Mixed Lebesgue Spaces Research Articles

Semi-Average Sampling for Shift-Invariant Signals in a Mixed Lebesgue Space

This article mainly studies the nonuniform and semi-average sampling of time-varying shift-invariant signals living in a mixed Lebesgue space under the condition that the generator of the shift-invariant subspace belongs to a hybrid-norm space of mixed form depending on the orders p and q. First, the sampling stability for two kinds of semi-average sampling functionals is established. Then, we give the corresponding iterative approximation projection algorithms with exponential convergence for recovering the time-based shift-invariant signals from the semi-average samples.

A sampling theory for non-decaying signals in mixed Lebesgue spaces

In this paper, we study the sampling theory for non-decaying signals in mixed Lebesgue spaces . First, we give some Riesz-type bounds, which connect the weighted discrete norms of the samples to the weighted continuous norms of original and interpolated signals. Then, we prove that the sampling and reconstruction through the Riesz-type bounds are stable when the generated kernel is in the appropriate mixed Wiener amalgam spaces.

Nonuniform average sampling in multiply generated shift-invariant subspaces of mixed Lebesgue spaces

In this paper, we study nonuniform average sampling problem in multiply generated shift-invariant subspaces of mixed Lebesgue spaces. We discuss two types of average sampled values: average sampled values [Formula: see text] generated by single averaging function and average sampled values [Formula: see text] generated by multiple averaging functions. Two fast reconstruction algorithms for these two types of average sampled values are provided.

Adaptive sampling of time-space signals in a reproducing kernel subspace of mixed Lebesgue space

The mixed Lebesgue space is a suitable tool for modeling and measuring signals living in time-space domains. And sampling in such spaces plays an important role for processing high-dimensional time-varying signals. In this paper, we first define reproducing kernel subspaces of mixed Lebesgue spaces. Then, we study the frame properties and show that the reproducing kernel subspace has finite rate of innovation. Finally, we propose a semi-adaptive sampling scheme for time-space signals in a reproducing kernel subspace, where the sampling in time domain is conducted by a time encoding machine. Two kinds of timing sampling methods are considered and the corresponding iterative approximation algorithms with exponential convergence are given.

Sampling and reconstruction in reproducing kernel subspaces of mixed Lebesgue spaces

In this paper, we study the sampling and average sampling problems in a reproducing kernel subspace of mixed Lebesgue space. Let V be an image of $$L^{p,q}({\mathbb {R}}^{d+1})$$ under idempotent integral operator defined by a kernel K satisfying certain decay and regularity conditions. Then, we prove that every f in V can be reconstructed uniquely and stably from its samples as well as from its average samples taken on a sufficiently small $$\gamma $$ -dense set. Further, we derive iterative reconstruction algorithms for reconstruction of f in V from its samples and average samples. We also obtain the error estimates in iterative reconstruction algorithm from noisy samples and iterative noise.

Average sampling and reconstruction of reproducing kernel signals in mixed Lebesgue spaces

In this paper, we mainly study the average sampling and reconstruction of signals in a reproducing kernel subspace of the mixed Lebesgue space Lp,q(Rm+n). First, the sampling stability for two kinds of average sampling functionals is considered. Then, the corresponding iterative approximation projection algorithms with exponential convergence are established. Finally, the asymptotic pointwise error estimates are given for reconstructing a signal from its average samples corrupted by additive random noise.

Sampling theorem on shift invariant subspaces in mixed Lebesgue spaces

Sampling theorem is the theoretical basis of modern communication system, and is also one of the most powerful basic tools in signal processing and image processing. In this paper, we give sampling theorem on shift invariant subspaces in mixed Lebesgue spaces Lp,q(ℝd+1). When the sampling density is large enough, this sampling theorem can reconstruct exactly the signals in shift invariant subspaces. Our results improve the result in [6].

Mixed Morrey spaces

We introduce mixed Morrey spaces and show some basic properties. These properties extend the classical ones. We investigate the boundedness in these spaces of the iterated maximal operator, the fractional integral operator and singular integral operator. Furthermore, as a corollary, we obtain the boundedness of the iterated maximal operator in classical Morrey spaces. We also establish a version of the Fefferman–Stein vector-valued maximal inequality and some weighted inequalities for the iterated maximal operator in mixed Lebesgue spaces.

Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces

We prove that bilinear fractional integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions. We also treat bilinear singular multiplier operators which preserve regularity and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed Lebesgue spaces.

The Lp,q-stability of the Shifts of Finitely Many Functions in Mixed Lebesgue Spaces Lp,q(ℝd+1)

The stability is an expected property for functions, which is widely considered in the study of approximation theory and wavelet analysis. In this paper, we consider the Lp,q-stability of the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1). We first show that the shifts ϕ(· − k) (k ∈ ℤd+1) are Lp,q-stable if and only if for any ξ ∈ ℝd+1, $$\sum\nolimits_{k \in \mathbb{Z}^{d + 1} } {\left| {\hat \varphi (\xi + 2\pi k)} \right|^2 > 0}$$ . Then we give a necessary and sufficient condition for the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1) to be Lp,q-stable which improves some known results.

Mixed problems for degenerate abstract parabolic equations and applications

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

A fast solution method for time dependent multidimensional Schrödinger equations

ABSTRACTIn this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrödinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high-order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200.

Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue spaces [formula omitted

In this paper, we study the nonuniform sampling and reconstruction problem in shift-invariant subspaces of mixed Lebesgue spaces. We first show that shift-invariant subspaces in mixed Lebesgue spaces Lp,q(Rd+1) can be well-defined. Then we propose that the sampling problem in shift-invariant subspaces of mixed Lebesgue spaces is well-posed. At last, the nonuniform samples {f(xj,yk):k,j∈J} of a function f belonging to a shift-invariant subspace of mixed Lebesgue spaces are proposed, and we give a fast reconstruction algorithm that allows exact reconstruction of f as long as the sampling set X={(xj,yk):k,j∈J} is sufficiently dense.

Asymptotic stability of solitary waves in generalized Gross–Neveu model

For the nonlinear Dirac equation in (1+1)D with scalar self-interaction (Gross–Neveu model), with quintic and higher order nonlinearities (and within certain range of the parameters), we prove that solitary wave solutions are asymptotically stable in the “even” subspace of perturbations (to ignore translations and eigenvalues ±2ωi). The asymptotic stability is proved for initial data in H1. The approach is based on the spectral information about the linearization at solitary waves which we justify by numerical simulations. For the proof, we develop the spectral theory for the linearized operators and obtain appropriate estimates in mixed Lebesgue spaces, with and without weights.

Leibniz’s Rule, Sampling and Wavelets on Mixed Lebesgue Spaces

Several new results about Leibniz’s rule, sampling, and wavelets in the context of mixed Lebesgue spaces are presented. The results obtained rely in part on vector-valued estimates for Calderon–Zygmund operators and the use of appropriate maximal functions and Littlewood–Paley estimates.

Generalized free time-dependent Schrödinger equation with initial data in Fourier Lebesgue spaces

Consider the solution of the free time-dependent Schro dinger equation with initial data f. It is shown by Sjogren and Sjolin that there exists f in the Sobolev space Hs (Rd), s = d/2 such that tangential convergence can not be widened to convergence regions. The author obtained in a previous paper the corresponding results for a generalized version of the Schrodinger equation, where −Δx is replaced by an operator \({\varphi(D)}\), with special conditions on \({\varphi}\). In this paper we show that similar results may be obtained for initial data in usual and mixed Fourier Lebesgue spaces. We also relax the conditions on \({\varphi}\).

CALDERÓN–ZYGMUND OPERATORS ON MIXED LEBESGUE SPACES AND APPLICATIONS TO NULL FORMS

The boundedness of Calderon–Zygmund operators is proved in the scale of the mixed Lebesgue spaces. As a consequence, the boundedness of the bilinear null forms on various space–time mixed Sobolev–Lebesgue spaces is shown.

The Ap condition and a positive, linear operator

SynopsisA positive, linear operator is exhibited which is bounded on the mixed weighted Lebesgue space if and only if the weight w satisfies the Ap condition of Muckenhoupt.

Lipschitz spaces and mixed Lebesgue spaces

It is shown that translation invariant linear operators which improve Lipschitz classes behave almost as well as the corresponding fractional Riesz transforms when applied to the mixed Lebesgue spaces. These results partially generalize some of the theorems concerning Riesz transforms and mixed Lebesgue classes due to Adams and Bagby, Lizorkin, and others.