Inequalities for nonuniform wavelet frames

Abstract

Abstract Gabardo and Nashed studied nonuniform wavelets by using the theory of spectral pairs for which the translation set Λ = { 0 , r / N } + 2 ⁢ ℤ {\Lambda=\{0,r/N\}+2\mathbb{Z}} is no longer a discrete subgroup of ℝ {\mathbb{R}} but a spectrum associated with a certain one-dimensional spectral pair. In this paper, we establish three sufficient conditions for the nonuniform wavelet system { ψ j , λ ⁢ ( x ) = ( 2 ⁢ N ) j / 2 ⁢ ψ ⁢ ( ( 2 ⁢ N ) j ⁢ x - λ ) , j ∈ ℤ , λ ∈ Λ } {\{\psi_{j,\lambda}(x)=(2N)^{j/2}\psi((2N)^{j}x-\lambda),\,j\in\mathbb{Z},\,% \lambda\in\Lambda\}} to be a frame for L 2 ⁢ ( ℝ ) {L^{2}(\mathbb{R})} . The proposed inequalities are stated in terms of Fourier transforms and hold without any decay assumptions on the generator of such a system.