CALDERON'S REPRODUCING FORMULA FOR HANKEL … · 2020. 1. 14. · [6] I.MarreroandJ.J.Betancor,Hankel convolution of generalized functions, Rendiconti di Matem- atica e delle sue

CALDERÓN’S REPRODUCING FORMULAFOR HANKEL CONVOLUTION

R. S. PATHAK AND GIREESH PANDEY

Received 27 March 2006; Accepted 27 March 2006

Calderon-type reproducing formula for Hankel convolution is established using the the-ory of Hankel transform.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction

Calderon’s formula [3] involving convolutions related to the Fourier transform is usefulin obtaining reconstruction formula for wavelet transform besides many other applica-tions in decomposition of certain function spaces. It is expressed as follows:

f (x)=∫∞

0

(φt ∗φt ∗ f

)(x)

dt

t, (1.1)

where φ :Rn→ C and φt(x)= t−nφ(x/t), t > 0. For conditions of validity of identity (1.1),we may refer to [3].

Hankel convolution introduced by Hirschman Jr. [5] related to the Hankel transformwas studied at length by Cholewinski [1] and Haimo [4]. Its distributional theory wasdeveloped by Marrero and Betancor [6]. Pathak and Pandey [8] used Hankel convolutionin their study of pseudodifferential operators related to the Bessel operator. Pathak andDixit [7] exploited Hankel convolution in their study of Bessel wavelet transforms. Inwhat follows, we give definitions and results related to the Hankel convolution [5] to beused in the sequel.

Let γ be a positive real number. Set

σ(x)= x2γ+1

2γ+1/2Γ(γ+ 3/2),

j(x)= Cγx1/2−γJγ−1/2(x),

Cγ = 2γ−1/2Γ(γ+

12

),

(1.2)

where Jγ−1/2 denotes the Bessel function of order γ− 1/2.

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2006, Article ID 24217, Pages 1–7DOI 10.1155/IJMMS/2006/24217

brought to you by COREView metadata, citation and similar papers at core.ac.uk

provided by Crossref

https://core.ac.uk/display/194631609?utm_source=pdf&utm_medium=banner&utm_campaign=pdf-decoration-v1

http://dx.doi.org/10.1155/S0161171206242170

2 Calderon’s formula

We define Lp,σ(0,∞), 1≤ p ≤∞, as the space of those real measurable functions φ on(0,∞) for which

‖φ‖p,σ =[∫∞

0

∣∣φ(x)∣∣pdσ(x)

]1/p

<∞, 1≤ p <∞,

‖φ‖∞,σ = ess sup0<x<∞

|φ(x)| <∞.(1.3)

For each φ∈ L1,σ(0,∞), the Hankel transform of φ is defined by

�(φ)= φ(x)=∫∞

0j(xt)φ(t)dσ(t), 0≤ x <∞. (1.4)

From [5, page 314], we know that φ is bounded and continuous on [0,∞) and

‖φ‖∞,σ ≤ ‖φ‖1,σ . (1.5)

If φ(x)∈ L1,σ(0,∞) and if φ(t)∈ L1,σ(0,∞), then, by inversion, we have [5, page 316]

φ(x)=∫∞

0j(xt)φ(t)dσ(t), 0 < x <∞. (1.6)

If φ(x) and ψ(x) are in L1,σ(0,∞), then the following Parseval formula also holds [10,page 127]:

∫∞0φ(t)ψ(t)dσ(t)=

∫∞0φ(x)ψ(x)dσ(x). (1.7)

To introduce Hankel convolution #, we define

D(x, y,z)=∫∞

0j(xt) j(yt) j(zt)dσ(t)

= 23γ−5/2[Γ(γ+

12

)]2[Γ(γ)π1/2]−1

(xyz)−2γ+1[Δ(xyz)]2γ−2

,

(1.8)

where Δ(xyz) denotes the area of triangle with sides x, y, z if such a triangle exists andzero otherwise. Clearly D(x, y,z)≥ 0 and is symmetric in x, y, z. Applying inversion for-mulae (1.6) to (1.8), we get

∫∞0D(x, y,z) j(zt)dσ(z)= j(xt) j(yt) 0 < x, y <∞, 0≤ t <∞. (1.9)

Now setting t = 0, we obtain∫∞

0D(x, y,z)dσ(z)= 1. (1.10)

Let p,q,r ∈ [1,∞) and 1/r = 1/p+ 1/q− 1. The Hankel convolution of φ ∈ Lp,σ(0,∞) andψ ∈ Lq,σ(0,∞) is defined by [5, page 311]

(φ#ψ)(x)=∫∞

0

∫∞0φ(y)ψ(z)D(x, y,z)dσ(y)dσ(z). (1.11)

R. S. Pathak and G. Pandey 3

By [9, page 179], the integral is convergent for almost all x ∈ (0,∞) and

‖φ#ψ‖r,σ ≤ ‖φ‖p,σ‖ψ‖q,σ . (1.12)

Moreover, if p =∞, then (φ#ψ)(x) is defined for all x ∈ (0,∞) and is continuous. If φ,ψ ∈L1,σ(0,∞), then from [5, page 314]

(φ#ψ)(t)= φ(t)ψ(t), 0≤ t <∞. (1.13)

In this paper, Hankel dilation �a is defined by

�aφ(x)= φa(x)= a−2γ−1φ(x

a

), a > 0. (1.14)

2. Calderon’s formula

In this section, we obtain Calderon’s reproducing identity using the properties of Hankeltransform and Hankel convolutions.

