A generalized convolution for finite Fourier transformations

A Generalized Convolution for Finite Fourier Transformations

Presented to the Society, April 16, 1948; received by the editors June 25, 1948. 1 The author wishes to thank Professor R. V. Churchill for his advice in the preparation of this paper. The content of this paper is part of a dissertation submitted in partial fulfillment of the requirements for the degree of doctor of philosophy in the University of Michigan. 2 The numbers in brackets refer to th...

A generalized Fourier transform and convolution on time scales

In this paper, we develop some important Fourier analysis tools in the context of time scales. In particular, we present a generalized Fourier transform in this context as well as a critical inversion result. This leads directly to a convolution for signals on two (possibly distinct) time scales as well as several natural classes of time scales which arise in this setting: dilated, closed under...

Integral Transforms of Fourier Cosine and Sine Generalized Convolution Type

GENERALIZATION OF TITCHMARSH'S THEOREM FOR THE GENERALIZED FOURIER-BESSEL TRANSFORM

In this paper, using a generalized translation operator, we prove theestimates for the generalized Fourier-Bessel transform in the space L2 on certainclasses of functions.

A Note on the Convolution Theorem for the Fourier Transform

In this paper we characterize those bounded linear transformations Tf carrying L1(R1) into the space of bounded continuous functions on R1 , for which the convolution identity T (f ∗ g) = Tf ·Tg holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.