# Generalized convolutions and the integral equations of the convolution type

01 Apr 2010-Complex Variables and Elliptic Equations (Taylor & Francis Group)-Vol. 55, Iss: 4, pp 331-345

TL;DR: In this article, six new generalized convolutions of the integral transforms of Fourier type were given, and a class of integral equations of convolution type by using the constructed convolutions was investigated.

Abstract: This article gives six new generalized convolutions of the integral transforms of Fourier type, and investigates a class of integral equations of convolution type by using the constructed convolutions Namely, the explicit solutions in L 1(ℝ d ) of a class of integral equations of convolution type are obtained

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TL;DR: In this article, the necessary and sufficient conditions for the solvability of two integral equations of convolution type were presented, the first equation generalizes from integral equations with the Gaussian kernel, and the second one contains the Toeplitz plus Hankel kernels.

26 citations

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TL;DR: In this paper, a class of singular integral equations of convolution type with Hilbert kernel was studied in the space $L^{2}[-pi, \pi], where the equations can be changed into either a system of discrete equations or a discrete jump problem depending on some parameter via the discrete Laurent transform.

Abstract: One class of singular integral equations of convolution type with Hilbert kernel is studied in the space $L^{2}[-\pi, \pi]$
in the article. Such equations can be changed into either a system of discrete equations or a discrete jump problem depending on some parameter via the discrete Laurent transform. We can thus solve the equations with an explicit representation of solutions under certain conditions.

21 citations

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TL;DR: The result in this paper improves the theory of integral equations and the classical boundary value problems for analytic functions.

Abstract: In this article, we study some classes of singular integral equations of convolution type with Cauchy kernels in the class of exponentially increasing functions. Such equations are transformed into Riemann boundary value problems on either a straight line or two parallel straight lines by Fourier transformation. We propose one method different from the classical one for the study of such problems and obtain the general solutions and the conditions of solvability. Thus, the result in this paper improves the theory of integral equations and the classical boundary value problems for analytic functions.

21 citations

### Cites background from "Generalized convolutions and the in..."

...Giang-Tuan [9] studied the Noether theory of convolution type SIEs with constant coefficients....

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TL;DR: In this paper, the authors considered integral equations of convolution type with the Toeplitz plus Hankel kernels firstly posed by Tsitsiklis and Levy (1981) and obtained a necessary and sufficient condition for the solvability and unique explicit L 2 -solution.

17 citations

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TL;DR: In this article, it was shown that any Hermite function or linear combination of those functions is a weight function of four explicit generalized convolutions for the Fourier cosine and sine transforms, and sufficient and necessary conditions for the solvability and explicit solutions of integral equations of convolution type are provided by using the constructed convolutions.

Abstract: In this paper, we show that arbitrary Hermite function or appropriate linear combination of those functions is a weight-function of four explicit generalized convolutions for the Fourier cosine and sine transforms. With respect to applications, normed rings on \({L^1(\mathbb{R}^d)}\) are constructed, and sufficient and necessary conditions for the solvability and explicit solutions in \({L^1(\mathbb{R}^d)}\) of the integral equations of convolution type are provided by using the constructed convolutions.

15 citations

### Cites background from "Generalized convolutions and the in..."

...1) generalizes from equations with Gaussian kernel which has the applications in Physics, Medicine and Biology (see [6,8,9,12])....

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##### References

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TL;DR: Titchmarsh's new book comes up to the high standard of the former Introduction to the Theory of Fourier Integrals by Prof E C Titchmarm as mentioned in this paper.

