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Dedicated to Professor V. I. Burenkov on the occasion of his 70th birthday We characterize the traces of vector-valued Besov and Lizorkin-Triebel spaces. Therefrom we derive the corresponding assertions for the vector-valued Sobolev spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$W^m_{p}({\mathbf R}^n,E)\,$\end{document}. Here we do not assume the UMD property for the Banach space E. [ABSTRACT FROM AUTHOR]

This article is devoted to nonlinear approximation and estimation via piecewise polynomials built on partitions into dyadic rectangles. The approximation rate is studied over possibly inhomogeneous and anisotropic smoothness classes that contain Besov classes. Highlighting the interest of such a result in statistics, adaptation in the minimax sense to both inhomogeneity and anisotropy of a related multivariate density estimator is proved. Besides, that estimation procedure can be implemented with a computational complexity simply linear in the sample size. [ABSTRACT FROM AUTHOR]

In this survey, results on the representation of a function as an absolutely convergent Fourier integral are collected, classified and discussed. Certain applications are also given. [ABSTRACT FROM AUTHOR]

Compressive sensing (CS) is an emerging approach for the acquisition of signals having a sparse or compressible representation in some basis. While the CS literature has mostly focused on problems involving 1-D signals and 2-D images, many important applications involve multidimensional signals; the construction of sparsifying bases and measurement systems for such signals is complicated by their higher dimensionality. In this paper, we propose the use of Kronecker product matrices in CS for two purposes. First, such matrices can act as sparsifying bases that jointly model the structure present in all of the signal dimensions. Second, such matrices can represent the measurement protocols used in distributed settings. Our formulation enables the derivation of analytical bounds for the sparse approximation of multidimensional signals and CS recovery performance, as well as a means of evaluating novel distributed measurement schemes. [ABSTRACT FROM PUBLISHER]

We characterize the anisotropic weak Hardy spaces HAp,∞(Rn) associated with an expansive matrix A by using square functions involving wavelets coefficients. [ABSTRACT FROM AUTHOR]

We consider a hydrogen-like atom in a quantized electromagnetic field which is modeled by means of a no-pair operator acting in the positive spectral subspace of the free Dirac operator minimally coupled to the quantized vector potential. We prove that the infimum of the spectrum of the no-pair operator is an evenly degenerate eigenvalue. In particular, we show that the bottom of its spectrum is strictly less than its ionization threshold. These results hold true, for arbitrary values of the fine-structure constant and the ultraviolet cut-off and for all Coulomb coupling constants less than the critical one of the Brown-Ravenhall model, 2/(2/π + π/2). For Coulomb coupling constants larger than the critical one, we show that the quadratic form of the no-pair operator is unbounded below. Along the way we discuss the domains and operator cores of the semi-relativistic Pauli-Fierz and no-pair operators, for Coulomb coupling constants less than or equal to the critical ones. [ABSTRACT FROM AUTHOR]

In this work we develop highly geometric Hardy spaces, for the full range 0< p≤1. These spaces are constructed over multi-level ellipsoid covers of ℝ that are highly anisotropic in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. This generalizes previous work on anisotropic Hardy spaces where the geometry of the space was 'fixed' over ℝ and extends Hardy spaces over spaces of homogeneous type, where the theory holds for p values that are 'close' to 1. [ABSTRACT FROM AUTHOR]

In this paper, the authors give the boundedness on Triebel-Lizorkin spaces for the parabolic singular integral with rough kernel and its commutator. [ABSTRACT FROM AUTHOR]

We investigate blow-up theory for doubly critical anisotropic problems in bounded domains of the Euclidean space. [ABSTRACT FROM AUTHOR]

This paper deals with entropy numbers for compact embeddings of Besov-Sobolev spaces with mixed integrability into Lebesgue spaces. [ABSTRACT FROM AUTHOR]

