scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Operational Properties of Two Integral Transforms of Fourier Type and their Convolutions

22 Oct 2009-Integral Equations and Operator Theory (Birkhäuser-Verlag)-Vol. 65, Iss: 3, pp 363-386
TL;DR: In this paper, the operational properties of two integral transforms of Fourier type were presented, and the formulation of convolutions for those transforms were derived and applied to linear partial differential equations and an integral equation with mixed Toeplitz-Hankel kernel.
Abstract: In this paper we present the operational properties of two integral transforms of Fourier type, provide the formulation of convolutions, and obtain eight new convolutions for those transforms. Moreover, we consider applications such as the construction of normed ring structures on \(L_{1}({\mathbb{R}})\), further applications to linear partial differential equations and an integral equation with a mixed Toeplitz-Hankel kernel.

Content maybe subject to copyright    Report

Citations
More filters
Journal ArticleDOI
TL;DR: In this article, the quadratic Fourier transform was examined by analyzing corresponding six subcases of the transform within a reproducing kernel Hilbert spaces framework, and the results showed that the transform can be expressed as a generalized quadrastic function for one order parameter in the ordinary Fourier transformation.
Abstract: In this paper we shall examine the quadratic Fourier transform which is introduced by the generalized quadratic function for one order parameter in the ordinary Fourier transform. This will be done by analyzing corresponding six subcases of the quadratic Fourier transform within a reproducing kernel Hilbert spaces framework.

33 citations


Cites background from "Operational Properties of Two Integ..."

  • ...It is worth saying that for any integral transform, in general, an inversion formula is very important (see [2, 3, 6, 7, 10, 12, 14, 15, 16, 17, 22, 23])....

    [...]

Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: Two novel convolutions for the fractional Fourier transforms are introduced, and natural algebraic properties of the corresponding multiplications such as commutativity, associativity and distributivity are proved, which may be useful in signal processing and other types of applications.
Abstract: In this paper we introduce two novel convolutions for the fractional Fourier transforms, and prove natural algebraic properties of the corresponding multiplications such as commutativity, associativity and distributivity, which may be useful in signal processing and other types of applications. We analyze a consequent comparison with other known convolutions, and establish necessary and sufficient conditions for the solvability of associated convolution equations of both the first and second kind in $$L^1({\mathbb {R}})$$L1(R) and $$L^2 ({\mathbb {R}})$$L2(R) spaces. An example satisfying the sufficient and necessary condition for the solvability of the equations is given at the end of the paper .

21 citations


Cites background from "Operational Properties of Two Integ..."

  • ...– Equations (16) and (17) in [23, Theorem 1] are in fact generalized convolution and product theorems (see [27,28])....

    [...]

  • ...Namely, a convolution transform, mathematically, is diagonalized by another transform; and in the new (momentum) representation a convolution turns into an operator of multiplication by a function (see [27,28])....

    [...]

Journal ArticleDOI
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

Journal ArticleDOI
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 "Operational Properties of Two Integ..."

  • ...Generalized convolution with weight-function is a nice idea based on the so-called factorizationidentity(see[ 11 ,13,16]).Thefollowingtheorempresentsthegeneralized convolutions with Hermitian weight-function....

    [...]

References
More filters
Book
01 Jan 1973

14,545 citations

Book
01 Jan 1954

4,736 citations


"Operational Properties of Two Integ..." refers background in this paper

  • ...We mention interesting properties of the transforms Fc ,F s (see [ 1 , 15, 18]):...

    [...]

BookDOI
01 Jan 1998
TL;DR: In this article, the analysis of linear partial differential operators i distribution theory and fourier rep are a good way to achieve details about operating certain products using instruction manuals, which are clearlybuilt to give step-by-step information about how you ought to go ahead in operating certain equipments.
Abstract: the analysis of linear partial differential operators i distribution theory and fourier rep are a good way to achieve details about operating certainproducts. Many products that you buy can be obtained using instruction manuals. These user guides are clearlybuilt to give step-by-step information about how you ought to go ahead in operating certain equipments. Ahandbook is really a user's guide to operating the equipments. Should you loose your best guide or even the productwould not provide an instructions, you can easily obtain one on the net. You can search for the manual of yourchoice online. Here, it is possible to work with google to browse through the available user guide and find the mainone you'll need. On the net, you'll be able to discover the manual that you might want with great ease andsimplicity

3,025 citations


"Operational Properties of Two Integ..." refers background in this paper

  • ..., (1971) Convolution Equations and Projection Methods for their Solutions, , Nauka, Moscow, (in Russian) Hochstadt, H., (1973) Integral Equations, , John Wiley & Sons, N. Y Hörmander, L., (1983) The Analysis of Linear Partial Differential Operators I, , Springer-Verlag, Berlin Kakichev, V.A., On the convolution for integral transforms (1967) Izv....

