The finite Hartley new convolutions and solvability of the integral equations with Toeplitz plus Hankel kernels

Anh, Pham Ky; Tuan, Nguyen Minh; Tuan, P. D.

doi:10.1016/j.jmaa.2012.07.041

Cited by 19 publications

(17 citation statements)

References 15 publications

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“…We give the following orthogonal projection operators:

P_{\pm} = \frac{1}{2} false(I \pm T false),

then the operators P + , P − map

L^{2} false(double-struckR false)

onto

L^{2} false(R^{+} false), L^{2} false(R^{-} false)

, respectively, and

L^{2} false(double-struckR false) = L^{2} false(R^{+} false) \overset{+}{true} L^{2} false(R^{-} false)

, where

R^{+} = false{x false| x > 0 false}, R^{-} = false{x false| x < 0 false}

. We introduce the following lemmas and (see Li and Anh et al).…”

Section: Definitions and Lemmasmentioning

confidence: 99%

Generalized boundary value problems for analytic functions with convolutions and its applications

2019

Math Methods in App Sciences

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In this paper, we first establish a locality theory for the Noethericity of generalized boundary value problems on the spaces Lpfalse(double-struckRfalse)3.0235ptfalse(1≤p<∞false). By means of this theory, of the classical boundary value theory, and of the theory of Fourier analysis, we discuss the necessary and sufficient conditions of the solvability and obtain the general solutions and the Noether conditions for one class of generalized boundary value problems. All cases as regards the index of the coefficients in the equations are considered in detail. Moreover, we apply our theoretical results to the solvability of singular integral equations with variable coefficients. Thus, this paper will be of great significance for the study of improving and developing complex analysis, integral equation, and boundary value theory.

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“…We give the following orthogonal projection operators:

P_{\pm} = \frac{1}{2} false(I \pm T false),

then the operators P + , P − map

L^{2} false(double-struckR false)

onto

L^{2} false(R^{+} false), L^{2} false(R^{-} false)

, respectively, and

L^{2} false(double-struckR false) = L^{2} false(R^{+} false) \overset{+}{true} L^{2} false(R^{-} false)

, where

R^{+} = false{x false| x > 0 false}, R^{-} = false{x false| x < 0 false}

. We introduce the following lemmas and (see Li and Anh et al).…”

Section: Definitions and Lemmasmentioning

confidence: 99%

Generalized boundary value problems for analytic functions with convolutions and its applications

2019

Math Methods in App Sciences

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“…For other convolutions and integral operators, while not being exhaustive, we refer the reader to [1,2,3,8,12,13,14,15,16,17,18,22,25,26,28]. In addition, it is relevant to have in mind that the factorization property of convolutions is crucial in solving corresponding convolution type equations [6,7,11,25].…”

Section: Introductionmentioning

confidence: 99%

New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications

2018

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We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadraticphase Fourier integral operators are also studied (including a Riemann-Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity).As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.

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“…The integral equation with the Toeplitz plus Hankel kernel is of the form

λ f false(x false) + true \int_{0}^{T} K false(x, τ false) f false(τ false) d τ = g false(x false), x > 0;

here,

λ \in double-struckC, T > 0, K false(x, τ false)

is the sum of a Toeplitz and a Hankel kernel, ie,

K (x, τ) = p (x - τ) + q (x + τ),

where p , q , g are given functions and f is unknown function (see previous studies).…”

Section: Introductionmentioning

confidence: 99%

Integral equation of Toeplitz plus Hankel's type and parabolic equation related to the Kontorovich‐Lebedev‐Fourier generalized convolutions

Tuan

Hoang²,

Hong

2018

Math Methods in App Sciences

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The finite Hartley new convolutions and solvability of the integral equations with Toeplitz plus Hankel kernels

Cited by 19 publications

References 15 publications

Generalized boundary value problems for analytic functions with convolutions and its applications

Generalized boundary value problems for analytic functions with convolutions and its applications

New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications

Integral equation of Toeplitz plus Hankel's type and parabolic equation related to the Kontorovich‐Lebedev‐Fourier generalized convolutions

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