Besov Function Research Articles

Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions

We study the approximation by tensor networks (TNs) of functions from classical smoothness classes. The considered approximation tool combines a tensorization of functions in $L^p([0,1))$, which allows to identify a univariate function with a multivariate function (or tensor), and the use of tree tensor networks (the tensor train format) for exploiting low-rank structures of multivariate functions. The resulting tool can be interpreted as a feed-forward neural network, with first layers implementing the tensorization, interpreted as a particular featuring step, followed by a sum-product network with sparse architecture. In part I of this work, we presented several approximation classes associated with different measures of complexity of tensor networks and studied their properties. In this work (part II), we show how classical approximation tools, such as polynomials or splines (with fixed or free knots), can be encoded as a tensor network with controlled complexity. We use this to derive direct (Jackson) inequalities for the approximation spaces of tensor networks. This is then utilized to show that Besov spaces are continuously embedded into these approximation spaces. In other words, we show that arbitrary Besov functions can be approximated with optimal or near to optimal rate. We also show that an arbitrary function in the approximation class possesses no Besov smoothness, unless one limits the depth of the tensor network.

Publisher Correction: Differentiability of logarithmic Besov functions in terms of capacities

Publisher Correction: Differentiability of logarithmic Besov functions in terms of capacities

A continuous-parameter Katznelson–Tzafriri theorem for algebras of analytic functions

We prove a continuous-parameter version of a theorem of Katznelson–Tzafriri type for power-bounded operators which have a bounded calculus for analytic Besov functions. We also show that the result can be extended to some operators which have functional c

Differentiability of logarithmic Besov functions in terms of capacities

Differentiability of logarithmic Besov functions in terms of capacities

Nonlinear wavelet-based estimation to spectral density for stationary non-Gaussian linear processes

Nonlinear wavelet-based estimators for spectral densities of non-Gaussian linear processes are considered. The convergence rates of mean integrated squared error (MISE) for those estimators over a large range of Besov function classes are derived, and it is shown that those rates are identical to minimax lower bounds in standard nonparametric regression model within a logarithmic term. Thus, those rates could be considered as nearly optimal. Therefore, the resulting wavelet-based estimators outperform traditional linear methods if the degree of smoothness of spectral densities varies considerably over the interval of interest, such as sharp spike, cusp, bump, etc., since linear estimators are not able to attain these rates. Unlike in classical nonparametric regression with Gaussian noise errors where thresholds are determined by normal distribution, we determine the thresholds based on a Bartlett type approximation of a quadratic form with dependent variables by its corresponding quadratic form with independent identically distributed (i.i.d.) random variables and Hanson-Wright inequality for quadratic forms in sub-gaussian random variables. The theory is illustrated with some numerical examples, and our simulation studies show that our proposed estimators are comparable to the current ones.

FUNCTIONAL CALCULI FOR SECTORIAL OPERATORS AND RELATED FUNCTION THEORY

AbstractWe construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.

The theory of Besov functional calculus: Developments and applications to semigroups

We extend and deepen the theory of functional calculus for semigroup generators, based on the algebra B of analytic Besov functions, which we initiated in a previous paper. In particular, we show that our construction of the calculus is optimal in several natural senses. Moreover, we clarify the structure of B and identify several important subspaces in practical terms. This leads to new spectral mapping theorems for operator semigroups and to wide generalisations of a number of basic results from semigroup theory.

RETRACTED ARTICLE: Art image simulation design of craft products based on virtual reality and human–computer interactive processing

At present, virtual reality technology is the most researched technology in various high-tech industries at home and abroad. Because virtual reality technology can produce three-dimensional animation that is very similar to reality, and with the help of the core technology of the technology simulation system, it provides consumers with an immersive information interaction experience, which greatly reduces the user’s learning of a certain new knowledge. Time and energy invested. One such virtual reality technology was used in the 2010 World Expo Online. Intelligent transportation is a combination of traditional road transportation infrastructure and modern high-tech information technology, and it is an important transportation mode for realizing future vehicle–road coordination. In terms of intelligent transportation, human–computer intelligent interaction technology determines the operating efficiency of the system and the corresponding service quality. In the process of actual use, the researchers developed LCD and audio decoder drivers for the system, and used the UDP protocol to establish a traffic information communication channel between the embedded system and the server, and through the Linux system as the basis. The principle of communication has added the function of human–computer intelligent interaction. This research clarified the basic principles of SMS and the design method of human–computer interaction system based on SMS technology. The researchers established the communication channel between the SMS module and the PC with the help of RS-232 technology, used the Visualc++ program development platform and multi-threaded operation technology to realize the function of sending and receiving messages in real time, and allowed the system to recognize the user's sending according to the set rules Instructions. Applying image recognition technology to actual production, constructing the time digital information image database, characteristic database and corresponding recognition database, this is also a preliminary attempt to apply image recognition technology in the field of stone carving art. Through the establishment of an automatic recognition system for tomb stone carvings in the Southern Song Dynasty, the high-tech image recognition technology is applied to the field of art, and the recognition accuracy of users in the process of recognizing historical pictures is improved, and the efficiency of recognizing historical damaged images is improved. The first is an image recognition zoom algorithm based on variational and wavelet transform. The Besov function is used to create a variational functional for the regularity of the historical image, and then the enlarged historical image is obtained by finding the minimum value of the variational functional, Convert the minimum value of this function to wavelet transform and use the program iteration method to solve; secondly, it is the historical image enlargement algorithm based on the curve and surface evolution, which uses the internal parameters of the system as the standard prior of the historical image Model, perform corresponding calculations on the model, and obtain the evolution equation of the gradient descent flow about the horizontal line of the image.

