On Banach Spaces Whose Unit Sphere Determines Polynomials

In this paper we study the problem of characterizing the real Banach spaces whose unit sphere determines polynomials, i.e., if two polynomials coincide in the unit sphere, is this sufficient to guarantee that they are identical？ We show that, in the frame of spaces with unconditional basis, non- reflexivity is a sufficient, although not necessary, condition for the above question to have an affirmative answer. We prove that the only lp^n spaces having this property are those with p irrational, while the only lp spaces which do not enjoy it are those with p an even integer. We also introduce a class of polynomial determining sets in any real Banach space.     

Wilson Bases and Modulation Spaces

We show that the recently discovered WILSON bases of exponential decay are unconditional bases for all modulation spaces on R, including the classical BESSEL potential spaces, the Segal algebra S_o, and the SCHWARTZ space. As a consequence we obtain new bases for spaces of entire functions. On the other hand, the WILSON bases are no unconditional bases for the ordinary L^p-spaces for p ≠ 2.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces

We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then give applications of these results to Hardy spaces by showing the spacesH _p (T ⁿ ) are mutually non-isomorphic for differing values ofn when 0<p<1. The research of the first two authors was partially supported by NSF-grant DMS 8901636.     

Ergodic Banach spaces

We show that any Banach space contains a continuum of non-isomorphic subspaces or a minimal subspace. We define an ergodic Banach space X as a space such that E₀ Borel reduces to isomorphism on the set of subspaces of X, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results. We show that an unconditional basis of a Banach space, of which every block-subspace is complemented, must be asymptotically c₀ or ?_p, and we deduce some new characterisations of the classical spaces c₀ and ?_p.     

The Haar wavelet characterization of weighted Herz spaces and greediness of the Haar wavelet basis

In this paper we consider the Haar wavelet on weighted Herz spaces. Our weight class, whose name is Ap-dyadic local, is the one defined by the first author (2007). We shall investigate the class of Ap-dyadic weights in connection with the maximal inequalities. After obtaining the properties of weights in the first half of the present paper, we consider the Haar wavelet on weighted Herz spaces in the latter half. We shall show that the Haar wavelet basis is an unconditional basis. We also show that the Haar wavelet is not greedy except for the trivial case, that is, the Haar wavelet is greedy if and only if the Herz space under consideration is a weighted Lp space.     

Jackson and smoothness theorems for freud weights

We prove Jackson, realization, and converse theorems for Freud weights inL _p, 0<p ≤ ∞. Even forp ≥ 1, our conditions on the weight in our Jackson theorems are far less restrictive than those previously imposed. Moreover, the method—of first approximating by a spline, and then by a polynomial—is new in this context, and of intrinsic interest, since it avoids the use of orthogonal polynomials for Freud weights. We establish some properties of the modulus of smoothness, valid inL _p for 0<p ≤ ∞. Since theK-functional is identically zero inL _p,p<1, the analysis of the modulus of continuity involves a different tool, namely, realization, which works inL _p for all 0<p ≤ ∞. We deduce Marchaud-type inequalities.     

Smoothness and Best Approximation with Respect to Freud-Type Weights,a New Approach

A new measure of smoothness is defined and related to best approximation by polynomials with respect to weighted L _p (R) with Freud-type weights. Other related norms are also discussed. Comparisons with the known measure of smoothness on weighted L _p spaces are obtained. Related K-functionals and realization functionals are introduced. The new measure of smoothness allows us to consider a more general class of function spaces, to achieve Marchaud, Jackson and Bernstein-type inequalities, and to relate it to expressions involving the coefficients of the expansion by orthogonal polynomials with respect to Freud-type weights. Some of the results are new for approximation by Hermite polynomials in the weighted L _p space with the weight \({e^{-x^{2}}}\) .     

Mean convergence of an extended Lagrange interpolation process on [0,+∞)

We study some extended Lagrange interpolation processes based on the zeros of the generalized Laguerre polynomials. We give necessary and sufficient conditions such that the convergence of these processes, in suitable L _p weighted spaces on the real semiaxis, is assured for 1<p<+∞.     

Holomorphic Jackson's theorems in polydiscs

The purpose of this article is to establish Jackson-type inequality in the polydiscs U^N of for holomorphic spaces X, such as Bergman-type spaces, Hardy spaces, polydisc algebra and Lipschitz spaces. Namely,

where is the deviation of the best approximation of fX by polynomials of degree at most k_j about the jth variable z_j with respect to the X-metric and is the corresponding modulus of continuity.     

An Extension of the Notion of Orthogonality to Banach Spaces

We extend the usual notion of orthogonality to Banach spaces. We show that the extension is quite rich in structure by establishing some of its main properties and consequences. Geometric characterizations and comparison results with other extensions are established. Also, we establish a characterization of compact operators on Banach spaces that admit orthonormal Schauder bases. Finally, we characterize orthogonality in the spaces l₂^p(C).     