Theorem 2.1. Let φ and ψ ∈ L1,σ(0,∞) be such that following admissibility condition holds:

∫∞0φ(ξ)ψ(ξ)

dσ(ξ)ξ2γ+1 = 1 (2.1)

for all ξ ∈ (0,∞). Then the following Calderon’s reproducing identity holds:

f (x)=∫∞

0

(f #φa#ψa

)(x)

dσ(a)a2γ+1 ∀ f ∈ L1(R). (2.2)

Proof. Taking Hankel transform of the right-hand side of (2.2), we get

�[∫∞

0

(f #φa#ψa

)(x)

dσ(a)a2γ+1

]

=∫∞

0f (ξ)φa(ξ)ψa(ξ)

dσ(a)a2γ+1 = f (ξ)

∫∞0φa(ξ)ψa(ξ)

dσ(a)a2γ+1

= f (ξ)∫∞

0φ(aξ)ψ(aξ)

dσ(a)a2γ+1 = f (ξ).

(2.3)

Now, by putting aξ = ω, we get

∫∞0φ(aξ)ψ(aξ)

dσ(a)a2γ+1 =

∫∞0φ(ω)ψ(ω)

dσ(ω/ξ)(ω/ξ)2γ+1

=∫∞

0φ(ω)ψ(ω)

dσ(ω)ω2γ+1

= 1.

(2.4)

Hence, the result follows.The equality (2.2) can be interpreted in the following L2- sense. �


Theorem 2.2. Suppose φ∈ L1,σ(0,∞) is real valued and satisfies

∫∞0

[φ(aξ)

]2 dσ(a)a2γ+1 = 1. (2.5)

For f ∈ L1,σ(0,∞)∩L2,σ(0,∞), suppose that

fε,δ(x)=∫ δε

(f #φa#φa

)(x)

dσ(a)a2γ+1 . (2.6)

Then ‖ f − fε,δ‖2,σ → 0 as ε→ 0 and δ→∞.

Proof. Taking Hankel transform of both sides of (2.6) and using Fubini’s theorem, we get

fε,δ(ξ)= f (ξ)∫ δε

[φ(aξ)

]2 dσ(a)a2γ+1 . (2.7)

By [5, page 311], we have

∥∥φa#φa# f∥∥

2,σ ≤∥∥φa#φa

∥∥1,σ‖ f ‖2,σ

≤ ∥∥φa∥∥21,σ‖ f ‖2,σ .

(2.8)

Now using above inequality and Minkowski’s inequality [2, page 41], we get

‖ f ‖22,σ =

∫∞0dσ(x)

∣∣∣∣∫ δε

(φa#φa# f

)(x)

dσ(a)a2γ+1

∣∣∣∣2

≤∫ δε

∫∞0

∣∣(φa#φa# f)(x)∣∣2dσ(x)

dσ(a)a2γ+1

≤∫ δε

∥∥(φa#φa# f )(x)∥∥

2,σdσ(a)a2γ+1

≤ ∥∥φa∥∥21,σ‖ f ‖2,σ

∫ δε

dt

t

= ∥∥φa‖21,σ‖ f ‖2,σ log

(δ

ε

).

(2.9)

Hence, by Parseval formula (1.7), we get

limε→0δ→∞

‖ f − fε,δ‖22,σ = lim

ε→0δ→∞

‖ f − fε,δ‖22,σ

= limε→0δ→∞

∫∞0

∣∣∣∣ f (ξ)(

1−∫ δε

[φ(aξ)

]2 dσ(a)a2γ+1

)∣∣∣∣2

dσ(x)= 0.(2.10)

Since | f (ξ)(1−∫ δε [φ(aξ)]2(dσ(a)/a2γ+1))|≤| f (ξ)|, therefore, by the dominated conver-gence theorem, the result follows.

