Hyperbolic wavelet thresholding methods and the curse of dimensionality through the maxiset approach

In this paper we compute the maxisets of some denoising methods (estimators) for multidimensional signals based on thresholding coefficients in hyperbolic wavelet bases. That is, we determine the largest functional space over which the risk of these estimators converges at a chosen rate. In the unidimensional setting, refining the choice of the coefficients that are subject to thresholding by pooling information from geometric structures in the coefficient domain (e.g., vertical blocks) is known to provide ‘large maxisets’. In the multidimensional setting, the situation is less straightforward. In a sense these estimators are much more exposed to the curse of dimensionality. However we identify cases where information pooling has a clear benefit. In particular, we identify some general structural constraints that can be related to compound functional models and to a minimal level of anisotropy.     

Bayesian Variable Selection on Model Spaces Constrained by Heredity Conditions

This article investigates Bayesian variable selection when there is a hierarchical dependence structure on the inclusion of predictors in the model. In particular, we study the type of dependence found in polynomial response surfaces of orders two and higher, whose model spaces are required to satisfy weak or strong heredity conditions. These conditions restrict the inclusion of higher-order terms depending upon the inclusion of lower-order parent terms. We develop classes of priors on the model space, investigate their theoretical and finite sample properties, and provide a Metropolis–Hastings algorithm for searching the space of models. The tools proposed allow fast and thorough exploration of model spaces that account for hierarchical polynomial structure in the predictors and provide control of the inclusion of false positives in high posterior probability models.     

Metrizably fibered generalized ordered spaces

In this paper we characterize generalized ordered spaces that are metrizably fibered in terms of certain quotient spaces and in terms of the existence of special open covers. We apply our results to give a new characterization of perfect generalized ordered spaces that have a σ-closed-discrete dense subset and to give examples of GO-spaces that are, or are not, metrizably fibered.     

Singular Integral Operators on Besov Spaces

Summary. We introduce generalized BESOV spaces in terms of mean oscillation and weight functions, following a recent work of Dorronsoro, and study the continuity of singular integral operators on them. Relations between these spaces and the BESOV spaces in terms of modulus of continuity are also studied. An application to pseudo-differential operators is given.     

Sharp Békollé estimates for the Bergman projection

We prove sharp estimates for the Bergman projection in weighted Bergman spaces in terms of the Békollé constant. Our main tools are a dyadic model dominating the operator and an adaptation of a method of Cruz-Uribe, Martell and Pérez.     

Characterizations of Banach spaces whose duals areL 1 spaces

We characterize those Banach spaces whose duals are isometric toL ₁ spaces in terms of the structure of the spaces of absolutely summing, integral, nuclear, and fully nuclear operators from or into these spaces.     

Product of Volterra Type Integral and Composition Operators on Weighted Fock Spaces

We characterize the bounded, compact, and Schatten class product of Volterra type integral and composition operators acting between weighted Fock spaces. Our results are expressed in terms of certain Berezin type integral transforms on the complex plane ?. We also estimate the norms and essential norms of these operators in terms of the integral transforms. All our results are valid for weighted composition operators when acting between the class of weighted Fock spaces considered.     

Points de namioka espaces normants applications à la théorie isométrique de la dualité

We study in this work various aspects of the isometric theory of duality. We show that in wide classes of Banach spaces, dual spaces are characterized by the existence of a retraction fromE″ ontoE. The predual of such spaces is then unique. We study the imbedding of regularly normed spaces into dual spaces. We better the known results on loss of regularity of the norm of dual spaces. We characterize the dual norms on an Asplund space in terms of “bad differentiability”.      

Matroidal approaches to rough sets via closure operators

This paper studies rough sets from the operator-oriented view by matroidal approaches. We firstly investigate some kinds of closure operators and conclude that the Pawlak upper approximation operator is just a topological and matroidal closure operator. Then we characterize the Pawlak upper approximation operator in terms of the closure operator in Pawlak matroids, which are first defined in this paper, and are generalized to fundamental matroids when partitions are generalized to coverings. A new covering-based rough set model is then proposed based on fundamental matroids and properties of this model are studied. Lastly, we refer to the abstract approximation space, whose original definition is modified to get a one-to-one correspondence between closure systems (operators) and concrete models of abstract approximation spaces. We finally examine the relations of four kinds of abstract approximation spaces, which correspond exactly to the relations of closure systems.     

