Necessary Conditions And Sufficient Conditions For Gabor Frames And Wavelet Frames

The concept of frame was introduced by R.J.Duffin and A.G.Schacffer to study nonharmonic fourier series in 1952.One of the important applications of frame is that we can reconstruct a function from its frame coefficients.These years,with the devel-opment of wavelet analysis,people give more and more attention to frames.In less than twenty years,frame theory is widely used in function theory,partial differential equa-tion,quantum mechanics,theory physics,signal processing,wavelet analysis,irregular sampling theory and many other fields.Many authors have studied deeply for frames.In this paper, we concentrate on two problems.First,we get several inequalities for multi-generated irregular Gabor frames of the form U1≤k≤r{eigk(x-μ):μ∈Δk,λ∈Λk,λ∈Λk}, whereΔk andΛk are arbitrary sequences of points in Rd and gk∈L2(R),1≤k≤Theorem 1 LetΔk andΛk be sequences of points in Rd and gk∈L2(Rd),1 k≤r.(ⅰ).IfU11 and b>0 be constants.(ⅰ).If{aj/2ψ(aj·-bk):j,k∈Z} is a frame for L2(R) with frame bounds A and B, then A≤Gα,n(ψ,ω)≤B,a,e.,(?)n≥1,α∈Λ\{0}.(ⅱ).Moreover, if a is an integer, then the above condition is a necessary and sufficient condition for {aj/2ψ(aj·-bk):j, k∈Z} to be a frame for L2(R). Where Gα,n(ψ,ω)=[Δ(k-l)α(ψ,ω+lα)]-n≤k,l≤n is a (2n+1)×(2n+1)matrix,Theorem 3 Letψ∈L2(Rd),a real expansive matrix A and a real nonsingular matrix B be given. (ⅰ).If{│detA│j/2ψ(Ajx-Bk):j∈Z,k∈Zd} forms a frame for L2(Rd) with frame bounds L and M, then(ⅱ).Moreover, if A is an integer matrix and AB=BA, then the above condition is a necessary and sufficient condition for {│detA│j/2ψ(Ajx - Bk):j∈Z,k∈Zd} to be a frame for L2(Rd). Where is a (2n+1)d×(2n + 1)d matrix,α∈Λ:={α∈Rd:there is j∈Z,k∈Zd such thatα= A*-jB*-1k}.