Formulation of Localized Cosine Bases that Preserve Polynomial Modulated Sinusoids

Abstract

We develop a class of compactly supported window functions V_m so that each polynomial p(x) of degree m - 1 as well as p(x) modulated with an integer frequency has a locally finite representation in terms of the local cosine basis ψ = {ψ^k _j (x) = Vm(x - j /2 - 1/4 ) cos((2k + 1)π(x - j))}. Explicit formulations for V_m and its Fourier transform are derived. It is shown that these window functions are non-negative, have minimal support of length m, and maximal smoothness of order 2m - 2. Furthermore, we determine the exact Riesz bounds of ψ. Smoothness and stability for these bases are superior as compared to other local cosine bases with similar properties in the literature. Consequently, the bases ψ are particularly useful for applications in signal and image processing.