The asymptotic behavior in a layer-like domain of weak solutions with nonzero fluxes for the Dirichlet problem of the stationary Navier-Stokes system is investigated. If the data decay sufficiently fast, then first three asymptotic terms of any solution which grows at infinity not 'too fast' have the same structure as those for the linear Stokes problem. The first asymptotic term which occurs due to the nonlinearity is of order |x|-3 at infinity in the velocity part; it is constructed explicitly. The results are obtained without any assumption on the smallness of the data. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study the anisotropic local Hardy spaces $h_{A}^{p}(\mathbb{R}^{n})$ 0< p≤1, associated with the expansive matrix A. We obtain an atomic characterization of the distributions in $h_{A}^{p}(\mathbb{R}^{n})$. Also we describe the dual spaces of our local Hardy anisotropic spaces as anisotropic Campanato type spaces. [ABSTRACT FROM AUTHOR]

This paper studies photon-limited spectral intensity estimation and proposes a spatially and spectrally adaptive, nonparametric method for estimating spectral intensities from Poisson observations. Specifically, our method searches through estimates defined over a family of recursive dyadic partitions in both the spatial and spectral domains, and finds the one that maximizes a penalized log likelihood criterion. The key feature of this approach is that the partition cells are anisotropic across the spatial and spectral dimensions, so that the method adapts to varying degrees of spatial and spectral smoothness, even when the respective degrees of smoothness are not known a priori. The proposed approach is based on the key insight that spatial boundaries and singularities exist in the same locations in every spectral band, even though the contrast or perceptibility of these features may be very low in some bands. The incorporation of this model into the reconstruction results in significant performance gains. Furthermore, for spectral intensities that belong to the anisotropic Holder-Besov function class, the proposed approach is shown to be near-minimax optimal. The upper bounds on the risk function, which is the expected squared Hellinger distance between the true intensity and the estimate obtained using the proposed approach, matches the best possible lower bound up to a log factor for certain degrees of spatial and spectral smoothness. Experiments conducted on realistic data sets show that the proposed method can reconstruct the spatial and the spectral inhomogeneities very well even when the observations are extremely photon-limited (i.e., less than 0.1 photon per voxel). [ABSTRACT FROM AUTHOR]

The article presents a numerical solution on spatial periodic problem using Lin-Reissner-Tsien equation of transonic gas dynamics. It provides a comprehensive process on the derivation of plane-parallel transient transonic gas flows equation. It advises the need to have knowledge on Cauchy and initial boundary-value problems in the Lin-Reissner-Tsien equation. It discusses several theorems, definitions, and examples related to the equation.

Anisotropic operators appear in several branches of applied sciences and, in particular, physics. They involve directional derivatives with distinct weights which create distortions in the ambient space. Anisotropic rescaling comes with the notion of asymptotically stable domains. We prove two results: one of geometric nature, the other one of analytic nature, where both guarantee that a given domain is asymptotically stable. We also discuss specific examples. [ABSTRACT FROM AUTHOR]

We investigate blow-up theory and prove sharp Sobolev asymptotics for a general class of anisotropic critical equations in bounded domains of $${\mathbb{R}^n}$$ . [ABSTRACT FROM AUTHOR]

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space. [ABSTRACT FROM AUTHOR]

Abstract  We study properties of anisotropic Triebel–Lizorkin spaces associated with general expansive dilations and doubling measures on $${\mathbb{R}}^n$$ using wavelet transforms. This paper is a continuation of (Bownik in J Geom Anal 2007, to appear, Trans Am Math Soc 358:1469–1510, 2006), where we generalized the isotropic methods of dyadic $$\varphi$$ -transforms of Frazier and Jawerth (J Funct Anal 93:34–170, 1990) to non-isotropic settings. By working at the level of sequence spaces, we identify the duals of anisotropic Triebel–Lizorkin spaces. We also obtain several real and complex interpolation results for these spaces. [ABSTRACT FROM AUTHOR]

In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $$ \mathcal{L}_\lambda $$ which arise naturally in the $$ \bar \partial _b $$ -complex. They introduced weighted Sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $$ \bar \partial _b $$ in these spaces using integral kernels and approximate inverses. In the 1990’s, Rumin introduced a differential complex for compact contact manifolds, showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator, and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper, we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping. [ABSTRACT FROM AUTHOR]

