Approximation of Functions by n-Separate Wavelets in the Spaces Lp(ℝ), 1 ≤ p ≤ ∞
Е. А. Плещева · Proceedings of the Steklov Institute of Mathematics · 2020
We consider the orthonormal bases of n -separate MRAs and wavelets constructed by the author earlier. The classical wavelet basis of the space L 2 (ℝ) is formed by shifts and compressions of a single function ψ . In contrast to the classical case, we consider a basis of L 2 (ℝ) formed by shifts and compressions of n functions ψ s , s = 1,..., n . The constructed n -separate wavelets form an orthonormal basis of L 2 (ℝ). In this case, the series \(\sum olimits_{s = 1}^n {\sum olimits_{j \in {\rm Z}} {\sum olimits_{k \in {\rm Z}} {f,\psi _{nj + s}^s >\psi _{nj + s}^s} } } \) converges to the function f in the space L 2 (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.