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The familiar Beurling theorem (an uncertainty principle), which is known for the Fourier transform pairs, has recently been proved by the author for the Kontorovich-Lebedev transform. In this paper analogs of the Beurling theorem are established for certain index transforms with respect to a parameter of the modified Bessel functions. In particular, we treat the generalized Lebedev-Skalskaya transforms, the Lebedev type transforms involving products of the Alacdoriald functions of different arguments and an index transform with the Nicholson kernel function. We also find inversion formulas for the Lebedev-Skalskaya operators of an arbitrary index and the Nicholson kernel transform.
Czasopismo
Rocznik
Tom
Strony
93--110
Opis fizyczny
Bibliogr. 16 poz.
Twórcy
autor
- University of Porto, Faculty of Sciences, Department of Pure Mathematics, Campo Alegre st., 687, 4169-007 Porto, Portugal, syakubov@fc.up.pt
Bibliografia
- [1] V.A. Ditkin, A.P. Prudnikov, Operational Calculus, Vysh, Shkola, Moscow (1975) [in Russian].
- [2] A. Erdelyi. W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. II. McGraw-Hill, New York, London and Toronto. 1953.
- [3] L. Hormander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark. Mat., 29 (1991), 237-240.
- [4] N.N. Lebedev, Sur une formule d'inversion, C.R.Acad. Sci. URSS 52 (1946), 655-658.
- [5] N.N. Lebedev, On a representation of an arbitrary function by integrals involving a square of the Macdonald function with an imaginary index. Siberian Mathem. Journ. 3 (1962) 2, 213-222 [in Russian].
- [6] N.N. Lebedev, I.P.Skalskaya, Certain integral transformations related to the Kontorovich-Lebedev transform, [in:] Problems of Math. Physics. Leningrad (1976), 68-79 [in Russian].
- [7] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series: Vol. I: Elementary Functions; Vol. 2: Special Functions, Gordon and Breach, New York, 1986.
- [8] I.N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 1972.
- [9] S.B. Yakubovich, Yu. F. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions (Kluwers Ser. Math, and Appl.: Vol. 287), Dordrecht, Boston, London, 1994.
- [10] S.B. Yakubovich, Index Transforms, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1996.
- [11] S.B. Yakubovich, On Lebedev type integral transformations associated with modified Bessel functions, Nieuw Arc. Wiskd. 17 (1999) 2, 219-227.
- [12] S.B. Yakubovich, The Lebedev-Skalskaya transform. Kluwer Encycl. of Math. Suppl. II (2000), 304-305.
- [13] S.B. Yakubovich, On the index integral transformation with Nicholson's function as the kernel, 3. Math. Anal. Appl. 269 (2002), 689-701.
- [14] S.B. Yakubovich, The Planchcrel and Hausdorff-Young type theorems for an index transformation, Journal for Analysis and its Applications 25 (2006), 193-204.
- [15] S.B. Yakubovich, Uncertainty principles for the Kontorovich-Lebedev transform, Math. Model. Anal. 13 (2008) 2, 289-302.
- [16] S.B. Yakubovich, A class of integral equations and index transformations related to the modified and incomplete Bessel functions. Preprint. CMUP. 2008, 19 pp.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHB-0002-0009