul. ()

Therefore, () becomes

emω(b)b–Dna–Dlb–μ–/(Bψφ)(b,a)

≤emc()Qμ

∞

ν=

(m)ν

ν!

∞



uμ+n+ν+

×

ν

r=

ν

r

u–Dν–ru–μ–/φ˜(u)cr+l,ρ +auρ–r–laruldu

≤emc()Qμ

∞

ν=

(m)ν

ν!

ν

r=

ν

r

cr+l,ρ +aρ–r–l +ar

× ∞



uμ+n+ν++l +uρ–r–lu–Dν–ru–μ–/φ˜(u)du. ()

SupposePis an integer not less than μ+ n+ l+ , then

uμ+n+ν++l≤( +u)P+ν. ()

Using (), the right-hand side of equation () can be bounded by

emc()_Q

μ

∞

ν=

ν

r=

ν

r

cr+l,ρ(m) ν

ν! ( +a)

(ρ–l)

×

∞



( +u)P+ν_{( +}_u₎(ρ–r–l)_u–_Dν–r

u–μ–/_φ˜₍_u₎_du

=emc()Qμ( +a)(ρ–l)

∞

ν=

ν

r=

ν

r

cr+l,ρ(m)

ν

ν!

×

∞



( +u)P+ν+ρ_{( +}_u₎–l_u–_Dν–r

u–μ–/_φ˜₍_u₎_du_.

Using inequality (), the above expression is bounded by

emc()Qμ,l,ρe–(

ρ–l)l/d_e(ρ–l)ω(a)/d ∞

ν=

ν

r=

ν

r

cr(m)

ν

ν! e

–(P+ρ+ν)l/d

× sup

u∈R+e

((P+ρ+ν)/d)ω(u)_u–_Dν–r

u–μ–/φ˜(u)

∞



du

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Using property (), we can estimate the right-hand side of () by

emc()Qμ,l,ρe–(P+

ρ–l)l/d_e(ρ–l)ω(a)/d ∞

ν=

ν

r=

ν

r

cr(m)

ν

ν! e

–(νl)/d_βμ

(P+ρ+ν)/d,(ν–r)(φ)

=Q_μ_,l,_ρe(ρ–l)ω(a)/d_exp_mc₍_{) – (}_P_{+ }_ρ_{– }_l₎_l_/_d ∞

ν=

((m)e–l/d₎ν ν! ( +c)

ν

× max

≤r≤ν

β_(P+μ _ρ₊_ν_)/d,(_ν_–r)(φ)/νν

=Qμ,l,ρe(

ρ–l)ω(a)/d_exp_mc₍_{) – (}_P_{+ }_ρ_{– }_l₎_l_/_d ∞

ν=

((m)e–l/d( +c))ν

ν!

× max

≤r≤ν

β_(P+μ _ρ₊_ν_)/d,(_ν_–r)(φ)/νν.

Therefore,

emω(b)–mω(a)b–Dna–Dlb–μ–/(Bψφ)(b,a)

≤Q_μ_,l,_ρexpmc() – (P+ ρ– l)l/d ×

∞

ν=

((m)e–l/d_{( +}_c₎₎ν ν!

max

≤r≤ν

β_(P+μ _ρ₊_ν_)/d,(_ν_–r)(φ)/νν,

wherem = (ρ–l)/d. Now, choosing

<

max

≤r≤νβ μ

(P+ρ+ν)/d,(ν–r)(φ)

–/ν

ν!me–l/d( +c)–,

we ﬁnd that the last series is convergent. Therefore,

emω(b)–mω(a)_b–_Dn

a–Dlb–μ–/₍_B

ψφ)(b,a)<∞.

Hence, (Bψφ)∈Hμω(R+).

