scholarly journals On the Lebedev transformation in Hardy's spaces

2004 ◽  
Vol 2004 (66) ◽  
pp. 3603-3616
Author(s):  
Semyon B. Yakubovich
Keyword(s):  
Hardy Space ◽  
Arbitrary Function ◽  
Bessel Functions ◽  
Type Theorem ◽  
Fourier Integral ◽  
Modified Bessel Functions ◽  
Weighted Hardy Space ◽  
Integral Type ◽  
Hardy’S Space ◽  
And Inversion

We establish the inverse Lebedev expansion with respect to parameters and arguments of the modified Bessel functions for an arbitrary function from Hardy's spaceH2,A,A>0. This gives another version of the Fourier-integral-type theorem for the Lebedev transform. The result is generalized for a weighted Hardy spaceH⌢2,A≡H⌢2((−A,A);|Γ(1+Rez+iτ)|2dτ),0<A<1, of analytic functionsf(z),z=Rez+iτ, in the strip|Rez|≤A. Boundedness and inversion properties of the Lebedev transformation from this space into the spaceL2(ℝ+;x−1dx)are considered. WhenRez=0, we derive the familiar Plancherel theorem for the Kontorovich-Lebedev transform.

HARMONIC ANALYSIS OF THE LEBEDEV-STIELTJES INTEGRALS

2009 ◽  
Vol 02 (02) ◽  
pp. 307-320
Author(s):  
SEMYON B. YAKUBOVICH
Keyword(s):  
Hilbert Space ◽  
Harmonic Analysis ◽  
Bessel Functions ◽  
Orthonormal System ◽  
Modified Bessel Functions ◽  
Parseval Equality ◽  
Bochner Technique ◽  
Bessel Inequality ◽  
Fourier Type ◽  
And Inversion

We expand the Bochner technique on the following Lebedev-Stieltjes integrals [Formula: see text] which are related to the Kontorovich-Lebedev transformation. Mapping and inversion properties are investigated. The Fourier type series with respect to an uncountable orthonormal system of the modified Bessel functions are considered in the Bohr type pre-Hilbert space. The Bessel inequality and Parseval equality are proved.

Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 284-292 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Lommel Functions ◽  
Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘basic converging factors’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

(p, q)-Extended Bessel and Modified Bessel Functions of the First Kind

2017 ◽  
Vol 72 (1-2) ◽  
pp. 617-632 ◽  
Author(s):  
Dragana Jankov Maširević ◽  
Rakesh K. Parmar ◽  
Tibor K. Pogány
Keyword(s):  
Bessel Functions ◽  
Modified Bessel Functions

Approximated Fundamental Frequency for Thin Circular Plates Clamped or Pinned at the Edge

2007 ◽  
Author(s):  
George Weiss
Keyword(s):  
Circular Plate ◽  
Fundamental Frequency ◽  
Bessel Functions ◽  
Unit Area ◽  
Continuous System ◽  
Modified Bessel Functions ◽  
Circular Plates ◽  
Numerical Parameter ◽  
Elliptical Plate ◽  
Distributed Load

Calculating the exact solution to the differential equations that describe the motion of a circular plate clamped or pinned at the edge, is laborious. The calculations include the Bessel functions and modified Bessel functions. In this paper, we present a brief method for calculating with approximation, the fundamental frequency of a circular plate clamped or pinned at the edge. We’ll use the Dunkerley’s estimate to determine the fundamental frequency of the plates. A plate is a continuous system and will assume it is loaded with a uniform distributed load, including the weight of the plate itself. Considering the mass per unit area of the plate, and substituting it in Dunkerley’s equation rearranged, we obtain a numerical parameter K02, related to the fundamental frequency of the plate, which has to be evaluated for each particular case. In this paper, have been evaluated the values of K02 for thin circular plates clamped or pinned at edge. An elliptical plate clamped at edge is also presented for several ratios of the semi–axes, one of which is identical with a circular plate.

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