A note on irregular wavelet frames

In this note, we give a necessary condition for irregular wavelet frames, which extends the results of Chui and Shi. We prove that if the system {ψm,n≔sm-1/2ψsm-1·-tn}m,n∈Z constitute a frame for L2(R) with frame bounds A, B, then ∑m∈Z|ψˆ(smξ)|2 is bounded up and below. Moreover, the upper and lower bounds depend only on the frame bounds A, B and density of the samples tn's, not rely on the samples sm's and the specific generator function ψ however.