Construction of Frames for Shift-Invariant Spaces

Abstract

We construct a sequence {ϕi(·-j)∣j∈ℤ, i=1,…,r} which constitutes a p-frame for the weighted shift-invariant space Vμp(Φ)={∑i=1r∑j∈ℤci(j)ϕi(·-j)∣{ci(j)}j∈ℤ∈ℓμp, i=1,…,r}, p∈[1,∞], and generates a closed shift-invariant subspace of Lμp(ℝ). The first construction is obtained by choosing functions ϕi, i=1,…,r, with compactly supported Fourier transforms ϕ^i, i=1,…,r. The second construction, with compactly supported ϕi, i=1,…,r, gives the Riesz basis.