A study of generating functions of sampling theorems and optimum interpolation functions minimizing some kinds of measures

Abstract

AbstractGiven an absolutely square‐integrable waveform bandpass limited to the interval |ω| ≤ ω0, ω0 > 0, and regularly spaced sample points tn = nτ0, (τ0 = π/ω1 > ω0; n = 0, ±1, ±2, …) the interpolation formula which corrects errors in the sample values of the original waveform is not uniquely determined. In this paper, we derive several theorems which establish the relation between the interpolation functions and a two‐variable function of angular frequency ω and time t called the generating function of the sampling theorem. First, we show that these interpolation functions may be considered as functions of time with coefficients obtained from the Fourier series expansion of the generating functions with respect to the angular frequency ω. This theorem depends on another already established theorem, for which a brief proof is given in the Appendix, for the reader's convenience, in the case that the relation is defined on a vector space. Next, under the condition that the generating function is absolutely continuous, we consider the problem of determining a generating function that minimizes the sum of the squares of the weighted interpolation functions. We show that the sum of the squares of the weighted interpolation functions as a measure provides a bound on the magnitude of the interpolation functions and also measures the rate of decrease. In this paper, under two conditions, which are satisfied in the sampling theorem, a generating function in the sampling theorem is determined which minimizes the above‐mentioned measure of the interpolation functions. Using these arguments, we also derive generating functions which minimize various weighting measures related to the interpolation functions ωn(t), such as when the weights are squares, (2k)‐th powers, and (1 + n2ω).