Two versions of pseudo-differential operators involving the Kontorovich–Lebedev transform in L 2 (ℝ + ; dx / x )

Math Methods in App Sciences

Mandal

2017

Self Cite

In this paper, we obtained some useful estimates for convolution corresponding to Kontorovich-Lebedev transform (KL-transform) in Lebesgue space. Some continuity theorems for translation, convolution, and KL-transform in test function space H(R + ) are discussed. Then an integral representation of pseudodifferential operator involving KL-transform is found out, and its estimates in Lebesgue space is obtained. At the end, some applications of KL-transform and its convolution are discussed.

“…Various estimates for convolution associated with KL‐transform were already discussed in previous studies . Prasad and Mandal estimated convolution of 2 functions

φ \in L^{p} false(R_{+}; x^{- 1} d x false)

and

ψ \in L^{q} false(R_{+}; x^{- 1} d x false)

L^{1} false(R_{+}; d x false)

, where

1 < p, q < \infty

. Here, we obtained an inequality of convolution for

φ, ψ \in L^{1} false(R_{+}; x^{- 1} d x false)

L^{1} false(R_{+}; d x false)

.…”

Section: Estimates Of Convolution and Some Resultsmentioning

confidence: 64%

R_{+} = false(0, \infty false)

, as given in previous studies():

false(frakturK φ false) false(τ false) = \int_{0}^{\infty} K_{i τ} false(x false) φ false(x false) x^{- 1} d x, τ \in R_{+},

where K i τ ( x ) is Macdonald function given as, p. 82 (21):

K_{i τ} false(x false) = \int_{0}^{\infty} e^{- x \cosh t} \cos false(τ t false) d t, x > 0, τ > 0 .

Clearly, from

false| K_{i τ} false(x false) false| \leq \int_{0}^{\infty} e^{- x \cosh t} d t = K_{0} false(x false),

Now from Erde'lyi et al,, p. 136 the asymptotic expansion of Macdonald function K b ( x ) for nonnegative real numbers b is given as

\begin{matrix} right K_{0} (x) & left \approx \log (\frac{2}{x}), x \to 0, & right \\ ... \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, from and ,

A_{x}^{'} false(x^{- 1} φ false(x false) false) = x^{- 1} A_{x} φ false(x false)

holds . It can be extended to

k \in N_{0}

times, and thus, we can write

{false(A_{x}^{'} false)}^{k} false(x^{- 1} φ false(x false) false) = x^{- 1} A_{x}^{k} φ false(x false) .

By using the operational formula for k =1 and A x K i τ ( x )=− τ 2 K i τ ( x ), we can write

false(frakturK A_{false(\cdot false)} φ false) false(τ false) = - τ^{2} false(frakturK φ false) false(τ false) .

The inversion formula of is defined as ():

φ false(x false) = K^{- 1} false(frakturK φ false) false(x false) = \frac{2}{π^{2}} \int_{0}^{\infty} K_{i τ} false(x false) false(frakturK φ false) false(τ false) τ \sinh false(π τ false) d τ, x \in R_{+} .

From references herein,() we have

\frac{2}{π^{2}} \int_{0}^{\infty} K_{i τ} false(x false) K_{i τ} false(y false) K_{i τ} false(z false) τ \sinh false(π τ false) d τ = T false(x, y, z false),

where ...…”

Section: Introductionmentioning

confidence: 99%

“…The translation operator

T_{x} φ false(y false)

is defined as ():

T_{x} φ false(y false) = T_{y} φ false(x false) = \int_{0}^{\infty} T false(x, y, z false) φ false(z false) z^{- 1} d z,

and the convolution operator is given by

false(φ ♯ ψ false) false(x false) = \int_{0}^{\infty} T_{x} φ false(y false) ψ false(y false) y^{- 1} d y

= \int_{0}^{\infty} \int_{0}^{\infty} T false(x, y, z false) φ false(z false) ψ false(y false) z^{- 1} y^{- 1} d z d y .

From Prasad and Mandal,, p. 304 we have

false(frakturK false(φ ♯ ψ false) false) false(τ false) = false(frakturK φ false) false(τ false) false(frakturK ψ false) false(τ false) .

…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Kontorovich‐Lebedev transform and its associated pseudodifferential operator

Math Methods in App Sciences

Mandal

2017

Self Cite

“…The representation of KL-transform and various relations related to it like translation, convolution, Plancherel's and Parseval's relation etc. have been expressed in many ways [1,5,22,23,27,[35][36][37][38][39]. Now we consider the class of all measurable functions L p (R + ; x −1 dx), of f on R + with norm given as:…”

Section: Introductionmentioning

confidence: 99%

Composition of wavelet transforms and wave packet transform involving Kontorovich-Lebedev transform

Mandal

Kumar

2021

Filomat

Self Cite

The main objective of this paper is to study the composition of continuous Kontorovich-Lebedev wavelet transform (KL-wavelet transform) and wave packet transform (WPT) based on the Kontorovich-Lebedev transform (KL-transform). Estimates for KL-wavelet and KL-wavelet transform are obtained, and Plancherel?s relation for composition of KL-wavelet transform and WPT-transform are derived. Reconstruction formula for WPT associated to KL-transform is also deduced and at the end Calderon?s formula related to KL-transform using its convolution property is obtained.

Pseudo‐differential operator in quaternion space

Kundu

Math Methods in App Sciences

2023

Self Cite

This paper introduces the quaternion Schwarz type space, and quaternion linear canonical transform (QLCT) mapping properties are also discussed. Further, the quaternion pseudo-differential operator (QPDO) associated with QLCT is described. Some of its characteristics, including estimates, boundedness, and integral representation in quaternion Sobolev type space, are derived. Some applications of QLCT, quaternion differential equations, are also discussed.