Article

 Title: A Parseval equation and a generalized finite Hankel transformation (English) Author: Betancor, Jorge J. Author: Flores, Manuel T. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 32 Issue: 4 Year: 1991 Pages: 627-638 . Category: math . Summary: In this paper, we study the finite Hankel transformation on spaces of ge\-ne\-ra\-lized functions by developing a new procedure. We consider two Hankel type integral transformations $h_\mu$ and $h_\mu ^{\ast }$ connected by the Parseval equation $$\sum_{n=0}^{\infty }(h_\mu f)(n)(h_\mu ^{\ast } \varphi )(n)= \int_{0}^{1}f(x)\varphi (x)\, dx.$$ A space $S_\mu$ of functions and a space $L_\mu$ of complex sequences are introduced. $h_\mu ^{\ast }$ is an isomorphism from $S_\mu$ onto $L_\mu$ when $\mu \geq -\frac{1}{2}$. We propose to define the generalized finite Hankel transform $h'_\mu f$ of $f\in S'_\mu$ by $$\langle (h'_\mu f), ((h_\mu ^{\ast } \varphi )(n))_{n=0}^{\infty }\rangle =\langle f,\varphi \rangle, \quad \text{for } \varphi \in S_\mu .$$ (English) Keyword: finite Hankel transformation Keyword: distribution Keyword: Parseval equation MSC: 44A15 MSC: 46F12 idZBL: Zbl 0763.46028 idMR: MR1159809 . Date available: 2009-01-08T17:47:42Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118442 . Reference: [1] Betancor J.J.: The Hankel-Schwartz transform for functions of compact support.Rend. Mat. Appl. 7 (3-4) (1987), 399-409. MR 0986009 Reference: [2] Betancor J.J.: A mixed Parseval's equation and a generalized Hankel transformation of distributions.Can. J. Math. XLI (2) (1989), 274-284. Zbl 0666.46046, MR 1001612 Reference: [3] Churchill R.V.: Fourier Series and Boundary Value Problems.McGraw Hill, New York, 1963. Zbl 0378.42001, MR 0149173 Reference: [4] Cinelli G.: An extension of the finite Hankel transform and applications.Int. J. Engng. 3 (1965), 539-559. Zbl 0151.17102, MR 0194853 Reference: [5] Dube L.S.: On finite Hankel transformation of generalized functions.Pacific J. Math. 62 (1976), 365-378. Zbl 0329.46044, MR 0410365 Reference: [6] Gelfand I.M., Shilov G.E.: Generalized functions.Vol. III, Academic Press, New York, 1967. MR 0217416 Reference: [7] Liu S.H.: Method of generalized finite Hankel transform.Z. Angew. Math. Mech. 51 (1971), 311-313. MR 0284772 Reference: [8] Méndez J.M.: A mixed Parseval equation and the generalized Hankel transformation.Proc. Amer. Math. Soc. 102 (1988), 619-624. MR 0928991 Reference: [9] Méndez J.M.: The finite Hankel-Schwartz transform.J. Korean Math. Soc. 26 (1) (1989), 647-655. MR 1005866 Reference: [10] Méndez J.M., Negrín J.R.: Fourier Bessel series expansions of generalized functions and finite Hankel transforms of distributions.Rev. Roum. de Math. Pures et Appl. XXXIV (7) (1989), 647-655. MR 1023593 Reference: [11] Pandey J.N., Pathak R.S.: Eigenfunction expansion of generalized functions.Nagoya Math. J. 72 (1978), 1-25. Zbl 0362.34018, MR 0514887 Reference: [12] Pathak R.S.: Orthogonal series representations for generalized functions.J. Math. Anal. Appl. 130 (1988), 316-333. Zbl 0647.46037, MR 0929938 Reference: [13] Pathak R.S., Singh O.P.: Finite Hankel transforms of distributions.Pacific J. Math. 99 (1982), 439-458. Zbl 0484.46039, MR 0658074 Reference: [14] Sneddon I.N.: On finite Hankel transforms.Phil. Mag. (7) 17 (1946), 16-25. MR 0018263 Reference: [15] Sneddon I.N.: The Use of Integral Transforms.Tata McGraw Hill, New Delhi, 1979. Zbl 0237.44001 Reference: [16] Titchmarsh E.C.: A class of expansions in series of Bessel functions.Proc. London Math. Soc. (2) 22 (1924), xiii-xvi. Reference: [17] Watson G.N.: Theory of Bessel Functions.2nd ed., Cambridge University Press, Cambridge, 1958. Zbl 0849.33001 Reference: [18] Zemanian A.H.: Orthonormal series expansions of certain distributions and distributional transform calculus.J. Math. Anal. Appl. 14 (1966), 263-275. Zbl 0138.37804, MR 0211259 Reference: [19] Zemanian A.H.: Generalized Integral Transformations.Interscience Publishers, New York, 1968. Zbl 0643.46029, MR 0423007 .

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