Regularization of the ill-posed problem of extrapolation with the Malvar-Wilson wavelets (Q2764818)

The problem under consideration is the extrapolation of a band-limited signal \(f\in L^2(R^d)\) with spectrum \({\mathcal F}f\) (Fourier transform) embedded in a convex compact \(B\subset R^{d}\) when \(f\) is known on a dilation \(A_{\lambda}=\lambda A\) of a compact convex body \(A \subset R^{d}\). Formally it means to solve the first kind equation \(K{f}=g\) where \(K=Q_{\lambda A}P_B\), \(Q_{A}f(t)=f(t)\), \(t\in A\), \(Q_{A}f(t)=0\), \(t\notin A\), \(P_{B}={\mathcal F}^{-1}Q_B\mathcal{F}\). The operator \(K\) is compact Hilbert-Schmidt type and the equation after some finite-range approximation can be solved by the SVD algorithm. However this way is rather complicated if the approximating algebraic equation system has a large dimension and authors propose a new more simple reduction method which is based, as well as the SVD, on the decomposition of the solution space \(L^{2}(R^d)=V_{+}\oplus V_{0}\oplus V_{-}\). The main theoretical result is an order estimation of the first subspace \(V_{+}\) dimension \(N\sim({\lambda/(2\pi)})^{d}|A||B|\) that ensures numerically stable invertibility of the restriction of \(K\) on this subspace. Respectively, the reducing problem can be solved by a projection method of Galerkin type. The orthogonal basis is the Malvar-Wilson discrete parameter wavelets \(\{w_{jk}(x)=\) \(\sqrt{2}w(x-k)\cos((j+1/2)(x-k)),\) \(j\in N\), \(k\in Z\}\) where \(w(x)\) is some window function. The matrix involved to solve the corresponding AES is almost diagonal. Two numerical examples of restoration of one and two-dimensional signals are presented.