Multivariate Haar systems in Besov function spaces

We determine all cases for which the -dimensional Haar wavelet system on the unit cube is a conditional or unconditional Schauder basis in the classical isotropic Besov function spaces , , , defined in terms of first-order -moduli of smoothness. We obtain similar results for the tensor-product Haar system , and characterize the parameter range for which the dual of is trivial for . Bibliography: 31 titles.

Uniqueness of Dissipative Solutions to the Complete Euler System

Dissipative solutions have recently been studied as a generalized concept for weak solutions of the complete Euler system. Apparently, these are expectations of suitable measure valued solutions. Motivated from Feireisl et al. (Commun Partial Differ Equ 44(12):1285–1298, 2019), we impose a one-sided Lipschitz bound on velocity component as uniqueness criteria for a weak solution in Besov space $$B^{\alpha ,\infty }_{p}$$ with $$\alpha >1/2$$ . We prove that the Besov solution satisfying the above mentioned condition is unique in the class of dissipative solutions. In the later part of this article, we prove that the one sided Lipschitz condition gives uniqueness among weak solutions with the Besov regularity, $$B^{\alpha ,\infty }_{3}$$ for $$\alpha >1/3$$ . Our proof relies on commutator estimates for Besov functions and the relative entropy method.

Wavelet Thresholding in Fixed Design Regression for Gaussian Random Fields

In this paper, we consider block thresholding wavelet estimators of spatial regression functions on stationary Gaussian random fields observed over a rectangular domain indexed with \({{\mathbb {Z}}}^2\), whose covariance function is assumed to satisfy some weak condition. We investigate their asymptotic rates of convergence under the mean integrated squared error when spatial regression functions belong to a large range of Besov function classes \(B^{s}_{p,q}({{\mathbb {R}}}^2)\). To do this, we derived a result showing the discrepancy between empirical wavelet coefficients and true wavelet coefficients is within certain small rate across above Besov function classes. Based on that, we are able to determine the rates of convergence of our estimators and the supremum norm error over above function classes. The obtained rates of convergence correspond to those established in the standard univariate nonparametric regression with short-range dependence. Therefore, those rates could be considered as sharp as possible. A mild simulation study is carried out to examine the finite sample performance of the proposed estimates.

Nonparametric estimation of a quantile density function under Lp risk via block thresholding method

Here we propose a new quantile density function estimator via block thresholding methods and investigate its asymptotic convergence rates under Lp risk with over Besov balls. We show that the considered estimator achieves optimal or near optimal rates of convergence according to the values of the parameter ν of the Besov classes . We show that this estimator attain optimal and nearly optimal rates of convergence over a wide range of Besov function classes, and in particular enjoys those faster rates without the extraneous logarithmic penalties that given in Chesneau et al. A simulation study shows new proposed estimator performs better at the tails than existing competitors.

On restrictions of Besov functions

In this paper, we study the smoothness of restrictions of Besov functions. It is known that for any f∈Bp,qs(RN) with q≤p we have f(⋅,y)∈Bp,qs(Rd) for a.e. y∈RN−d. We prove that this is no longer true when p<q. Namely, we construct a function f∈Bp,qs(RN) such that f(⋅,y)∉Bp,qs(Rd) for a.e. y∈RN−d. We show that, in fact, f(⋅,y) belong to Bp,q(s,Ψ)(Rd) for a.e. y∈RN−d, a Besov space of generalized smoothness, and, when q=∞, we find the optimal condition on the function Ψ for this to hold. The natural generalization of these results to Besov spaces of generalized smoothness is also investigated.

