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

Under certain assumptions we show that a wavelet frame {(Aj, bj,k)ψ}j,k∈Z:={|det Aj|−1/2ψ(Aj−1j(x−bj,k))}j,k∈Z in L2(Rd) remains a frame when the dilation matrices Aj and the translation parameters bj, k are perturbed. As a special case of our result, we obtain that if {T(Aj, AjBn)ψ}j∈Z, n∈Zd is a frame for an expansive matrix A and an invertible matrix B, then {T(A'j, AjBλn) }j∈Z,n∈Zd is a frame if kA−jA’j−I‖2≤εand‖λn− n‖∞≤n for sufficiently smallε, n>0.

We show that if an irregular multi-generated wavelet system forms a frame, then both the time parameters and the logarithms of scale parameters have finite upper Beurling densities, or equivalently, both are relatively uniformly discrete. Moreover, if generating functions are admissible, then the logarithms of scale parameters possess a positive lower Beurling density. However, the lower Beurling density of the time parameters may be zero. Additionally, we prove that there are no frames generated by dilations of a finite number of admissible functions

In this paper we give sufficient conditions for irregular Gabor systems to be frames. We show that for a large class of window functions, every relatively uniformly discrete sequence in R2 with sufficiently high density will generate a Gabor frame. Explicit frame bounds are given. We also study the stability of irregular Gabor frames and show that every Gabor frame with arbitrary time-frequency parameters is stable if the window function is nice enough. Explicit stability bounds are given.

In this paper, we study the reconstruction of functions in shift invariant subspaces from local averages with symmetric averaging functions. We present an average sampling theorem for shift invariant subspaces and give quantitative results on the aliasing error and the truncation error. We show that every square integrable function can be approximated by its average sampling series. As special cases we also obtain new error bounds for the case of regular sampling. Examples are given.

In this paper, we study the reconstruction of functions in spline subspaces from local averages. We present an average sampling theorem for shift invariant subspaces generated by cardinal B-splines and give the optimal upper bound for the support length of averaging functions. Our result generalizes an earlier result by Aldroubi and Gr