Stability of localized operators

Abstract

Let p , 1 p ∞, be the space of all p-summable sequences and C a be the convolution operator associated with a summable sequence a. It is known that the p -stability of the convolution operator C a for different 1 p ∞ are equivalent to each other, i.e., if C a has p -stability for some 1 p ∞ then C a has q -stability for all 1 q ∞. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the p -stability (or L p -stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.

Stability of Localized Integral Operators on WeightedLpSpaces

“…The same discretization has been used in [14,19] to establish Wiener's lemma and stability for localized integral operators on unweighted function spaces L p , 1 ≤ p < ∞. In the second part of this section, we consider the boundedness and off-diagonal decay property of discretization matrices A n , n ∈ Z.…”

“…As we always assume in the paper that the integral operator T in (1.2) has its kernel with certain off-diagonal decay, its discretization matrices A n , n ∈ Z, have similar off-diagonal decay, see Proposition 2.7 of the fifth subsection. For N ≥ 1 and [14] to establish the equivalence of stability of a localized integral operator on unweighted function space L p for different exponent 1 ≤ p < ∞.…”

“…We remark that for the operator zI − T , it is established in [14] the equivalence of its stability on unweighted function spaces…”

“…We are working on the problem whether or not the above inclusion is indeed an equality when the kernel of the operator T satisfies (1.9). The reader may refer to [1,2,5,8,11,12,13,14] for spectra σ p,w (T ) of various integral operators T , and [10,17,18,19] for its connection to Wiener's lemma for infinite matrices.…”

Frames and their associated $\emph{H}_{{\kern-2pt}\emph{F}}^{\emph{p}}$ -subspaces

“…Typical examples include the classical Wiener space characterization with respect to Fourier basis [3], the Hardy space and the Sobolev spaces characterization with respect to wavelet bases [8,15], and the modulation space characterization with respect to Gabor frame bases [12]. All these (potential) applications led to some recent systematic research on localized bases or frames (see for example [1,2,6,10,11,14,19,20] and the references therein), and most of the work so far is mainly focused on localized bases with reference to a given orthonormal/Riesz basis or on localized frames with special structure such as Gabor and wavelet frames. In this paper, motivated by a recent work of Heil and Larson [14] we investigate the general theory of subspaces of a Hilbert space that can be realized as a copy (through the coefficient expansion with respect to a given frame) of a subspace of the p -space.…”