Approximation of Functions by n-Separate Wavelets in the Spaces L^p(ℝ), 1 ≤ p ≤ ∞

Abstract

We consider the orthonormal bases of n-separate MRAs and wavelets constructed by the author earlier. The classical wavelet basis of the space L²(ℝ) is formed by shifts and compressions of a single function ψ. In contrast to the classical case, we consider a basis of L²(ℝ) formed by shifts and compressions of n functions ψ^s, s = 1,...,n. The constructed n-separate wavelets form an orthonormal basis of L²(ℝ). In this case, the series $\sum\nolimits_{s = 1}^n {\sum\nolimits_{j \in {\rm Z}} {\sum\nolimits_{k \in {\rm Z}} {f,\psi _{nj + s}^s >\psi _{nj + s}^s} } }$ converges to the function f in the space L²(ℝ). We write additional constraints on the functions ϕ^s and ψ^s, s = 1,..., n, that provide the convergence of the series to the function f in the spaces L^p(ℝ), 1 ≤ p <- ∞, in the norm and almost everywhere.