On a generalized Hankel type convolution of generalized functions
The authors consider the Hankel-type transformation given by \[ F(\tau)=(h_{\mu,\nu} f)(\tau)=\nu\tau^{-1}\int_0^{\infty}(x\tau)^{\nu}J_{\mu}[(x\tau)^{\nu}]f(x)dx \qquad (\tau>0). \] For \(\nu>0\) and \(\mu>-1/2\), they define a space \(H_{\mu,\nu}\) of test functions, analogous to that introduced by \textit{A. H. Zemanian} [SIAM J. Appl. Math. 14, 561-576 (1966; Zbl 0154.13803)], where \(h_{\mu,\nu}\) is an automorphism. Motivated by the Parseval relation \[ \int_0^{\infty}xf(x)g(x)dx=\int_0^{\infty}\tau F(\tau)G(\tau)d\tau, \] they extend \(h_{\mu,\nu}\) by transposition to the dual \(H_{\mu,\nu}^{* '}\) of \(H_{\mu,\nu}^{*}\), the space \(H_{\mu,\nu}\) skewed by the factor \(x\). The classical Hankel convolution introduced by \textit{I. I. Hirschman} [J. Anal. Math. 8, 307-336 (1961; Zbl 0099.31301)], \textit{D. T. Haimo} [Trans. Am. Math. Soc. 116, 330-375 (1965; Zbl 0135.33502)] and \textit{F. M. Cholewinski} [Mem. Am. Math. Soc. 58 (1965; Zbl 0137.30901)] is then extended to a generalized convolution, defined by \[ \langle x^{\nu-1}f*\sigma(x),x\phi(x) \rangle = \langle f(x),x^{\nu}\sigma*\phi(x)\rangle \] whenever \(f\in H_{\mu,\nu}^{* '}\), \(\phi\in H_{\mu,\nu}\) and \(\sigma\) is a distribution of compact support in \((0,\infty)\). This convolution is shown to satisfy the exchange formula with respect to the generalized \(h_{\mu,\nu}\)-transformation. Algebraic and differentiability properties of this generalized convolution are established. The existence of an identity element is discussed. The paper under review is a straightforward extension of one by \textit{J. de Sousa Pinto} [SIAM J. Math. Anal. 16, 1335-1346 (1985; Zbl 0592.46038)] where the same results are obtained for \(\nu=1\) and \(\mu=0\). There is an extensive literature by J.J. Betancor and the reviewer concerning the definition and study of a generalized Hankel convolution in distribution spaces wider than that considered by de Sousa Pinto.
scientific article; zbMATH DE number 1740621