Sampling Theorem and explicit interpolation formula for non-decaying unbounded signals

TL;DR

This paper extends the classical sampling theorem to unbounded signals with sublinear growth, providing explicit interpolation formulas and methods for calculating coefficients for signals with high polynomial growth.

Contribution

It establishes an analog sampling theorem for non-decaying signals and derives explicit interpolation formulas with coefficient decay rates, applicable to signals with high polynomial growth.

Findings

Derived an explicit interpolation formula for unbounded signals

Established coefficient decay rate of approximately 1/k^2

Provided a method for calculating coefficients for signals with high polynomial growth

Abstract

The paper establishes an analog Whittaker-Shannon-Kotelnikov sampling theorem for unbounded non-decaying band-limited signals. An explicit interpolation formula is obtained for signals sublinear growth with rate of growth less than 1/2. At any time, the rate of decay for the $k$th coefficients of this formula is $\sim 1/k^2$. In addition, the paper obtains a method for calculating the coefficients of the interpolation formula applicable to signals with arbitrarily high rate of polynomial growth.