A Parseval equation and a generalized finite Hankel transformation

Jorge J. Betancor; Manuel T. Flores

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 4, page 627-638
  • ISSN: 0010-2628

Abstract

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In this paper, we study the finite Hankel transformation on spaces of generalized functions by developing a new procedure. We consider two Hankel type integral transformations h μ and h μ * connected by the Parseval equation ∑ n = 0 ∞ ( h μ f ) ( n ) ( h μ * ϕ ) ( n ) = ∫ 0 1 f ( x ) ϕ ( x ) d x . A space S μ of functions and a space L μ of complex sequences are introduced. h μ * is an isomorphism from S μ onto L μ when μ ≥ - 1 2 . We propose to define the generalized finite Hankel transform h μ ' f of f ∈ S μ ' by 〈 ( h μ ' f ) , ( ( h μ * ϕ ) ( n ) ) n = 0 ∞ 〉 = 〈 f , ϕ 〉 , for ϕ ∈ S μ .

How to cite

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Betancor, Jorge J., and Flores, Manuel T.. "A Parseval equation and a generalized finite Hankel transformation." Commentationes Mathematicae Universitatis Carolinae 32.4 (1991): 627-638. <http://eudml.org/doc/247325>.

@article{Betancor1991,
abstract = {In this paper, we study the finite Hankel transformation on spaces of generalized 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 $\mu \ge -\frac\{1\}\{2\}$. We propose to define the generalized finite Hankel transform $h^\{\prime \}_\mu f$ of $f\in S^\{\prime \}_\mu $ by \[ \langle (h^\{\prime \}\_\mu f), ((h\_\mu ^\{\ast \} \varphi )(n))\_\{n=0\}^\{\infty \}\rangle =\langle f,\varphi \rangle , \quad \text\{for \} \varphi \in S\_\mu . \]},
author = {Betancor, Jorge J., Flores, Manuel T.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {finite Hankel transformation; distribution; Parseval equation; Bessel function of first kind; generalized finite Hankel transform},
language = {eng},
number = {4},
pages = {627-638},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A Parseval equation and a generalized finite Hankel transformation},
url = {http://eudml.org/doc/247325},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Betancor, Jorge J.
AU - Flores, Manuel T.
TI - A Parseval equation and a generalized finite Hankel transformation
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 4
SP - 627
EP - 638
AB - In this paper, we study the finite Hankel transformation on spaces of generalized 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 $\mu \ge -\frac{1}{2}$. We propose to define the generalized finite Hankel transform $h^{\prime }_\mu f$ of $f\in S^{\prime }_\mu $ by \[ \langle (h^{\prime }_\mu f), ((h_\mu ^{\ast } \varphi )(n))_{n=0}^{\infty }\rangle =\langle f,\varphi \rangle , \quad \text{for } \varphi \in S_\mu . \]
LA - eng
KW - finite Hankel transformation; distribution; Parseval equation; Bessel function of first kind; generalized finite Hankel transform
UR - http://eudml.org/doc/247325
ER -

References

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