We introduce and study anisotropic Triebel-Lizorkin spaces associated with general expansive dilations and doubling measures on ℝn with the use of wavelet transforms. This work generalizes the isotropic methods of dyadic ϕ-transforms of Frazier and Jawerth to nonisotropic settings. We extend results involving boundedness of wavelet transforms, almost diagonality, smooth atomic and molecular decompositions to the setting of doubling measures. We also develop localization techniques in the endpoint case of p = ∞, where the usual definition of Triebel-Lizorkin spaces is replaced by its localized version. Finally, we establish nonsmooth atomic decompositions in the range of 0 < p ≤ 1, which is analogous to the usual Hardy space atomic decompositions. [ABSTRACT FROM AUTHOR]

Let ( X, Y) be a pair of normed spaces such that X ⊂ Y ⊂ L 1[0, 1] n and { e k } k be an expanding sequence of finite sets in ℤ n with respect to a scalar or vector parameter k, k ∈ ℕ or k ∈ ℕ n . The properties of the sequence of norms $$\{ \left\| {S_{e_k } (f)} \right\|x\} _k $$ of the Fourier sums of a fixed function f ∈ Y are studied. As the spaces X and Y, the Lebesgue spaces L p [0, 1], the Lorentz spaces L p,q [0, 1], L p,q [0, 1] n , and the anisotropic Lorentz spaces L p, q*[0, 1] n are considered. In the one-dimensional case, the sequence { e k } k consists of segments, and in the multidimensional case, it is a sequence of hyperbolic crosses or parallelepipeds in ℤ n . For trigonometric polynomials with the spectrum given by step hyperbolic crosses and parallelepipeds, various types of inequalities for different metrics in the Lorentz spaces L p,q [0, 1] n and L p, q*[0, 1] n are obtained. [ABSTRACT FROM AUTHOR]

An integral equation of the first kind of convolution type is considered. A method closely related to Tikhonov regularization is used for constructing regularized approximate solutions that under certain assumptions converge to the exact solution of this ill-posed problem. V I Burenkov, I F Dorofeev and A S Pankratov obtained solutions of this problem for the isotropic case by using Nikol'skiĭ-Besov spaces of functions possessing some common smoothness of fractional order as a means of characterizing the smoothness properties of the exact solution and the error on the right-hand side. The main aim of the present work is to obtain similar results for the anisotropic case, thus allowing the right-hand side error, the exact and regularized solutions to be considered as members of a wider class of functions than in the former case. A complete investigation of the anisotropic case is carried out. Some of the results obtained are also new for the isotropic case. [ABSTRACT FROM AUTHOR]

We study in fairly general measure spaces ( X, μ) the (non–linear) potential theory of L p sub–Markovian semigroups which are given by kernels having a density with respect to the underlying measure. In terms of mapping properties of the operators we provide sufficient conditions for the existence (and regularity) of such densities. We give various (dual) representations for several associated capacities and, in the corresponding abstract Bessel potential spaces, we study the role of the truncation property. Examples are discussed in the case of ℝ n , where abstract Bessel potential spaces can be identified with concrete function spaces. [ABSTRACT FROM AUTHOR]

In this paper, which is the sequel to [16], we study inverse estimates of the Bernstein type for nonlinear approximation with structured redundant dictionaries in a Banach space. The main results are for blockwise incoherent dictionaries in Hilbert spaces, which generalize the notion of joint block-diagonal mutually incoherent bases introduced by Donoho and Huo. The Bernstein inequality obtained for such dictionaries is proved to be sharp, but it has an exponent that does not match that of the corresponding Jackson inequality. [ABSTRACT FROM AUTHOR]

Abstract. In this work we develop the theory of weighted anisotropic Besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms. This study extends the isotropic Littlewood- Paley methods of dyadic f-transforms of Frazier and Jawerth [19, 21] to non-isotropic settings. [ABSTRACT FROM AUTHOR]

In this paper we construct compactly supported biorthogonal wavelet bases of the interval, with dilation factor M. Next, the natural MRA on the cube arising from the tensor product of a multilevel decomposition of the unit interval is developed. New Jackson and Bernstein type inequalities are proved, providing a characterization for anisotropic Sobolev spaces. [ABSTRACT FROM AUTHOR]

