Geodesic
Generalized Convolutions for the Fourier Integral Transforms and Applications
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 1 (2008) no. 4, pp. 371-379.

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This paper provides some generalized convolutions for the Fourier integral transforms and treats the applications. Namely, there are six generalized convolutions with weight-function for the Fourier integral transforms. As for applications, the normed ring structures on $L^1(\mathbb R^d)$ are constructed, and the explicit solution in $L^1(\mathbb R^d)$ of the integral equations with the mixed Toeplitz–Hankel kernel are obtained.
Keywords: generalized convolution, normed ring, integral equation of convolution type. 
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Bui Thi Giang; Nguyen Minh Tuan. Generalized Convolutions for the Fourier Integral Transforms and Applications. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 1 (2008) no. 4, pp. 371-379. http://geodesic.mathdoc.fr/item/JSFU_2008_1_4_a1/

[1] J. W. Brown, R. V. Churchill, Fourier Series and Boundary Value Problems, 7 edition, McGraw-Hill Science/Engineering/Math., Springer, Berlin–Heidelberg–New York–London, 2006, Hardcover, 384 p. | MR

[2] N. Ya. Vilenkin, “Matrix Elements of the Indecomposable Unitary Representations for Motion Group of the Lobachevsky's Space and Generalized Mehler–Fox”, Dokl. Akad. Nauk. USSR, 118:2 (1958), 219–222 (Russian) | MR | Zbl

[3] V. A. Kakichev, “On the Convolution for Integral Transforms”, Izv. AN BSSR Ser. Fiz. Mat., 1967, no. 2, 48–57 (Russian) | Zbl

[4] V. A. Kakichev, “On the Matrix Convolutions for Power Series”, Izv. Vyssh. Uchebn. Zaved. Ser. Mat., 1990, no. 2, 53–62 (Russian) | MR | Zbl

[5] V. A. Kakichev, N. X. Thao, V. K. Tuan, “On the Generalized Convolutions for Fourier Cosine and Sine Transforms”, East-West J. Math., 1:1 (1998), 85–90 | MR | Zbl

[6] F. Al-Musallam, V. K. Tuan, “A Class of Convolution Transforms”, Frac. Calc. Appl. Anal., 3:3 (2000), 303–314 | MR | Zbl

[7] L. E. Britvina, “Generalized Convolutions for the Hankel Transform and Related Integral Operators”, Math. Nachr., 280:9–10 (2007), 962–970 | DOI | MR | Zbl

[8] L. E. Britvina, “A Class of Integral Transforms Related to the Fourier Cosine Convolution”, Integral Transforms Spec. Funct., 16:5–6 (2005), 379–389 | DOI | MR | Zbl

[9] V. A. Kakichev, Polyconvolution, Taganrogskij Radio-Tekhniticheskij Univ., Taganrog, 1997

[10] N. X. Thao, N. M. Khoa, “On the Generalized Convolution with a Weight Function for the Fourier Sine and Cosine Transforms”, Integral Transforms Spec. Funct., 17:9 (2006), 673–685 | DOI | MR | Zbl

[11] N. X. Thao, V. K. Tuan, N. T. Hong, “Generalized Convolution Transforms and Toeplitz Plus Hankel Integral Equation”, Frac. Calc. App. Anal., 11:2 (2008), 153–174 | MR | Zbl

[12] V. K. Tuan, “Integral Transform of Fourier Type in a New Class of Functions”, Dokl. Akad. Nauk BSSR, 29:7 (1985), 584–587 (Russian) | MR | Zbl

[13] M. A. Naimark, Normed Rings, P. Noordhoff N. V., Groningen, Netherlands, 1959 | MR | Zbl

[14] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1991 | MR | Zbl