Abstract: SINCE the publication of Prof Zygmund's “Trigonometric Series” in 1935, there has been considerable demand for another book dealing with trigonometric integrals Prof Titchmarsh's book meets this demand He is already well known to students of mathematics by his text-book on the theory of functions, and his new book comes up to the high standard of the former Introduction to the Theory of Fourier Integrals By Prof E C Titchmarsh Pp x + 390 (Oxford: Clarendon Press; London: Oxford University Press, 1937) 17s 6d net

746 citations

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01 Jan 1938

TL;DR: In this article, the Fourier-Bessel series has been used to solve boundary value problems in the context of Fourier Cosine and Sine Equations of physics, including the following:

Abstract: Preface 1 Fourier Series Piecewise Continuous Functions Fourier Cosine Series Examples Fourier Sine Series Examples Fourier Series Examples Adaptations to Other Intervals 2 Convergence of Fourier Series One-Sided Derivatives A Property of Fourier Coefficients Two Lemmas A Fourier Theorem A Related Fourier Theorem Examples Convergence on Other Intervals A Lemma Absolute and Uniform Convergence of Fourier Series The Gibbs Phenomenon Differentiation of Fourier Series Integration of Fourier Series 3 Partial Differential Equations of Physics Linear Boundary Value Problems One-Dimensional Heat Equation Related Equations Laplacian in Cylindrical and Spherical Coordinates Derivations Boundary Conditions Duhamel's Principle A Vibrating String Vibrations of Bars and Membranes General Solution of the Wave Equation Types of Equations and Boundary Conditions 4 The Fourier Method Linear Operators Principle of Superposition Examples Eigenvalues and Eigenfunctions A Temperature Problem A Vibrating String Problem Historical Development 5 Boundary Value Problems A Slab with Faces at Prescribed Temperatures Related Temperature Problems Temperatures in a Sphere A Slab with Internally Generated Heat Steady Temperatures in Rectangular Coordinates Steady Temperatures in Cylindrical Coordinates A String with Prescribed Initial Conditions Resonance An Elastic Bar Double Fourier Series Periodic Boundary Conditions 6 Fourier Integrals and Applications The Fourier Integral Formula Dirichlet's Integral Two Lemmas A Fourier Integral Theorem The Cosine and Sine Integrals Some Eigenvalue Problems on Undounded Intervals More on Superposition of Solutions Steady Temperatures in a Semi-Infinite Strip Temperatures in a Semi-Infinite Solid Temperatures in an Unlimited Medium 7 Orthonormal Sets Inner Products and Orthonormal Sets Examples Generalized Fourier Series Examples Best Approximation in the Mean Bessel's Inequality and Parseval's Equation Applications to Fourier Series 8 Sturm-Liouville Problems and Applications Regular Sturm-Liouville Problems Modifications Orthogonality of Eigenfunctions adn Real Eigenvalues Real-Valued Eigenfunctions Nonnegative Eigenvalues Methods of Solution Examples of Eigenfunction Expansions A Temperature Problem in Rectangular Coordinates Steady Temperatures Other Coordinates A Modification of the Method Another Modification A Vertically Hung Elastic Bar 9 Bessel Functions and Applications The Gamma Function Bessel Functions Jn(x) Solutions When v = 0,1,2,... Recurrence Relations Bessel's Integral Form Some Consequences of the Integral Forms The Zeros of Jn(x) Zeros of Related Functions Orthogonal Sets of Bessel Functions Proof of the Theorems Two Lemmas Fourier-Bessel Series Examples Temperatures in a Long Cylinder A Temperature Problem in Shrunken Fittings Internally Generated Heat Temperatures in a Long Cylindrical Wedge Vibration of a Circular Membrane 10 Legendre Polynomials and Applications Solutions of Legendre's Equation Legendre Polynomials Rodrigues' Formula Laplace's Integral Form Some Consequences of the Integral Form Orthogonality of Legendre Polynomials Normalized Legendre Polynomials Legendre Series The Eigenfunctions Pn(cos theta) Dirichlet Problems in Spherical Regions Steady Temperatures in a Hemisphere 11 Verification of Solutions and Uniqueness Abel's Test for Uniform Convergence Verification of Solution of Temperature Problem Uniqueness of Solutions of the Heat Equation Verification of Solution of Vibrating String Problem Uniqueness of Solutions of the Wave Equation Appendixes Bibliography Some Fourier Series Expansions Solutions of Some Regular Sturm-Liouville Problems Some Fourier-Bessel Series Expansions Index

537 citations