Based on distribution-theoretical definitions of L 2 and Sobolev spaces given by Werner in [P. Werner (1970). A distribution-theoretical approach to certain Lebesgue and Sobolev spaces. J. Math. Anal. Appl. , 29 , 19-78.] real interpolation, Besov type spaces and approximation spaces with respect to multiresolution approximations are considered. The key for the investigation are generalized moduli of smoothness introduced by Haf in [H. Haf (1992). On the approximation of functionals in Sobolev spaces by singular integrals. Applicable Analysis , 45 , 295-308.]. Those moduli of smoothness allow to connect the concept of L 2 -functionals with more recent developments in multiscale analysis, see e.g. [W. Dahmen (1995). Multiscale analysis, approximation, and interpolation spaces. In: C.K. Chui and L.L. Schumaker (Eds.), Approximation Theory VIII, Vol. 2 : Wavelets and Multilevel Approximation , pp. 47-88.]. In particular, we derive wavelet characterizations for the Sobolev spaces introduced by Werner and establish stable wavelet decompositions of L 2 -functionals. Generalizations to more general spaces of functionals and applications are also mentioned. [ABSTRACT FROM AUTHOR]

Let µ be a positive Radon measure in R n with compact support G. Let Q jm be cubes with side-length 2 - j+1 originating from the canonical tiling of R n where j\in N 0 and m\in Z n. If ? \in R, 0 < p \le 8, 0 < q \le 8, then µ ? pq is the mixed l q?l p-quasi-norm of the sequence 2 j ?). Quantities of this type are considered in fractal geometry (multifractal formalism) and in the theory of the function spaces B jm). Quantities of this type are considered in fractal geometry (multifractal formalism) and in the theory of the function spaces B s pq ( R n) and F s is an equivalent quasi-norm in some of these spaces (µ-property). If |G| = 0, then pq ( R n). In Theorem 1 we deal with the question when µ ? pq is an equivalent quasi-norm in some of these spaces (µ-property). If |G| = 0, then S µ consists of those points ( t, s) in the t? s-diagram in Figure 1 for which µ belongs to B s p 8 ( R n) with pt = 1. Theorem 2 deals with the interrelation of S µ and µ ? pq. Some applications to truncated Riesz potentials, Bessel potentials and Fourier transforms of µ are given. [ABSTRACT FROM AUTHOR]

We consider differential operators of type \[ Au(x) &=& u(x) + (-1)^{t_1}\frac{\partial ^{2t_1} u(x)}{\partial x_1^{2t_1}}+ (-1)^{t_2}\frac{\partial ^{2t_2} u(x)}{\partial x_2^{2t_2}} , x&=& (x_1,x_2)\in \R ^2 , \] and Sierpinski carpets $\G \subset \R ^2$ . The aim of the paper is to investigate spectral properties of the fractal differential operator $ A^{-1} \circ tr ^{\Gamma} $ acting in the anisotropic Sobolev space $W^{(t_1,t_2)}_2({\Bbb R}^2)$ where $tr^{\Gamma}$ is closely related to the trace operator $tr_{\Gamma}$ . [ABSTRACT FROM AUTHOR]

Using an appropriate scaling group for the 3-D Schrödinger–Poisson equation and the equivalence between the Schrödinger formalism and the Wigner representation of quantum mechanics it is proved that, when time goes to infinity, the limit of the rescaled self-consistent potential can be identified as the Coulomb potential. As a consequence, Schrödinger–Poisson and Wigner–Poisson systems are asymptotically simplified and their long-time behavior is explained through the solutions of the corresponding linear limit problems. [ABSTRACT FROM AUTHOR]

We propose a new method of nonparametric estimation which is based on locally constant smoothing with an adaptive choice of weights for every pair of data points. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on some simulated univariate and bivariate examples and compare it with other nonparametric methods. Finally we discuss applications of this procedure to magnetic resonance and satellite imaging. [ABSTRACT FROM AUTHOR]

We study parabolic equations of the divergent form with degeneration. We have obtained an estimate for the maximum of modulus of generalized solutions of the first boundary-value problem with a zero on the parabolic boundary. [ABSTRACT FROM AUTHOR]