    [...]

  • ..., (1971) Convolution Equations and Projection Methods for their Solutions, , Nauka, Moscow, (in Russian) Hochstadt, H., (1973) Integral Equations, , John Wiley & Sons, N. Y Hörmander, L., (1983) The Analysis of Linear Partial Differential Operators I, , Springer-Verlag, Berlin Kakichev, V.A., On the convolution for integral transforms (1967) Izv. ANBSSR, Ser. Fiz. Mat., (2), pp. 48-57. , (in Russian) Kakichev, V.A., Thao, N.X., Tuan, V.K., On the generalized convolutions for Fourier cosine and sine transforms (1998) EastWest Jour. Math., 1 (1), pp. 85-90 Naimark, M.A., (1959) Normed Rings, , P. Noordhoff Ltd., Groningen, Netherlands Rudin, W., (1991) Functional Analysis, , McGraw-Hill, N. Y Smith, S.W., (2002) Digital Signal Processing: A Practical Guide for Engineers and Scientists, , ISBN 0-7506-7444-X (e-book) Sneddon, I., (1951) Fourier Transforms, , McGraw-Hill, New York-Toronto-London Thao, N....

    [...]

  • ..., (1971) Convolution Equations and Projection Methods for their Solutions, , Nauka, Moscow, (in Russian) Hochstadt, H., (1973) Integral Equations, , John Wiley & Sons, N....

    [...]

  • ..., (1971) Convolution Equations and Projection Methods for their Solutions, , Nauka, Moscow, (in Russian) Hochstadt, H., (1973) Integral Equations, , John Wiley & Sons, N. Y Hörmander, L., (1983) The Analysis of Linear Partial Differential Operators I, , Springer-Verlag, Berlin Kakichev, V.A., On the convolution for integral transforms (1967) Izv. ANBSSR, Ser. Fiz. Mat., (2), pp. 48-57. , (in Russian) Kakichev, V.A., Thao, N.X., Tuan, V.K., On the generalized convolutions for Fourier cosine and sine transforms (1998) EastWest Jour. Math., 1 (1), pp. 85-90 Naimark, M.A., (1959) Normed Rings, , P. Noordhoff Ltd., Groningen, Netherlands Rudin, W., (1991) Functional Analysis, , McGraw-Hill, N....

    [...]

  • ..., (1971) Convolution Equations and Projection Methods for their Solutions, , Nauka, Moscow, (in Russian) Hochstadt, H., (1973) Integral Equations, , John Wiley & Sons, N. Y Hörmander, L., (1983) The Analysis of Linear Partial Differential Operators I, , Springer-Verlag, Berlin Kakichev, V.A., On the convolution for integral transforms (1967) Izv. ANBSSR, Ser. Fiz. Mat., (2), pp. 48-57. , (in Russian) Kakichev, V.A., Thao, N.X., Tuan, V.K., On the generalized convolutions for Fourier cosine and sine transforms (1998) EastWest Jour. Math., 1 (1), pp. 85-90 Naimark, M.A., (1959) Normed Rings, , P. Noordhoff Ltd., Groningen, Netherlands Rudin, W., (1991) Functional Analysis, , McGraw-Hill, N. Y Smith, S.W., (2002) Digital Signal Processing: A Practical Guide for Engineers and Scientists, , ISBN 0-7506-7444-X (e-book) Sneddon, I....

    [...]

Book
01 Jan 1937

2,577 citations


"Operational Properties of Two Integ..." refers background in this paper

  • ...and the corresponding Hermite function φn by φn(x) = (−1)e 1 2x 2 ( d dx )n e−x 2 (see [18])....

    [...]

  • ...1, Theorem 57 in [18] and the factorization identities of the convolutions, we obtain Tk(φ1 ∗ φ0) = γ2(−φ1)φ0 = −γ2φ0φ1, Tk(φ0 ∗ φ1) = γ2φ0(iφ1) = iγ2φ0φ1....

    [...]

  • ...Moreover, Fc, Fs are isometric operators in L2[0,+∞) satisfying the identities: F 2 c = I, F 2 s = I (see [2, 18])....

    [...]

  • ...2) (see Sneddon [15], Titchmarsh [18])....

    [...]

  • ...These transforms and the Fourier integral transform have been studied for a long time, and applied to many fields of mathematics (see Hörmander [9], Rudin [13], or [18])....

    [...]