Formulation of localized cosine bases that preserve polynomial modulated sinusoids

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 integerfrequency has a locally finite representation in terms of the local cosine basis ¥ = {¥ _j ^k (x) = V _m (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 exactRiesz bounds of ¥. Smoothness and stability for these bases are superior as compared to other local cosine bases with similar properties in the literature. Consequently, the bases ¥ are particularly useful for applications in signal and image processing.