The reproducing identity (2.2) holds in the pointwise sense under different sets of niceconditions. �


Theorem 2.3. Suppose f , f ∈ L1,σ(0,∞). Let φ∈ L1,σ(0,∞) be real valued and satisfies

∫∞0

[φ(aξ)

]2 dσ(a)a2γ+1 = 1, ξ ∈R−{0}. (2.11)

Then

limε→0δ→∞

∫ δε

(f #φa#φa

)(x)

dσ(a)a2γ+1 = f (x). (2.12)

Proof. Let

fε,δ(x)=∫ δε

(f #φa#φa

)(x)

dσ(a)a2γ+1 . (2.13)

By [5, page 311], we have

∥∥φa#φa# f∥∥

1,σ ≤∥∥φa#φa

∥∥1,σ

∥∥ f ∥∥1,σ

≤ ∥∥φa∥∥21,σ

∥∥ f ∥∥1,σ .(2.14)

Now,

∥∥ fε,δ∥∥

1,σ =∫∞

0dσ(x)

∣∣∣∣∣∫ δε

(φa#φa# f

)(x)

dσ(a)a2γ+1

∣∣∣∣∣

≤∫ δε

∫∞0

∣∣(φa#φa# f)(x)∣∣dσ(x)

dσ(a)a2γ+1

≤∫ δε

∥∥(φa#φa# f)(x)∥∥

1,σdσ(a)a2γ+1

≤ ∥∥φa∥∥21,σ‖ f ‖1,σ

∫ δε

dt

t

= ∥∥φa∥∥21,σ‖ f ‖1,σ log

(δ

ε

).

(2.15)

Therefore, fε,δ ∈ L1(0,∞). Also using Fubini’s theorem and taking Hankel transform of(2.13), we get

fε,δ(ξ)=∫∞

0j(xξ)

(∫ δε

(φa#φa# f

)(x)

dσ(a)a2γ+1

)dσ(x)

=∫ δε

∫∞0j(xξ)

(φa#φa# f

)(x)dσ(x)

dσ(a)a2γ+1

=∫ δεφa(ξ)φa(ξ) f (ξ)

dσ(a)a2γ+1

= f (ξ)∫ δε

[φ(aξ)

]2 dσ(a)a2γ+1 .

(2.16)


Therefore, by (2.11), | fε,δ(ξ)| ≤ | f (ξ)|. It follows that fε,δ ∈ L1,σ(0,∞). By inversion, wehave

f (x)− fε,δ(x)=∫∞

0j(xξ)

[f (ξ)− fε,δ(ξ)

]dσ(ξ), x ∈ (0,∞). (2.17)

Putting

hε,δ(ξ;x)= j(xξ)[f (ξ)− fε,δ(ξ)

]

= j(xξ) f (ξ)[

1−∫ δε

[φ(aξ)

]2 dσ(a)a2γ+1

],

(2.18)

we get

f (x)− fε,δ(x)=∫∞

0j(xξ)

[f (ξ)− fε,δ(ξ)

]dσ(ξ)

=∫∞

0hε,δ(ξ;x)dσ(ξ).

(2.19)

Now using (2.11) in (2.18), we get

limε→0δ→∞

hε,δ(ξ;x)= 0, ξ ∈R−{0}. (2.20)

Since |hε,δ(ξ;x)| ≤ | f (ξ)|, the Lebsegue dominated convergence theorem yields

limε→0δ→∞

[f (x)− fε,δ(x)

]= 0, ∀x. (2.21)

�

Acknowledgment

This work is supported by CSIR (New Delhi), Grant no. 9/13(04)/2003/ EMR-I.

References

[1] F. M. Cholewinski, A Hankel convolution complex inversion theory, Memoirs of the AmericanMathematical Society 58 (1965), 67.

[2] H. L. Elliott and M. Loss, Analysis, Narosa Publishing House, New Delhi, 1997.[3] M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces,

CBMS Regional Conference Series in Mathematics, vol. 79, American Mathematical Society,Rhode Island, 1991.

[4] D. T. Haimo, Integral equations associated with Hankel convolutions, Transactions of the Ameri-can Mathematical Society 116 (1965), no. 4, 330–375.

[5] I. I. Hirschman Jr., Variation diminishing Hankel transforms, Journal d’Analyse Mathematique 8(1960/1961), 307–336.

[6] I. Marrero and J. J. Betancor, Hankel convolution of generalized functions, Rendiconti di Matem-atica e delle sue Applicazioni. Serie VII 15 (1995), no. 3, 351–380.

[7] R. S. Pathak and M. M. Dixit, Continuous and discrete Bessel wavelet transforms, Journal of Com-putational and Applied Mathematics 160 (2003), no. 1-2, 241–250.


[8] R. S. Pathak and P. K. Pandey, A class of pseudo-differential operators associated with Bessel oper-ators, Journal of Mathematical Analysis and Applications 196 (1995), no. 2, 736–747.

[9] K. Trimeche, Generalized Wavelets and Hypergroups, Gordon and Breach, Amsterdam, 1997.[10] A. H. Zemanian, Generalized Integral Transformations, Pure and Applied Mathematics, vol. 18,

John Wiley & Sons, New York, 1968.

R. S. Pathak: Department of Mathematics, Banaras Hindu University, Varanasi 221005, IndiaE-mail address: [email protected]

Gireesh Pandey: Department of Mathematics, Banaras Hindu University, Varanasi 221005, IndiaE-mail address: gp2 [email protected]

mailto:[email protected]

mailto:[email protected]

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

Documents

CALDERON'S REPRODUCING FORMULA FOR HANKEL … · 2020. 1. 14. · [6] I.MarreroandJ.J.Betancor,Hankel convolution of generalized functions, Rendiconti di Matem- atica e delle sue