This paper deals with an alternative proof of Beurling-Lax theorem by adopting a constructive approach instead of the isomorphism technique which was used in the original proof.
In this paper, starting from a function analytic in a neighborhood of the unit disk and based on Bessel functions, we construct a family of generalized multivariate sinc functions, which are radial and named radial Bessel-sinc (RBS) functions being time-frequency atoms with nonlinear phase. We obtain a recursive formula for the RBS functions in R d with d being odd. Based on the RBS function, a corresponding sampling theorem for a class of non-bandlimited signals is established. We investigate a class of radial functions and prove that each of these functions can be extended to become a monogenic function between two parallel planes, where the monogencity is taken to be of the Clifford analysis sense.
Starting from piecewise constant functions, a novel family of generalized symmetric B-splines, with realizable ideal low-pass filters, are constructed. The first order generalized B-spline low-pass filter is closely related to functions analytic in a neighborhood of the unit disc and the generalized sinc functions. The properties of this kind of low-pass filters are investigated. The behavior of the generalized B-spline low-pass filter related to normalized Gaussian distribution is considered.
