The continuous fractional Bessel wavelet transformation

Prasad, Akhilesh; Mahato, Ashutosh; Singh, V. K.; Dixit, M. M.

doi:10.1186/1687-2770-2013-40

Cited by 13 publications

(10 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Wavelets are generated by translation and dilation of a single‐function ψ called mother wavelet, which satisfies the condition()

\int_{- \infty}^{\infty} \frac{false| normalΨ false(ω false) |^{2}}{false| ω false|} d ω < \infty,

known as admissibility condition for mother wavelet, and Ψ denotes the Fourier transform of ψ . Previously, wavelet transform associated with various integral transform like Fourier transform, fractional Fourier transform, Bessel transform,() fractional Bessel transform, () and LCT has been studied by several authors in various function spaces like in L p space, W space, S space, generalized Sobolev space, etc. Motivated by the above works, we define the linear canonical Hankel wavelet (LCH‐wavelet) ψ n , m , A of any function

ψ \in L_{μ, ν, α}^{2} false(I false)

by using the LCH‐translation and dilation

D_{m}

for LCH‐transform as follows:

\begin{matrix} alignleft \\ align-1 ψ_{n, m, A} & align-2 = {scriptD}_{m} (τ_{n}^{A} ψ) (t) = {scriptD}_{m} ψ^{A} (n, t) \\ align-1 & align-2 = m^{- \frac{1}{2} - 2 ν + 2 α} e^{\frac{i β a}{2 b} (\frac{1}{m^{2 ν}} - 1) (t^{2 ν} + n^{2 ν})} ψ^{A} (\frac{n}{m}, \frac{t}{m}) \\ align-1 & align-2 = m^{- \frac{1}{2} - 2 ν + 2 α} e^{\frac{i β a}{2 b} (\frac{1}{m^{2 ν}} - 1)} \end{matrix}

…”

Section: Continuous Linear Canonical Hankel Wavelet Transformmentioning

confidence: 99%

“…known as admissibility condition for mother wavelet, and Ψ denotes the Fourier transform of . Previously, wavelet transform associated with various integral transform like Fourier transform, 21 fractional Fourier transform, 22 Bessel transform, [23][24][25] fractional Bessel transform, 8,26,27 and LCT 28 has been studied by several authors in various function spaces like in L p space, W space, S space, generalized Sobolev space, etc. Motivated by the above works, we define the linear canonical Hankel wavelet (LCH-wavelet) n,m,A of any function ∈ L 2 , , (I) by using the LCH-translation (2.3) and dilation  m for LCH-transform as follows:…”

Section: Continuous Linear Canonical Hankel Wavelet Transformmentioning

confidence: 99%

See 1 more Smart Citation

Wavelet transform associated with linear canonical Hankel transform

Kumar

Mandal

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

The main goal of this paper is to study about the continuous as well as discrete wavelet transform in terms of linear canonical Hankel transform (LCH‐transform) and discuss some of its basic properties. Parseval's relation and reconstruction formula of continuous linear canonical Hankel wavelet transform (CLCH‐wavelet transform) is obtained. Moreover, semidiscrete and discrete LCH‐wavelet transform are also discussed.

show abstract

“…Wavelets are generated by translation and dilation of a single‐function ψ called mother wavelet, which satisfies the condition()

\int_{- \infty}^{\infty} \frac{false| normalΨ false(ω false) |^{2}}{false| ω false|} d ω < \infty,

ψ \in L_{μ, ν, α}^{2} false(I false)

by using the LCH‐translation and dilation

D_{m}

for LCH‐transform as follows:

\begin{matrix} alignleft \\ align-1 ψ_{n, m, A} & align-2 = {scriptD}_{m} (τ_{n}^{A} ψ) (t) = {scriptD}_{m} ψ^{A} (n, t) \\ align-1 & align-2 = m^{- \frac{1}{2} - 2 ν + 2 α} e^{\frac{i β a}{2 b} (\frac{1}{m^{2 ν}} - 1) (t^{2 ν} + n^{2 ν})} ψ^{A} (\frac{n}{m}, \frac{t}{m}) \\ align-1 & align-2 = m^{- \frac{1}{2} - 2 ν + 2 α} e^{\frac{i β a}{2 b} (\frac{1}{m^{2 ν}} - 1)} \end{matrix}