A trace theorem for Besov functions in spaces of homogeneous type

The aim of this paper is to prove a trace theorem for Besov functions in the metric setting, generalizing a known result from A. Jonsson and H. Wallin in the Euclidean case. We show that the trace of a Besov space defined in a ‘big set’ X is another Besov space defined in the ‘small set’ F ⊂ X. The proof is divided in three parts. First we see that Besov functions in F are restrictions of functions of the same type (but greater regularity) in X, that is we prove an extension theorem and mention examples where this theorem holds. Next, as an auxiliary result that can also be interesting on its own, we show that the interpolation between certain potential spaces gives a Besov space. Finally, to obtain that Besov functions in X can in fact be restricted to F, a restriction theorem, we first prove that this result holdsfor functions in the potential space, and then by the interpolation result previously shown, it must hold in the Besov case. For the interpolation and restriction theorems, we make additional assumptions on the spaces X and F, and on the order of regularity of the functions involved. We include an interesting example of our trace theorem, not covered by the classical one.

Method of fundamental solutions for mixed and crack type problems in the classical theory of elasticity

We analyse some new aspects concerning application of the fundamental solution method to the basic three-dimensional boundary value problems, mixed transmission problems, and also interior and interfacial crack type problems for steady state oscillation equations of the elasticity theory. First we present existence and uniqueness theorems of weak solutions and derive the corresponding norm estimates in appropriate function spaces. Afterwards, by means of the columns of Kupradze’s fundamental solution matrix special systems of vector functions are constructed explicitly. The linear independence and completeness of these systems are proved in appropriate Sobolev–Slobodetskii and Besov function spaces. It is shown that the problem of construction of approximate solutions to the basic and mixed boundary value problems and to the interior and interfacial crack problems can be reduced to the problems of approximation of the given boundary vector functions by elements of the linear spans of the corresponding complete systems constructed by the fundamental solution vectors. By this approach the approximate solutions of the boundary value and transmission problems are represented in the form of linear combinations of the columns of the fundamental solution matrix with appropriately chosen poles distributed outside the domain under consideration. The unknown coefficients of the linear combinations are defined by the approximation conditions of the corresponding boundary and transmission data.

Orlicz spaces and the large scale geometry of Heintze groups

We consider an Orlicz space based cohomology for metric (measured) spaces with bounded geometry. We prove the quasi-isometry invariance for a general Young function. In the hyperbolic case, we prove that the degree one cohomology can be identified with an Orlicz–Besov function space on the boundary at infinity. We give some applications to the large scale geometry of homogeneous spaces with negative curvature (Heintze groups). As our main result, we prove that if the Heintze group is not of Carnot type, any self quasi-isometry fixes a distinguished point on the boundary and preserves a certain foliation on the complement of that point.

Wavelet-based estimation of regression function with strong mixing errors under fixed design

ABSTRACTWe consider wavelet-based non linear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes Bsp, q. The theory is illustrated with some numerical examples.A new ingredient in our development is a Bernstein-type exponential inequality, for a sequence of random variables with certain mixing structure and are not necessarily bounded or sub-Gaussian. This moderate deviation inequality may be of independent interest.

Wavelets‐based smoothness comparisons for volume data

In this study, the authors describe an objective smoothness assessment method for volume data. The metric can predict the extent of the difference in smoothness between a reference model, which may not be of perfect quality, and a distorted version. The proposed metric is based on the wavelet characterisation of Besov function spaces. The comparison of Besov norms between two models can resolve the global and local differences in smoothness between them. Experimental results from volume datasets with smoothing and sharpening operations demonstrate its effectiveness. By comparing direct volume rendered images, the experimental results show that the proposed smoothness index correlates well with human perceived vision. Finally, the metric can help the analyse compression distortions when they compare volume data with different smoothness.

Compactness of embeddings of function spaces on quasi-bounded domains and the distribution of eigenvalues of related elliptic operators. Part II

We prove the asymptotic behaviour of eigenvalues of elliptic self-adjoint differential operators defined on a wide class of quasi-bounded domains. The estimates are based on corresponding asymptotic behaviour of entropy numbers of Sobolev embeddings of Sobolev and Besov function spaces defined on the quasi-bounded domains. We consider also the inverse problem i.e. we identify the class of functions that can describe the asymptotic behaviour of eigenvalues of Dirichlet Laplacian of some quasi-bounded domain.

Nonparametric regression on random fields with random design using wavelet method

We consider non-linear wavelet-based estimators of spatial regression functions with (known) random design on strictly stationary random fields, which are indexed by the integer lattice points in the \(N\)-dimensional Euclidean space and are assumed to satisfy some mixing conditions. We investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients and show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, wavelet estimators still achieve nearly optimal convergence rates for random fields and provide explicitly the extraordinary local adaptability.