On the Convolution Theorem of the Mehler‐Fock‐Transform for a Class of Generalized Functions (II)

Abstract

SynopsisWe extend in this paper the r a d t s of part (I) concerning the convolution structure of the ~-F o c~-t r~o r m of order zero to the case of ~m-FocK-transform of order n 2 0. This transform applies to generalized functions from spaces introduced by Bwaou. With these fe8ulf9 the operational calculus of this transform is completed.The M&mmn-Foc%s-transform of natural order n applied to ordinary functions obeying certain restrictions of their behaviour when the argumenk lands towards 1 or ca is defined by 00(1.1)where the kernel is a cone function of order n defined by 1 2^ n! := -(t2 -1)nln nP1(n + 112 + i z n + 1/.2it; ?z + 1;with the Gaussian hypergeometric function 3,. The operational properties of this tramformaiiion are represented in [7] chapter 7. There given applications to the solution of bonndq d u e problems too. In [lf Boaem investigated in hi5 thesis the IVIZHLg~-Boag-lxamForm of generaked functions and its applications. As well in the classical case of ordinary functions as in the caae of generalized functions a convolution theorem seems to be desirable. (Concerning a convolution in &-spaces see the references quoted in [a].) After the investigatbn of the convolution structure connected with the JS~HLEB-Fm-transform of order zero in part (I), (41, we present in *his paper similar results for the general ~-F~-t r a m f o m of order 11 already ske-without prooh in [3]. In view of the detailed exposition in the former paper we intend to expose only

On the convolution theorem of the finite Legendre transform in a Zemanian space

“…As the next step, we shall see that last expression defines a function H(x) = (g(y),TMy)) (3.5) which belongs to A (-1,1). And we achieve this objective with the aid of the procedure employed by H. J. Glaeske and A. Hess in [5] and [6] with regard to the Mehler-Fock transform.…”

“…(3)(4)(5)(6) For it, we fix arbitrarily χ in -1 < χ < 1. Notice that the successive derivatives of the translation operator are ir,,,/ χ^ψ…”

“…A' (-1,1) symbolizes the dual space of 1,1). In view of the above considerations, (1.1) can be rewritten /{vp(i)}(n) = Φ(η) = (vp(X),P"(X)) , η = 0,1,2,..., (1.5) for any <p(x) 6 1,1). The corresponding inversion formula is given by holds for every nonnegative integer k. D(-1,1) represents the space of infinitely differentiate functions with compact supports contained in (-1,1) and its dual D'(-1,1) is the space of Schwaxtz distributions.…”

The Wavelet Transform

“…Convolution for Watson transform [77], K μ -and I μ -transforms [98], Kontorovich-Lebedev transform [24], Mehler-Fock transform [25], Fourier-Jacobi transform [89] can similarly be obtained as special case of (7.2.9).…”

Mehler-Fock transforms of generalized functions via the method of adjoints

“…Related studies on Mehler-Fock transforms of generalized functions have been carried out in [3], [4], [5], [8], amongst others.…”