3 The spaceGω_μ

The Bessel-diﬀerential operatorSμis deﬁned by

Sμ,x=

d

dx+

( – μ₎

x . ()

From [, p.] we know that for anyφ∈Hμ(R+),

hμ(Sμφ) = –yhμφ ()

and

Sr_μ_,xφ(x) =

r

j=

cjxj+μ+/

x–Dr+jx–μ–/_φ₍_x₎_, _()

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Now, assume thatωis a function inM.Afunctionφ∈C∞(R+) is said to be in the space

Gω

μif for everyμ∈R,n∈N, andmis a positive real number,

Aμ

m,n(φ) = sup x∈(R+)

emω(x)_Sn

μφ(x)<∞. ()

The family{Aμm,n}m∈R,n∈Nof seminorms generates the topology ofGμω.

Definition . The set of all infinitely differentiable functions (Bψφ)(b,a) onR+satisfying

the condition

δμ_n,m,m(Bψφ) =sup

a,b

emω(b)–mω(a)Snμ,b(Bψφ)(b,a)<∞, ∀n∈N,

is denoted byGω

μ(R+), whereμ,m,m ∈R.

Theorem . The Bessel wavelet transform Bψis a continuous linear map of Gωμ(R+)into

Gω

μ(R+)forμ≥–/.

Proof As in the proof of Theorem ., using inequality (), we have

emω(b)Sμn,b(Bψφ)(b,a)

≤emc()

∞

ν=

(m)ν

ν! b

ν

n

j=

|cj|bj+μ+/b–D

n+j

b–μ–/(Bψφ)(b,a)

=emc()

∞

ν= n

j=

|cj|

(m)ν

ν! b

j+ν+μ+/_b–_Dn+j

b–μ–/(Bψφ)(b,a). ()

Assume that ≤μ+ / <p, wherepis a positive integer, thenbμ+/ _≤_{( +}_b₎μ+/_≤

( +b)p, and the right-hand side of equation () is bounded by

emc()

∞

ν= n

j=

|cj|

(m)ν

ν! ( +b)

j+ν+p_b–_Dn+j_b–μ–/₍_B

ψφ)(b,a)

=emc()

∞

ν= n

j=

|cj|

(m)ν

ν!

j+ν+p k=

j+ν+p k

bk

×b–Dn+jb–μ–/hμ

_˜

ψ(au)φ˜(u)(b).

Using Zemanian’s technique, equation () and the Leibnitz-type formula (), the last expression can be estimated by

emc()

∞

ν= n

j=

|cj|

(m)ν

ν!

j+ν+p k=

j+ν+p k

×

∞



uμ+(n+j)+k+u–Dku–μ–/ψ˜(au)φ˜(u) ×(bu)–(μ+n+j)_J

μ+k+(n+j)(bu)du

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≤emc()Qμ ∞ ν= n j= j+ν+p

k=

|cj|

(m)ν

ν!

j+ν+p k

× ∞



uμ+(n+j)+k+

k s= k s

u–Dk–su–μ–/φ˜(u)u–Dsψ˜(au)du.

In view of estimate (), the above expression can be bounded by

emc()_Q

μ ∞ ν= n j= j+ν+p

k=

|cj|

(m)ν

ν!

j+ν+p k

∞



uμ+(n+j)+k+

× k s= k s

u–_Dk–s_u–μ–/_φ˜₍_u₎_as_cs,ρ_{ +}_a_uρ–s_du

≤emc()Qμ

∞ ν= n j= j+ν+p

k=

|cj|

(m)ν

ν!

j+ν+p k

∞



uμ+(n+j)+k+

× k s= k s

u–Dk–su–μ–/φ˜(u) +ascs,ρ +aρ–s +uρ–sdu

≤emc()Qμ

∞ ν= n j= j+ν+p

k= k

s=

|cj|

(m)ν

ν!

j+ν+p k

k s

 +aρcs,ρ

×

∞



uμ+(n+j)+k+_{( +}_u₎(ρ–s)_u–_Dk–s

u–μ–/_φ˜₍_u₎_du_.

