Linear Approximation and Reproduction of Polynomials by Wilson Bases

Abstract

Wilson bases are constituted by trigonometric functions multiplied by translates of a window function with good time frequency localization. In this article we investigate the approximation of functions from Sobolev spaces by partial sums of the Wilson basis expansion. In particular, we show that the approximation can be improved if polynomials are reproduced. We give examples of Wilson bases, which reproduce linear functions with the lowest-frequency term only.

Formulation of Localized Cosine Bases that Preserve Polynomial Modulated Sinusoids

“…[9,11,12,14]). For some applications (e.g., in image compression) it is important that a smooth function has a good local approximation by using a very small number of terms, i.e., using only small indices k in (2.6).…”

“…For some applications (e.g., in image compression) it is important that a smooth function has a good local approximation by using a very small number of terms, i.e., using only small indices k in (2.6). In this case, the reconstruction of polynomials by only the low frequency elements of the Wilson bases is of essential importance (see [9,11,12]). The optimal case is that only the DC terms ψ 0 j , j ∈ Z, are necessary to represent polynomials up to a certain degree.…”

Spline Modulation of Sinusoids for Signal Representation

“…If we choose a translate w = N m (· − y), y ∈ R, as a window for the Wilson system, we also have the polynomial reproduction property, namely each p ∈ m−1 has a representation p(x) = j a j ψ 0 2j (x), x ∈ R, for some coefficients a j . It is known from [11,12] that Wilson bases with this property have very good approximation properties.…”

“…Wilson bases without the two-overlapping restriction have been investigated by several authors [8,11,16,17,19]. In [12], window functions which reproduce linear functions with only the first frequency term were constructed, allowing the overlap of three windows at the same time. In [9,11,12], it has been shown that the reproduction of polynomials results in very good approximation properties.…”

Direct and Inverse Approximation Theorems for Local Trigonometric Bases

“…e e and Meyer [24] and the Wilson bases of Daubechies et al [12]. Approximation properties of such bases have been investigated in [2,4].…”