Since radial positive definite functions on R remain positive definite when restricted to the sphere, it is natural to ask for properties of the zonal series expansion of such functions which relate to properties of the Fourier-Bessel transform of the radial function. We show that the decay of the Gegenbauer coefficients is determined by the behavior of the Fourier-Bessel transform at the origi...

This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf’s) and probability mass functions (pmf’s) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interprete...

We develop the nth order Fourier-Bessel series expansion of 1-D functions in the interval (0,α). Hence we establish the sampling theorem for a function with α-bandlimited nth order Hankel transform. The latter statement implies that the function is also Fourier transform αbandlimited. The samples’ locations are given by the roots of nth order Bessel functions. In addition, the sampling distance...

A method to test for nonlinearity in time series, without the need to apply the Fourier transform, is proposed. This method therefore avoids the drawbacks of previously proposed surrogate techniques associated with the estimation of the signal's power spectrum. The test addressed by this algorithm is that the data are generated by a stationary linear system. To achieve this, the algorithm takes...

This paper addresses the deconvolution of an image that has been obtained by superimposing many copies of an underlying unknown image of interest. The superposition is assumed to not be exact due to noise, and is described using an error distribution in position, orientation, or both. We assume that a good estimate of the error distribution is known. The most natural approach to take for the pu...

Ramanujan sums are exponential sums with exponent defined over the irreducible fractions. Until now, they have been used to provide converging expansions to some arithmetical functions appearing in the context of number theory. In this paper, we provide an application of Ramanujan sum expansions to periodic, quasiperiodic and complex time series, as a vital alternative to the Fourier transform....

We propose a new method for analysis of the sampling and reconstruction conditions of signals by use of the multiple-parameter fractional Fourier transform (MPFRFT). It is shown that the MPFRFT may provide a novel understanding of sampling process. The proposed sampling theorem generalizes classical Shannon sampling theorem and Fourier series expansion, and provides a fullreconstruction procedu...