Characterization Of Besov Spaces Research Articles

Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

For both localized and periodic initial data, we prove local existence in classical energy space H^{s}, s>frac{3}{2}, for a class of dispersive equations u_t + (n(u))_x + L u_x = 0 with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators L whose symbol is of temperate growth, and n(cdot ) in the local Sobolev space H^{s+2}_{mathrm {loc}}(mathbb {R}). In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semigroup methods and a new composition result for Besov spaces. In particular, we extend a previous result for Nemytskii operators on Besov spaces on mathbb {R} to the periodic setting by using the difference–derivative characterization of Besov spaces.

Algebra properties for Besov spaces on unimodular Lie groups

We consider the Besov space $B^{p,q}_\alpha (G)$ on a unimodular Lie group $G$ equipped with a sublaplacian $\varDelta $. Using estimates of the heat kernel associated with $\varDelta $, we give several characterizations of Besov spaces, and show an

A Characterization of Besov Spaces of Para-Accretive Type and Its Application

There are two folds in this article. One fold is to characterize the Besov spaces of para-accretive type , which reduces to the classical Besov spaces when the para-accretive function is constant, by using a discrete Calderon-type reproducing formula and Plancherel-Polya-type inequality associated to a para-accretive function b in Rn. The other is to show that a generalized singular integral operator T with extends to be bounded from for and , where e is the regularity exponent of the kernel of T.

Approximation based on elliptic splines

In this paper, we study the elliptic splines which include the well-known polyharmonic B-splines. We analyze their Fourier transforms, decay behaviors and polynomial reproducing properties. We also study the order of approximation in Sobolev spaces and consider their characterizations of Besov spaces by the scale projection operators, quasi-interpolation operators and wavelet operators.

Besov regularity for the Stokes and the Navier‐Stokes system in polyhedral domains

In this paper we study the regularity of solutions to the Stokes and the Navier‐Stokes system in polyhedral domains contained in ℝ3. We consider the scale Bsτ(Lτ), 1/τ = s/3 + 1/2 of Besov spaces which determines the approximation order of adaptive numerical wavelet schemes and other nonlinear approximation methods. We show that the regularity in this scale is large enough to justify the use of adaptive methods. The proofs of the main results are performed by combining regularity results in weighted Sobolev spaces with characterizations of Besov spaces by wavelet expansions.

Characterization of Besov Spaces on Nested Fractals by Piecewise Harmonic Functions

In the present paper we characterize the Besov spaces B^s_{pq} (\Gamma, \mu) on nested fractals in terms of the coefficients of functions with respect to the piecewise harmonic basis.

Interpolatory curl-free wavelets on bounded domains and characterization of Besov spaces

Based on interpolatory Hermite splines on rectangular domains, the interpolatory curl-free wavelets and its duals are first constructed. Then we use it to characterize a class of vector-valued Besov spaces. Finally, the stability of wavelets that we constructed are studied.MR(2000) Subject Classification: 42C15; 42C40.

CHARACTERIZATIONS OF BESOV SPACES IN THE UNIT BALL

In this paper we obtain some new characterization of Besov spaces on the unit ball of Cn. These characterization are also completely new even in the settings of the unit disk.

$L^p$ Bernstein estimates and approximation by spherical basis functions

The purpose of this paper is to establish L p error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L p Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L p norm of the function itself. An important step in its proof involves measuring the L p stability of functions in the approximating space in terms of the l p norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the L P norm. Finally, we give a new characterization of Besov spaces on the n-sphere in terms of spaces of SBFs.

First order [formula omitted]-variations and Besov spaces

Based on the notion of first order dyadic p -variation, we give a new characterization of Besov spaces B p , q s ( [ 0 , 1 ] ) for 0 < s < 1 , 1 ≤ p , q ≤ + ∞ and s > 1 / p . We also give results in the case where p < 1 . Hence we provide simple tools that enable us to derive new regularity properties for the trajectories of various continuous time stochastic processes.