Trivariate Cr Polynomial Macroelements

Trivariate C^r macroelements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1,2, these spaces reduce to well-known macroelement spaces used in data fitting and in the finite-element method. We determine the dimension of these spaces, and describe stable local minimal determining sets and nodal minimal determining sets. We also show that the spaces approximate smooth functions to optimal order.     

The characteristic of convexity of a Banach space and normal structure

We present some sufficient conditions for normal structure of Banach spaces and their dual spaces in terms of the characteristic of convexity, the James constant, and the coefficient of weak orthogonality. Many known results are improved and strengthened. We also show that some of our results are sharp.     

Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation

Restricted non-linear approximation is a type of N-term approximation where a measure ν on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel–Lizorkin spaces by showing the Temlyakov property in this setting. Moreover, new interpolation results for Triebel–Lizorkin and Besov spaces are obtained.     

Besov-Morrey spaces: Function space theory and applications to non-linear PDE

This paper is devoted to the analysis of function spaces modeled on Besov spaces and their applications to non-linear partial differential equations, with emphasis on the incompressible, isotropic Navier-Stokes system and semi-linear heat equations. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call Besov-Morrey or BM spaces. We obtain equivalent representations in terms of the Weierstrass semigroup and wavelets, and various embeddings in classical spaces. We then establish pseudo-differential and para-differential estimates. Our results cover non-regular and exotic symbols. Although the heat semigroup is not strongly continuous on Morrey spaces, we show that its action defines an equivalent norm. In particular, homogeneous BM spaces belong to a larger class constructed by Grzegorz Karch to analyze scaling in parabolic equations. We compare Karch's results with those of Kozono and Yamazaki and generalize them by obtaining short-time existence and uniqueness of solutions for arbitrary data with subcritical regularity. We exploit pseudo-differential calculus to extend the analysis to compact, smooth, boundaryless, Riemannian manifolds. BM spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator.

     

Anisotropic Mixed-Norm Hardy Spaces

We introduce and explore Hardy spaces defined by mixed Lebesgue norms and anisotropic dilations. We prove that the definitions of these spaces in terms of smooth, non-tangential, auxiliary, grand, and Poisson maximal operators coincide. We also study the relation between anisotropic mixed-norm Hardy spaces and mixed-norm Lebesgue spaces.     

On Lipschitz-type maximal functions and their smoothness spaces

In a recent monograph (cf. No. 293 of the Memoirs of the Amer. Math. Soc. 47 (1984)) DeVore and Sharpley study maximal functions of integral type and their related smoothness spaces. One of their central results gives an embedding theorem for the smoothness spaces in terms of Besov spaces. In this paper we consider the related problem when the Besov spaces are substituted by the so-called A-spaces introduced by Popov (take the τ-modulus instead of the ω-modulus). We will define Lipschitz-type maximal functions whose smoothness spaces satisfy a corresponding embedding theorem in terms of A-spaces. By well-known results new insights can only be expected for functions satisfying low order smoothness conditions and, therefore, only function spaces generated by first order differences are considered.     

Besov Spaces with General Weights

We introduce Besov spaces with general smoothness. These spaces unify and generalize the classical Besov spaces. We establish the $\varphi $-transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev embeddings. We establish the smooth atomic, molecular and wavelet decomposition of these function spaces. A characterization of these function spaces in terms of the difference relations is given.     

On the dirichlet problem for an improperly elliptic equation

We consider the problem of solvability of an inhomogeneous Dirichlet problem for a scalar improperly elliptic differential equation with complex coefficients in a bounded domain. A model case where the unit disk is chosen as the domain and the equation does not contain lower terms is studied. We prove that the classes of Dirichlet data for which the problem has a unique solution in the Sobolev space are spaces of functions with exponentially decreasing Fourier coefficients.