Wavelets of Wilson type with arbitrary shapes (Q1969037)

Let \(\{ \omega_j \}_{j\in \mathbb Z}\) be a family of complex-valued functions on \(\mathbb R\) and consider a sinusoidal family of functions given by \(T_{j,k}(x)= \sqrt{2} \cos(k+ 1/2)\pi (x-j)\), \(j \in {\mathbb Z}\), \(k=0,1,\dots\) (resp. \(T_{j,k}(x)=\sqrt{2} \sin(k+ 1/2)\pi (x-j) \), or an appropriate combination of sine and cosine). Define a two-parameter family of functions by \(\psi_{j,k}(x)= \omega_j(x) T_{j,k}(x)\). Equivalent conditions for \(\{ \psi_{j,k}\}\) to be a Schauder basis for \(L^2(\mathbb R)\) are given. Under extra conditions on \(\omega_j\), a condition for the existence of a dual Schauder basis of the same type is given. Other assumptions on \(\omega_j\) imply that \(\{ \psi_{j,k}\}\) is a frame if and only if \(\{ \psi_{j,k}\}\) is a Riesz basis.