…”

Section: Continuous Linear Canonical Hankel Wavelet Transformmentioning

confidence: 99%

Section: Continuous Linear Canonical Hankel Wavelet Transformmentioning

confidence: 99%

Wavelet transform associated with linear canonical Hankel transform

Kumar

Mandal

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…(6) Wavelets are considered to be the set of elements constructed from translation and dilation of a single function ∈ L 2 (R). 1,2,4 In the similar way, Pathak and Dixit 16 and Upadhyay et al 17 introduced the Bessel wavelet and the fractional Bessel wavelet by other works 11,14,16,18 as b,a, , which is defined below: × (b∕a, t∕a), b ≥ 0, a > 0, (7) where  a represents the dilation of a function. As per other studies, 1,2,5,11,14,16,19 the fractional wavelet transform W of ∈ L 2 , (I) associated with the wavelet ∈ L 2 , (I) defined by means of the integral transform:…”

Section: Introductionmentioning

confidence: 99%

“…The fractional Hankel translation

τ_{t}^{θ}

of ψ is given by the following:

\begin{matrix} right (τ_{t}^{θ} ψ) (ω) & left = ψ^{θ} (t, ω) & right \\ right & left = C_{ν, μ, θ} \int_{0}^{\infty} ψ (z) D_{ν, μ}^{θ} (t, ω, z) e^{\frac{i}{2} z^{2} \cot θ} z^{μ + ν + 1} d z . \end{matrix}

Wavelets are considered to be the set of elements constructed from translation and dilation of a single function

ψ \in L^{2} false(double-struckR false)

. In the similar way, Pathak and Dixit and Upadhyay et al introduced the Bessel wavelet and the fractional Bessel wavelet by other works as ψ b , a , θ , which is defined below:

\begin{matrix} right ψ_{b, a, θ} (t) & left = {scriptD}_{a} (τ_{b}^{θ} ψ) (t) = {scriptD}_{a} ψ^{θ} (b, t) = a^{- 2 μ - 2} e^{\frac{i}{2} (\frac{1}{a^{2}} - 1) t^{2} \cot θ} e^{\frac{i}{2} (\frac{1}{a^{2}} + 1) b^{2} \cot θ} & right \\ right & left \times ψ^{θ} (b / a, t / a), b \geq 0, a > 0, \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

Continuity of the fractional Hankel wavelet transform on the spaces of type S

Mahato

Singh

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

In this present article, we study the fractional Hankel transform and its inverse on certain Gel'fand‐Shilov spaces of type S. The continuous fractional wavelet transform is defined involving the fractional Hankel transform. The continuity of fractional Hankel wavelet transform is discussed on Gel'fand‐Shilov spaces of type S. This article goes further to discuss the continuity property of fractional Hankel transform and fractional Hankel wavelet transform on the ultradifferentiable function spaces.

show abstract

“…We have checked it to find the solution of differential equations with variable coefficients [15], and have found it has a strong point managing differential equations with variable coefficients compared with existing other integral transforms. On the other hand, efforts to find solutions of differential equations with variable coefficients using integral transforms have been pursued [4,[14][15][17][18][19]. In this article, we have checked several representations of Euler-Cauchy equation with an initial condition and the validity of the solution of it using the differentiation of transforms.…”

Section: Introductionmentioning

confidence: 99%

Several Representations of the Euler-Cauchy Equation With Respect to Integral Transforms

Cho¹,

Kim²

2013

Int. J. of Pure and Appl. Math.

View full text Add to dashboard Cite

Abstract:The subjects on the solution of differential equations with variable coefficients have aroused the interest of many researchers. Existing many integral transforms are playing a role to get the solution of it. In this sense, we have researched about several forms of Euler-Cauchy equation, using Laplace transform. Related to the topic, the proposed idea can be also applied to other transforms (Sumudu/Elzaki, etc.).

show abstract

The continuous fractional Bessel wavelet transformation

Cited by 13 publications

References 11 publications

Wavelet transform associated with linear canonical Hankel transform

Wavelet transform associated with linear canonical Hankel transform

Continuity of the fractional Hankel wavelet transform on the spaces of type S

Several Representations of the Euler-Cauchy Equation With Respect to Integral Transforms

Contact Info

Product

Resources

About