Besov Classes Research Articles

Average Widths and Optimal Recovery of Multivariate Besov Classes in Orlicz Spaces

In this paper, we study the average Kolmogorov σ–widths and the average linear σ–widths of multivariate isotropic and anisotropic Besov classes in Orlicz spaces and give the weak asymptotic estimates of these two widths. At the same time, we also give the asymptotic property of the optimal recovery of isotropic Besov classes in Orlicz spaces.

Solving a Dirichlet problem on unbounded domains via a conformal transformation

In this paper, we solve the p-Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure satisfying a Poincaré inequality. This is accomplished by studying a class of transformations that have been recently shown to render the domain bounded while maintaining uniformity. These transformations conformally deform the metric and measure in a way that depends on the distance to the boundary of the domain and, for the measure, a parameter p. We show that the transformed measure is doubling and the transformed domain supports a Poincaré inequality. This allows us to transfer known results for bounded uniform domains to unbounded ones, including trace results and Adams-type inequalities, culminating in a solution to the Dirichlet problem for boundary data in a Besov class.

Estimates for the entropy numbers of the Nikol'skii–Besov classes of functions with mixed smoothness in the space of quasi‐continuous functions

AbstractWe obtained order estimates for the entropy numbers of the Nikol'skii–Besov classes of functions with mixed smoothness in the metric of the space of quasi‐continuous functions . We also showed that for , , , , the estimate of the corresponding asymptotic characteristic is exact in order.

Complexity of Monte Carlo integration for Besov classes on the unit sphere

Complexity of Monte Carlo integration for Besov classes on the unit sphere

Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball

We consider the numerical integrationINTd(f)=∫Bdf(x)wμ(x)dx for the weighted Sobolev classes BWp,μr and the weighted Besov classes BBτr(Lp,μ) in the randomized case setting, where wμ,μ≥0, is the classical Jacobi weight on the ball Bd, 1≤p≤∞, r>(d+2μ)/p, and 0<τ≤∞. For the above two classes, we obtain the orders of the optimal quadrature errors in the randomized case setting are n−r/d−1/2+(1/p−1/2)+. Compared to the orders n−r/d of the optimal quadrature errors in the deterministic case setting, randomness can effectively improve the order of convergence when p>1.

Kolmogorov Widths of the Nikol’skii–Besov Classes of Periodic Functions of Many Variables in the Space of Quasicontinuous Functions

Kolmogorov Widths of the Nikol’skii–Besov Classes of Periodic Functions of Many Variables in the Space of Quasicontinuous Functions

Estimate for the Order of Orthoprojection Width of the Nikol’skii–Besov Class in the Metric of Anisotropic Lorentz Spaces

Estimate for the Order of Orthoprojection Width of the Nikol’skii–Besov Class in the Metric of Anisotropic Lorentz Spaces

Distributed nonparametric function estimation: Optimal rate of convergence and cost of adaptation

Distributed minimax estimation and distributed adaptive estimation under communication constraints for Gaussian sequence model and white noise model are studied. The minimax rate of convergence for distributed estimation over a given Besov class, which serves as a benchmark for the cost of adaptation, is established. We then quantify the exact communication cost for adaptation and construct an optimally adaptive procedure for distributed estimation over a range of Besov classes. The results demonstrate significant differences between nonparametric function estimation in the distributed setting and the conventional centralized setting. For global estimation, adaptation in general cannot be achieved for free in the distributed setting. The new technical tools to obtain the exact characterization for the cost of adaptation can be of independent interest.

Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space (X,dX,μX) satisfying a 2-Poincaré inequality. Given a bounded domain Ω⊂X with μX(X∖Ω)>0, and a function f in the Besov class B2,2θ(X)∩L2(X), we study the problem of finding a function u∈B2,2θ(X) such that u=f in X∖Ω and Eθ(u,u)≤Eθ(h,h) whenever h∈B2,2θ(X) with h=f in X∖Ω. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on Ω, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extends the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

With a view toward fractal spaces, by using a Korevaar–Schoen space approach, we introduce the class of bounded variation (BV) functions in the general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry–Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $$L^1$$ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in $$L^1$$ . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

Kolmogorov Widths of the Besov Classes $$B^1_{1,\theta}$$ and Products of Octahedra

We find the decay orders of the Kolmogorov widths of some Besov classes related to $$W^1_1$$ (the behavior of the widths for the class $$W^1_1$$ remains unknown): $$d_n(B^1_{1,\theta}[0,1],L_q[0,1])\asymp n^{-1/2}\log^{\max\{1/2,1-1/\theta\}}n$$ for $$2<q<\infty$$ and $$1\le\theta\le\infty$$ . The proof relies on the lower bound for the width of a product of octahedra in a special norm (maximum of two weighted $$\ell_{q_i}$$ norms). This bound generalizes B. S. Kashin’s theorem on the widths of octahedra in $$\ell_q$$ .

Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol’skii–Besov spaces

We have obtained the exact-order estimates for some approximative characteristics of the Sobolev classes $$ {\mathbbm{W}}_{p,\alpha}^r $$ and Nikоl’skii–Besov classes $$ {\mathbbm{B}}_{p,\theta}^r $$ of periodic functions of one and several variables in the norm of the space B∞,1. Properties of the linear operators realizing the orders of the best approximation for the classes $$ {\mathbbm{B}}_{\infty, \theta}^r $$ in this space by trigonometric polynomials generated by a set of harmonics with “numbers” from step hyperbolic crosses are investigated.

Minimax bounds for Besov classes in density estimation

We study the problem of density estimation on [0,1] under Lp norm. We carry out a new piecewise polynomial estimator and prove that it is simultaneously (near)-minimax over a very wide range of Besov classes Bπ,∞α(R). In particular, we may deal with unbounded densities and shed light on the minimax rates of convergence when π<p and α∈(1∕π−1∕p,1∕π].

Estimates of the best approximations of the functions of the Nikol’skii–Besov class in the generalized space of Lorentz

In this paper, we consider the generalized Lorentz space of periodic functions of several variables and the Nikol’skii–Besov space of functions. The article establishes a sufficient condition for a function to belong from one generalized Lorentz space to another space in terms of the difference of the partial sums of the Fourier series of a given function. Exact in order estimates of the best approximation by trigonometric polynomials of functions of the Nikol’skii–Besov class are obtained.

Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol'skii-Besov spaces

We have obtained the exact-order estimates for some approximative characteristics of the Sobolev classes $\mathbb{W}^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}$ and Nikоl'skii--Besov classes $\mathbb{B}^{\boldsymbol{r}}_{p,\theta}\ $ of periodic functions of one and several variables in the norm of the space $B_{\infty, 1}$. Properties of the linear operators realizing the orders of the best approximation for the classes $\mathbb{B}^{\boldsymbol{r}}_{\infty, \theta}$ in this space by trigonometric polynomials generated by a set of harmonics with $``$numbers$"$ from step hyperbolic crosses are investigated.

Functions of perturbed pairs of noncommuting contractions

We consider functions of pairs of noncommuting contractions on Hilbert space and study the problem as to which functions we have Lipschitz type estimates in Schatten–von Neumann norms. We prove that if belongs to the Besov class of analytic functions in the bidisc, then we have a Lipschitz type estimate for functions of pairs of not necessarily commuting contractions in the Schatten–von Neumann norms for . On the other hand, we show that for functions in , there are no such Lipschitz type estimates for , nor in the operator norm.

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

With a view toward fractal spaces, by using a Korevaar–Schoen space approach, we introduce the class of bounded variation (BV) functions in the general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry–Emery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $$L^1$$ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in $$L^1$$ . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

Randomized approximation numbers on Besov classes with mixed smoothness

We study the Kolmogorov and the linear approximation numbers of the Besov classes [Formula: see text] with mixed smoothness in the norm of [Formula: see text] in the randomized setting. We first establish two discretization theorems. Then based on them, we determine the exact asymptotic orders of the Kolmogorov and the linear approximation numbers for certain values of the parameters [Formula: see text]. Our results show that the linear randomized methods lead to considerably better rates than those of the deterministic ones for [Formula: see text].

Functions of noncommuting operators under perturbation of class Sp

AbstractIn this article we prove that for , there exist pairs of self‐adjoint operators and and a function f on the real line in the homogeneous Besov class such that the differences and belong to the Schatten–von Neumann class Sp but . A similar result holds for functions of contractions. We also obtain an analog of this result in the case of triples of self‐adjoint operators for any .

Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities

We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. As a highlight of the paper, we obtain a far reaching Lp-analogue, p≥1, of the Sobolev inequality that was proved for p=2 by N. Varopoulos under the assumption of ultracontractivity for the heat semigroup. The case p=1 is of special interest since it yields isoperimetric type inequalities.