On a Class of Generalized Sampling Functions (Analysis in Theory and Applications, 2014)
In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrable or Lebegue integrable on the real number line. They are in fact the generalization of the classic sinc function. Two approaches of constructing the generalized sampling functions are reviewed. Their properties such as cardinality, orthogonality, and decaying properties are discussed. The interactions of those functions and Hilbert transformer are also discussed.
Keywords: Generalized sampling function; sinc function; non-bandlimited signal; sampling theorem; Hilbert transform.