Can Evren Yarman: Slepian functions and generalization of Shannon sampling theorem for R-bandlimited functions (joint with appl. and comp. mathematics)

R-bandlimited functions, as defined by Slepian, are a generalization of band-limited functions in 1D to multiple dimensions (multi-D) whose multi-D Fourier transform is supported within a region R. Similarly, Slepian functions are defined as a multi-D generalization of prolate-spheroidal wave functions. They are eigenfunctions of restriction of the R-bandlimited projection operator to a compact support. R-bandlimitedness can be expressed using the Fourier integral or convolution integral. We generalized Shannon sampling theorem through discretization of the convolution representation when the compact support of restriction is similar to R. We start with an approximation of the convolution kernel as a discrete sum of the Fourier basis that is accurate within a region similar to R. Substitution of this approximation into the convolution representation leads to a discretization of the Fourier integral representation of R-bandlimited functions. We use properties of the Slepian functions to show that the discretization of the Fourier integral representation is equivalent to discretization of the convolution integral with similar quadratures. In this approach, the concept of aliasing in 1D is replaced by accuracy of the approximation of the convolution kernel of the R-bandlimited projection operator.