Chaotic dynamics of the heat semigroup on the Damek-Ricci spaces

We consider the rank one Riemannian symmetric spaces of noncompact type and their non-symmetric generalization, namely the Damek-Ricci spaces. We show that the heat semigroup generated by a certain perturbation of the Laplace-Beltrami operator of these spaces is chaotic on their L _p -spaces when p > 2. The range of p and the corresponding perturbation are sharp. A precursor to this result is due to Ji and Weber [19] where it was shown that under identical conditions the heat operator is subspace-chaotic on the Riemannian symmetric spaces, which is weaker than it being chaotic. We also extend the results to the Lorentz spaces L _p,q , which are generalizations of the Lebesgue spaces. This enables us to point out that the chaoticity degenerates to subspace-chaoticity only when q = ∞.     

Unconditional reproducing kernel bases in de Branges spaces

We describe unconditional bases of the form {k(z,λ _n ):λ _n ∈Λ}, Λ∩?=? in de Branges spaces, where k is the reproducing kernel.     

Discrete time analytic semigroups and the geometry of Banach spaces

We study different notions of discrete maximal regularity for discrete-time abstract Cauchy problems in Banach spaces. First we look at l ₂-discrete maximal regularity and show that Hilbert spaces are the only Banach spaces, among spaces with an unconditional basis, in which the analyticity of the associated discrete-time semigroup is a sufficient condition to obtain this kind of regularity. We then turn to different notions of regularity, in a l ₁ and in a l _∞ sense. We link the existence of particular semigroups such that the associated Cauchy problem has one of these maximal regularities to the geometry of the underlying Banach space (more precisely, to the existence of a complemented subspace isomorphic to c ₀ or l ₁). Finally, we give some elements to compare these regularities.     

Wavelet bases in the Lebesgue spaces on the field of p-adic numbers

In this paper we prove that p-adic wavelets form an unconditional basis in the space L ^r (? _p ⁿ ) and give the characterization of the space L ^r (? _p ⁿ ) in terms of Fourier coefficients of p-adic wavelets.Moreover, the Greedy bases in the Lebesgue spaces on the field of p-adic numbers are also established.     

Vector-valued multiparameter singular integrals and pseudodifferential operators

We consider multiparameter singular integrals and pseudodifferential operators acting on mixed-norm Bochner spaces Lp₁,…,pN(Rn₁×?×RnN;X) where X is a UMD Banach space satisfying Pisier's property (α). These geometric conditions are shown to be necessary. We obtain a vector-valued version of a result by R. Fefferman and Stein, also providing a new, inductive proof of the original scalar-valued theorem. Then we extend a result of Bourgain on singular integrals in UMD spaces with an unconditional basis to a multiparameter situation. Finally we carry over a result of Yamazaki on pseudodifferential operators to the Bochner space setting, improving the known vector-valued results even in the one-parameter case.     

On the primariness of the Poulsen simplex space

We show that the Poulsen simplex space is primary by investigating special monotone bases inL ₁-predual spaces.     

On the convergence of random polynomials and multilinear forms

We consider different kinds of convergence of homogeneous polynomials and multilinear forms in random variables. We show that for a variety of complex random variables, the almost sure convergence of the polynomial is equivalent to that of the multilinear form, and to the square summability of the coefficients. Also, we present polynomial Khintchine inequalities for complex gaussian and Steinhaus variables. All these results have no analogues in the real case. Moreover, we study the Lp-convergence of random polynomials and derive certain decoupling inequalities without the usual tetrahedral hypothesis. We also consider convergence on “full subspaces” in the sense of Sjögren, both for real and complex random variables, and relate it to domination properties of the polynomial or the multilinear form, establishing a link with the theory of homogeneous polynomials on Banach spaces.     

Gaussian characterizations of certain Banach spaces

The general form of characteristic functionals of Gaussian measures in spaces of type 2 and cotype 2 is found. Under the condition of existence of an unconditional basis this problem is solved for spaces not containing l_∞ⁿ uniformly. The solution uses the language of absolutely summing operators. For each of mentioned space classes it is shown that the results hold in them only. We consider also the equivalent problems on extension of a weak Gaussian distribution and convergence of Gaussian series. Some limit theorems are formulated.     

GENERALIZED WEIGHTED SOBOLEV SPACES AND APPLICATIONS TO SOBOLEV ORTHOGONAL POLYNOMIALS Ⅱ

We present a definition of general Sobolev spaces with respect to arbitrary measures ,W^k,p(Ω,μ) for 1≤p≤∞,In[RARP] we proved that these spaces are complete under very light conditions.Now we prove that if we consider certain general types of measures,then Cc^∞(R) is dense in these spaces,As an application to Sobolev orthogonal polynomials,we study the boundedness of the multiplication poerator,THis gives an estimation of the zeroes of Sobolev orthogonal polynomials.