# zbMATH — the first resource for mathematics

Convolutions for the Fourier transforms with geometric variables and applications. (English) Zbl 1226.43003
The authors consider generalised convolutions in the sense of Kakichev associated with the Euclidean Fourier transform and some of its variants: the Fourier transform composed with a shift, with dilations, with inversion, and the Fourier-cosine and Fourier-sine transforms. The authors show that $$L^1(\mathbb R^n)$$ equipped with these generalised convolutions is a normed ring and determine when it is commutative. As application, necessary and sufficient conditions for solving integral equations of convolution types, as well as the solutions, are given.

##### MSC:
 43A32 Other transforms and operators of Fourier type 44A35 Convolution as an integral transform 44A15 Special integral transforms (Legendre, Hilbert, etc.)
Full Text:
##### References:
 [1] Al-Musallam, A class of convolution transforms, Frac. Calc. Appl. Anal. 3 (3) pp 303– (2000) [2] G. Arfken Mathematical Methods for Physicists, 3rd edition (Academic Press, Orlando, 1985). [3] H. Bateman A. Erdely Tables of Integral Transforms (MC Gray-Hill, New York-Toronto-London, 1954). [4] S. Bochner K. Chandrasekharan Fourier Transforms (Princeton University Press, Princeton, 1949). [5] Britvina, Generalized convolutions for the Hankel transform and related integral operators, Math. Nachr. 280 (9-10) pp 962– (2007) · Zbl 1126.44005 [6] Britvina, A class of integral transforms related to the Fourier cosine convolution, Integral Transforms Spec. Funct. 16 (5-6) pp 379– (2005) · Zbl 1085.42003 [7] J. W. Brown R. V. Churchill Fourier Series and Boundary Value Problems, seven-th edition (McGraw-Hill Science/Engineering/Math, 2006). [8] Cho, A spherical dose model for radio surgery plan optimization, Biol. Med. Phys. Biomed. Eng. 43 pp 3145– (1998) [9] R. V. Churchill Fourier Series and Boundary Value Problems (McGraw-Hill, New York, 1941). [10] Garcia-Vicente, Experimental determination of the convolution kernel for the study of the spatial response of a detector, Med. Phys. 25 pp 202– (1998) [11] Garcia-Vicente, Exact analytical solution of the convolution integral equation for a general profile fitting function and Gaussian detector kernel, Biol. Med. Phys. Biomed. Eng. 45 (3) pp 645– (2000) [12] Giang, Convolutions of the Fourier-cosine and Fourier-sine integral transforms and integral equations of the convolution type, Herald of Polotsk State Uni. L (9) pp 7– (2008) [13] Giang, Generalized convolutions for the Fourier integral transforms and applications, Journal of Siberian Federal Univ. Math. & Phys. 1 (4) pp 371– (2008) [14] Giang, Generalized onvolutions for the integral transforms of Fourier type and applications, Frac. Calc. Appl. Anal. 12 (3) pp 253– (2009) [15] I. S. Gohberg I. A. Feldman Convolution Equations and Projection Methods for their Solutions (Nauka, Moscow, 1971) (in Russian). [16] H. Hochstadt Integral Equations (John Wiley & Sons, Inc., 1973). [17] Hömander, L2- estimates for Fourier integral operators with complex phase, Ark. Mat. 21 (1) pp 283– (1983) [18] L. Hömander The Analysis of Linear Partial Differential Operators I (Springer-Verlag, Berlin - Heidelberg - Berlin - Tokyo, 1983). [19] Kakichev, On the convolution for integral transforms, Izv. ANBSSR, Ser. Fiz. Mat. 2 pp 48– (1967) [20] Kakichev, On the matrix convolutions for power series, Izv. Vuzov. Mat. 2 pp 53– (1990) · Zbl 0708.47022 [21] V. A. Kakichev Polyconvolution, Taganskij Radio-tekhnicheskij Universitet (1997). ISBN: 5-230-24745-2 (in Russian). [22] Kakichev, On the generalized convolutions for Fourier cosine and sine transforms, East-West J. Math. 1 (1) pp 85– (1998) · Zbl 0935.42004 [23] M. A. Naimark Normed Rings (P. Noordhoff N. V., Groningen, Netherlands, 1959). [24] W. Rudin Functional Analysis (McGraw-Hill, New York-Hamburg-London-Paris-Tokyo, 1991). [25] Thao, On the generalized convolution with a weight function for the Fourier sine and cosine transforms, Integral Transforms Spec. Funct. 17 (9) pp 673– (2006) · Zbl 1102.44010 [26] N. X. Thao N. T. Hai Convolution for Integral Transforms and their Applications (Computer Center of the RAS, Moscow, 1997). [27] Thao, Generalized convolution transforms and Toeplitz plus Hankel integral equation, Frac. Calc. App. Anal. 11 (2) pp 153– (2008) · Zbl 1154.44002 [28] Thao, Integral transforms of Fourier cosine and sine generalized convolution type, Int. J. Math. Sci. 17 (2007) · Zbl 1201.42010 [29] Thierry, The Wiener lemma and certain of its generalizations, Bull. Amer. Math. Soc. (N. S.), New Ser. 24 (1) pp 1– (1991) [30] E. C. Titchmarsh Introduction to the Theory of Fourier Integrals (Chelsea Publising Company, New York, N. Y., 1986). [31] Tuan, Generalized convolutions relative to the Hartley transforms with applications, Sci. Math. Jpn. 70 (1) pp 77– (2009) · Zbl 1182.44005 [32] Tuan, Integral transform of Fourier type in a new class of functions, Dokl. Akad. Nauk BSSR 29 (7) pp 584– (1985) [33] Vilenkin, Matrix elements of the indecomsable unitary representations for motion group of the Lobachevskii’s space and generalized Mehler-Fox transforms, Dokl. Akad. Nauk. UzSSR 118 (2) pp 219– [34] V. S. Vladimirov Generalized Functions in Mathematical Physics (Mir Pub., Moscow, 1979).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.