Suppose thatNis a positive integer not less than μ+ n+p+ , then the above expression can be estimated by

emc()Qμ

∞ ν= n j= j+ν+p

k= k

s=

|cj|

(m)ν

ν!

j+ν+p k

k s

cs,ρ( +a)ρ

×

∞



( +u)N+ν( +u)(ρ–s)u–Dk–su–μ–/φ˜(u)du.

Using the inequality ( +u)≤e–l/deω(u)/d_{from (), the right-hand side of the above}

ex-pression can be bounded by

emc()_Q

μ,ρ

∞ ν= n j= j+ν+p

k= k

s=

|cj|

(m)ν

ν!

j+ν+p k

k s

cse–(ρl)/d_e(ρω(a))/d_e–(N+ν+ρ)l/d

× sup

u∈R+e

(N+ν+ρ)ω(u)/d_u–_Dk–s

u–μ–/φ˜(u)

∞



du

( +u)s. ()

Using property (), we can estimate the right-hand side of () by

emc()Qμ,ρe–(N+

ρ)l/d_e(ρω(a))/d ∞

ν= j+ν+p

k= n

j=

j+ν+p k

|cj|

(m)ν

ν! e

–lν/d_{( +}_c₎k

× max

≤s≤kβ

μ

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=e(ρω(a))/dQμ,ρexp

mc() – (N+ ρ)l/d

∞

ν= j+ν+p

k= n

j=

j+ν+p k

|cj|{(

m)e–l/d}ν

ν!

×( +c)kmax

≤s≤kβ

μ

(N+ν+ρ)/d,(k–s)(φ)

/νν

.

Therefore,

emω(b)–mω(a)Sμn,b(Bψφ)(b,a)

≤Q_μ_,_ρexpmc() – (N+ ρ)l/d

∞

ν= j+ν+p

k= n

j=

j+ν+p k

|cj|

×{(m)e–l/d}ν

ν!

( +c)kmax

≤s≤kβ

μ

(N+ν+ρ)/d,(k–s)(φ)

/νν

<∞

as the inﬁnite series can be made convergent by choosing

<

( +c)kmax

≤s≤kβ

μ

(N+ν+ρ)/d,(k–s)(φ)

–/ν

ν!me–l/d–,

wherem = ρ/d.

This completes the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

AP has identified the problems and suggested the solution, AM participated in the proof of the Theorems and MMD participated in the solution to find the Bessel wavelet transform. All authors read and approved the final manuscript.

Author details

1_{Department of Applied Mathematics, Indian School of Mines, Dhanbad, 826004, India.}2_{Department of Mathematics,}

NERIST, Nirjuli, India.

Acknowledgements

This work is supported by Indian School of Mines, Dhanbad, under grant No. 613002 /ISM JRF/Acad/2009.

Received: 17 November 2012 Accepted: 8 January 2013 Published: 22 January 2013

References

1. Belhadj, M, Betancor, JJ: Hankel transformation and Hankel convolution of tempered distributions. Rocky Mt. J. Math.

3(4), 1171-1203 (2001)

2. Pathak, RS, Shrestha, KK: The Hankel transform of Gevrey ultradistributions. Integral Transforms Spec. Funct.11(1), 61-72 (2001)

3. Pathak, RS, Prasad, A: The pseudo-diﬀerential operatorhμ,aon some Gevrey spaces. Indian J. Pure Appl. Math.34(8), 1225-1236 (2003)

4. Haimo, DT: Integral equations associated with Hankel convolutions. Trans. Am. Math. Soc.116, 330-375 (1965) 5. Pathak, RS, Dixit, MM: Bessel wavelet transform on certain function and distribution spaces. J. Anal. Appl.1, 65-83

(2003)

6. Zemanian, AH: Generalized Integral Transformations. Interscience, New York (1968)

doi:10.1186/1029-242X-2013-29

Cite this article as:Prasad et al.:Continuity of the Bessel wavelet transform on certain Beurling-type function spaces.