On generalized Jacobi matrices and orthogonal polynomials.
A two-sided sequence with values in a complex unital Banach algebra is a cosine sequence if it satisfies for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence is bounded if . A (bounded) group decomposition for a cosine sequence is a representation of c as for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying , respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, here referred...
There are known wavelets with exponential decay on infinity [2,3,4] and wavelets with compact support [5]. But these functions have finite smoothness. It is known that there do not exist infinitely differentiable compactly supported wavelets.
Let . We investigate the characterization problem which asks for a classification of all the triples such that the Weyl-Heisenberg system is a frame for . It turns out that the answer to the problem is quite complicated, see Gu and Han (2008) and Janssen (2003). Using a dilation technique, one can reduce the problem to the case where and only let and vary. In this paper, we extend the Zak transform technique and use the Fourier analysis technique to study the problem for the case of...