Periodic orthogonal splines and wavelets (Q1908135): Difference between revisions

In this interesting paper, the authors develop a general approach for non-stationary multiresolution of periodic functions. For a multiresolution \(\{V_k: k\geq 0\}\), necessary and sufficient conditions for \(\bigcup_{k\geq 0} V_k\) to be dense in \(L^2_{2\pi}\) and characterizations of a function \(\phi_k\) for which \(\phi_k(\cdot - 2\pi j2^{- k})\), \(j= 0, 1,\dots, 2^k- 1\), form a basis of \(V_k\) are given. By using so-called orthogonal splines various examples of multiresolutions are provided. The construction of corresponding periodic wavelets and Riesz stability are discussed in detail. Periodic polynomial spline wavelets and the trigonometric polynomial wavelets from Chui-Mhaskar are seen to follow as a special case. The general ``orthogonal spline bases'' give rise to algorithms in which the equations are the finite Fourier transforms of the classical wavelet decomposition and reconstruction equations. For Chui-Wang spline wavelets a comparison of frequency-based algorithms investigated in this paper with time-based algorithms shows that the frequency-based algorithms can be a useful alternative.