Macaulay-Owen, P. Parseval’s theorem for Hankel transforms. (English) Zbl 0063.03688 Proc. Lond. Math. Soc. (2) 45, 458-474 (1939). From the text: The Hankel repeated integral \[ f(x) = \int_0^\infty J_\nu(xy)y\,dy \ \int_0^\infty J_\nu(yt)tf(t)\,dt \tag{1} \] gives rise to the reciprocity formulae \[ f(x) = \int_0^\infty \sqrt{(xy)} J_\nu(xy) F(y)\,dy, \tag{1.1} \] \[ F(x) = \int_0^\infty \sqrt{(xy)} J_\nu(xy) f(y)\,dy, \tag{1.2} \] connecting two functions \(f\) and \(F\), and each of the two functions so connected is said to be the Hankel transform of the other and we write \(f=F^*\), or \(F = f^*\). Parseval’s theorem for Hankel transforms can be stated in the following form: If \(F\) and \(f\) are two functions satisfying the relations (1.1) and (1.2), and \(G\) and \(g\) are similarly related, so that \(F=f^*\) and \(G=g^*\), then \[ J = \int_0^\infty F(x)G(x)\,dx = \int_0^\infty f(x)g(x)\,dx = j. \tag{P} \] The object of this paper is to attempt a systematic discussion of the formula (P) on the lines set out by Hardy and Titchmarsh in their treatment of the Fourier cosine case. Cited in 12 Documents MSC: 44A15 Special integral transforms (Legendre, Hilbert, etc.) 42A99 Harmonic analysis in one variable PDFBibTeX XMLCite \textit{P. Macaulay-Owen}, Proc. Lond. Math. Soc. (2) 45, 458--474 (1939; Zbl 0063.03688) Full Text: DOI