Density conditions including necessary ones and sufficient ones for irregular wavelet systems to be frames are studied in this paper. We give a definition of Beurling density for the case of wavelet frames and show that for irregular wavelet systems to be frames, the time-scale parameters must be relatively uniformly discrete. We prove that for a nice wavelet function, every relatively uniformly discrete time-scale sequence with sufficiently high density will generate a frame. Explicit frame bounds are given. We also study the stability of wavelet frames and show that every wavelet frame with arbitrary time-scale parameters is stable provided the wavelet function is nice enough. Explicit stability bounds are given. Numerical examples show that our results are sharper than some known ones.

We study the construction of wavelet frames and Gabor frames with irregular time-scale and time-frequency parameters respectively. We give simple and sufficient conditions which ensure an irregular discrete wavelet system or Gabor system to be a frame. Explicit frame bounds are given. We also study the stability of wavelet frames and Gabor frames and give explicit stability bounds. Several known results are considerably improved. Examples are given.

Regular and irregular sampling theorems for multivariate shift invariant subspaces are studied. We give a characterization of regular points and an irregular sampling theorem, which covers many known results, e.g., Kadec's 1/4-theorem. We show that some subspaces may not have a regular point. We also present a reconstruction algorithm which is slightly different from the known one but is more efficient. We study the aliasing error and prove that every smooth square integrable function can be approximated by its sampling series.

In this paper, we show that every band-limited function can be reconstructed by its local averages near certain points. We give the optimal upper bounds for the support length of averaging functions with respect to both regular and irregular sampling points. Our results improve an earlier result by Gr

Finding general and verifiable conditions which imply that Gabor systems are (resp. cannot be) Gabor frames is among the core problems in Gabor analysis. In their paper on atomic decompositions for coorbit spaces [H.G.Feichtinger and K.Gr