Nonstationary tight wavelet frames, II: unbounded intervals

Wiener’s lemma for infinite matrices

“…The upper bound estimate (4.1) for A −1 is important to obtain a scale-independent Riesz and frame bounds in the study of non-stationary multiresolution analysis, wavelets on intervals (domains), and diffusion wavelets on manifolds, where the index sets X and the quasi-metrics ρ vary according to the scales ( [9,11,10,12,26,31,37]). For the case that p = ∞, X is a relatively-separated subset of R d , and the weight w is of the form w(x, y) = (1 + |x − y|) s with s > d, the upper bound estimate (4.1) is not mentioned in [26], but can be obtained by keeping track of the constants in the argument.…”

“…In the study of spline approximation and projection ( [15,16]), wavelets and affine frames ( [9,26]), Gabor frame ( [3,23,24]), and non-uniform sampling, the associated matrix algebras are extremely non-commutative but they are still expected to have the same property as the commutative matrix algebra W. We are motivated by the above expectation, and by the importance of Wiener's lemma in the study of (Gabor) frames on non-uniform grids and of non-uniform sampling and the reconstruction problem. For instance, we apply Wiener's lemma established in this paper to show the well-localization of dual (tight) frame generators of a locally finitely-generated space ( [37]), and robustness and finite implementation of an average (ideal) sampling and the reconstruction process ( [38]).…”

Hybrid spline frames

“…The multiresolution setting of hybrid splines and our construction of the corresponding tight frames are different from those studied by Chui, He and Stöckler in [8,9]. In [9] they developed a theory together and a general construction of nonstationary tight frames for L 2 (I), where I = [0, ∞) or R, using the kernels of certain approximation operators on the multiresolution subspaces.…”

“…In particular, they studied in greater detail multiresolution subspaces of spline functions with arbitrary nested knot sequences on R and on the semi-infinite interval [0, ∞), where 0 is considered a multiple knot with maximum multiplicity. However, explicit construction of nonstationary tight frames for specific cases remains an interesting and challenging problem, even in the special case of cubic spline frames derived from the general non-stationary construction in [9].…”

“…In [9] they developed a theory together and a general construction of nonstationary tight frames for L 2 (I), where I = [0, ∞) or R, using the kernels of certain approximation operators on the multiresolution subspaces. In particular, they studied in greater detail multiresolution subspaces of spline functions with arbitrary nested knot sequences on R and on the semi-infinite interval [0, ∞), where 0 is considered a multiple knot with maximum multiplicity.…”