A Modified and a Finite Index Weber Transforms

Abstract

This paper introduces, by way of constructing, specific finite and infinite integral transforms with Bessel functions $J_{\nu }$ and $Y_{\nu }\ddot{a}$ in their kernels. The infinite transform and its reciprocal look deceptively similar to the known Weber transform and its reciprocal, respectively, but fundamentally differ from them. The new transform enjoys an operational property that makes it useful for applications to some problems in differential equations with non-constant coefficients. The paper gives a characterization of the image of some spaces of square integrable functions with respect to some measure under the infinite and finite transforms.