×
Tuan, Vu Kim

Integral transforms of Fourier cosine convolution type. (English) Zbl 0920.46035

J. Math. Anal. Appl. 229, No. 2, 519-529 (1999).
The author investigates integral transforms of the type \[ g(x)= \int^\infty_0 [k(x+ y)+ k(| x-y|)] f(y)dy,\quad x\in\mathbb{R}_+ \] in spaces \(L_p(\mathbb{R}_+)\), \(1\leq p\leq 2\). It is proved that this transformation is a bounded operator from \(L_p(\mathbb{R}_+)\), \(1\leq p\leq 2\) into \(L_q(\mathbb{R}_+)\), \(p^{-1}+ q^{-1}= 1\). Furthermore, a Parseval formula, Watson and Plancherel theorems are proved. Particular cases and examples are considered.
Reviewer: H.-J.Glaeske (Jena)

Cited in 22 Documents

MSC:

46F12 Integral transforms in distribution spaces
44A15 Special integral transforms (Legendre, Hilbert, etc.)
42A85 Convolution, factorization for one variable harmonic analysis

Keywords:

integral transforms; Parseval formula; Watson and Plancherel theorems
Cite Review PDF
Full Text: DOI

References:

[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables (1964), Natl. Bur. of Standards: Natl. Bur. of Standards Washington · Zbl 0171.38503
[2] Fox, C., The G- and H-functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc., 98, 395-429 (1961) · Zbl 0096.30804
[3] Glaeske, H.-J.; Tuan, V. K., Mapping properties and composition structure of convolution transforms, Serdica, 16, 143-150 (1990) · Zbl 0757.44005
[4] Glaeske, H.-J.; Tuan, V. K., Mapping properties and composition structure of a class of integral transforms, (Halle, Boundary Value and Initial Value Problems in Complex Analysis: Studies in Complex Analysis and Its Applications to Partial Differential Equations 1 (1991), Longman: Longman Harlow), 209-220 · Zbl 0793.44005
[5] Glaeske, H.-J.; Tuan, V. K., Mapping properties and composition structure of multidimensional integral transforms, Math. Nachr., 152, 179-190 (1991) · Zbl 0729.44005
[6] Hirschman, I. I.; Widder, D. V., The Convolution Transforms (1955), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0128.33503
[7] X. T. Nguyen, V. A. Kakichev, V. K. Tuan, On the generalized convolutions for Fourier cosine and sine transforms, East-West Math. J.; X. T. Nguyen, V. A. Kakichev, V. K. Tuan, On the generalized convolutions for Fourier cosine and sine transforms, East-West Math. J. · Zbl 0935.42004
[8] Sneddon, I. N., The Use of Integral Transforms (1972), McGraw-Hill: McGraw-Hill New York · Zbl 0265.73085
[9] Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Space (1971), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0232.42007
[10] Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals (1986), Chelsea: Chelsea New York · Zbl 0601.10026
[11] Tuan, V. K., Integral transformations of Fourier type in a new class of functions (Russian), Dokl. Akad. Nauk BSSR, 29, 584-587 (1985) · Zbl 0569.44001
[12] Tuan, V. K., On the theory of generalized integral transforms in a certain function space, Soviet Math. Dokl., 33, 103-106 (1986) · Zbl 0604.44002
[13] Tuan, V. K., Generalized integral transformations of convolution type in some space of functions, Complex Analysis and Applications ’85, Varna, 1985 (1986), Bulgar. Acad. Sci: Bulgar. Acad. Sci Sofia, p. 720-735 · Zbl 0618.44004
[14] V. K. Tuan, Integral Transforms and Their Composition Structure (Russian), Byelorussian State University, Minsk, 1987; V. K. Tuan, Integral Transforms and Their Composition Structure (Russian), Byelorussian State University, Minsk, 1987
[15] Tuan, V. K., New classes of integral transforms with respect to an index, Soviet Math. Dokl., 37, 317-321 (1988) · Zbl 0688.44004
[16] Tuan, V. K.; Nguyen, T. H., On a class of Watson multidimensional integral transforms, Soviet Math. Dokl., 43, 508-510 (1991) · Zbl 0778.44004
[17] Tuan, V. K.; Yakubovich, S. B., A criterion for the unitarity of a two-sided integral transformation (in Russian), Ukraı̈n. Mat. Zh., 44, 697-699 (1992) · Zbl 0765.44002
[18] Tuan, V. K.; Dinh, T. D., On a class of multidimensional Watson integral transforms, Integral Transforms Special Functions, 1, 301-312 (1993) · Zbl 0827.44002
[19] Watson, G. N., General transforms, Proc. London Math. Soc. (2), 35, 159-199 (1933) · Zbl 0007.06401
[20] Watson, G. N., A Treatise on the Theory of Bessel Functions (1944), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0063.08184
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.