Intrinsic characterizations of Besov spaces on Lipschitz domains

AbstractThe aim of this paper is to study the equivalence between quasi‐norms of Besov spaces on domains. We suppose that the domain Ω ⊂ ℝn is a bounded Lipschitz open subset in ℝn. First, we define Besov spaces on Ω as the restrictions of the corresponding Besov spaces on ℝn. Then, with the help of equivalent and intrinsic characterizations (the Peetre‐type characterization 3.10 and the characterization via local means 3.13) of these spaces, we get another equivalent and intrinsic quasi‐norm using, this time, generalized differences and moduli of smoothness. We extend the well‐known characterization of Besov spaces on ℝn described in Theorem 2.4 to the case of Lipschitz domains.

Besov regularity of edge singularities for the Poisson equation in polyhedral domains

AbstractThis paper is concerned with the regularity of the solutions to the Poisson equation in polyhedral domains Ω contained in R3. Especially, we consider the specific scale Bτα(Lτ(Ω)),1/τ = α/3 + ½, of Besov spaces. The regularity of the variational solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and non‐linear numerical schemes. It is well‐known that in polyhedral domains different types of singularities according to edges and vertices occur. In this paper, we shall primarily be concerned with the Besov regularity of edge singularities. By the usual reduction procedure, these singularity functions are studied in an unfinite cylinder R × ϒ, where ϒ denotes a bounded polygonal domain contained in R2. We show that the singularity functions are much smoother in the specific Besov scale than in the usual L2–Sobolov scale which justifies the use of adaptive schemes. The proofs are based on specific representations of the solutions and on characterizations of Besov spaces by wavelet expansions. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Besov regularity for elliptic boundary value problems in polygonal domains

This paper is concerned with the regularity of the solutions to elliptic boundary value problems in polygonal domains Ω contained in R 2. Especially, we consider the specific scale B τ α(L τ(Ω)) , 1 τ = α 2 + 1 p , of Besov spaces. The regularity of the variational solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. The proofs are based on specific representations of the solutions which were, e.g., derived by Grisvard [1], and on characterizations of Besov spaces by wavelet expansions.

Besov Regularity for Interface Problems

This paper is concerned with the Besov regularity of the solutions to interface problems in a sector S of the unit disc in R2. We investigate the smoothness of the solutions as measured in the specific scale Bsτ(Lτ(S)), 1/τ = s/2 + 1/p, of Besov spaces which determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. The proofs are based on representations of the solution spaces which were derived by Kellogg [15] and on characterizations of Besov spaces by wavelet expansions. In der vorliegenden Arbeit beschäftigen wir uns mit der Besov-Regularität der Lösungen von Interface-Problemen in einem Sektor S der Einheitskreisscheibe im R2. Wir untersuchen die Glattheit der Lösungen in der speziellen Besov-Skala Bsτ(Lτ(S)), 1/τ = s/2 + 1/p, welche die Approximationsordnung bestimmt, die mittels adaptiver numerischer Verfahren maximal erreicht werden kann. Die Beweise beruhen auf Darstellungen der Lösungsräume, die von Kellogg [15] hergeleitet worden waren, und auf Charakterisierungen von Besov-Räumen mittels Wavelet-Entwicklungen.

Besov regularity for second order elliptic boundary value problems with variable coefficients

This paper is concerned with some theoretical foundations for adaptive numerical methods for elliptic boundary value problems. The approximation order that can be achieved by such an adaptive method is determined by certain Besov regularity of the weak solution. We study Besov regularity for second order elliptic problems in bounded domains in ℝ d . The investigations are based on intermediate Schauder estimates and on some potential theoretic framework. Moreover, we use characterizations of Besov spaces by wavelet expansions.

Multilevel preconditioning

This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the Bramble-Pasciak-Xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shift-invariant spaces, in particular, covering those induced by various types of wavelets.