Convolution kernels based on thin-plate splines

Quasi-interpolation using radial basis functions has become a popular method for constructing approximations to continuous functions in many space dimensions. In this paper we discuss a procedure for generating kernels for quasi-interpolation, using functions which have series expansions involving terms liker logr. It is shown that such functions are suitable if and only if is a positive even integer and the spatial dimension is also even.     

Kernel based approximation in Sobolev spaces with radial basis functions

In this paper, we study several radial basis function approximation schemes in Sobolev spaces. We obtain an optional error estimate by using a class of smoothing operators. We also discussed sufficient conditions for the smoothing operators to attain the desired approximation order. We then construct the smoothing operators by some compactly supported radial kernels, and use them to approximate Sobolev space functions with optimal convergence order. These kernels can be simply constructed and readily applied to practical problems. The results show that the approximation power depends on the precision of the sampling instrument and the density of the available data.     

Boundedness of oscillatory singular integral with rough kernels on Triebel-Lizorkin spaces

In this paper, we will prove the Triebel-Lizorkin boundedness for some oscillatory singular integrals with the kernel (x) satisfying a condition introduced by Grafakos and Stefanov. Our theorems will be proved under various conditions on the phase function, radial and nonradial. Since the L p boundedness of these operators is not complete yet, the theorems extend many known results.     

On the Design of Optimization Strategies Based on Global Response Surface Approximation Models

Striking the correct balance between global exploration of search spaces and local exploitation of promising basins of attraction is one of the principal concerns in the design of global optimization algorithms. This is true in the case of techniques based on global response surface approximation models as well. After constructing such a model using some initial database of designs it is far from obvious how to select further points to examine so that the appropriate mix of exploration and exploitation is achieved. In this paper we propose a selection criterion based on the expected improvement measure, which allows relatively precise control of the scope of the search. We investigate its behavior through a set of artificial test functions and two structural optimization problems. We also look at another aspect of setting up search heuristics of this type: the choice of the size of the database that the initial approximation is built upon.     

Tight frame expansions of multiscale reproducing kernels in Sobolev spaces

Multiscale kernels are a new type of positive definite reproducing kernels in Hilbert spaces. They are constructed by a superposition of shifts and scales of a single refinable function and were introduced in the paper of R. Opfer [Multiscale kernels, Adv. Comput. Math. (2004), in press]. By applying standard reconstruction techniques occurring in radial basis function- or machine learning theory, multiscale kernels can be used to reconstruct multivariate functions from scattered data. The multiscale structure of the kernel allows to represent the approximant on several levels of detail or accuracy. In this paper we prove that multiscale kernels are often reproducing kernels in Sobolev spaces. We use this fact to derive error bounds. The set of functions used for the construction of the multiscale kernel will turn out to be a frame in a Sobolev space of certain smoothness. We will establish that the frame coefficients of approximants can be computed explicitly. In our case there is neither a need to compute the inverse of the frame operator nor is there a need to compute inner products in the Sobolev space. Moreover we will prove that a recursion formula between the frame coefficients of different levels holds. We present a bivariate numerical example illustrating the mutiresolution and data compression effect.     

Weighted reproducing kernels and Toeplitz operators on harmonic Bergman spaces on the real ball

For the standard weighted Bergman spaces on the complex unit ball, the Berezin transform of a bounded continuous function tends to this function pointwise as the weight parameter tends to infinity. We show that this remains valid also in the context of harmonic Bergman spaces on the real unit ball of any dimension. This generalizes the recent result of C. Liu for the unit disc, as well as the original assertion concerning the holomorphic case. Along the way, we also obtain a formula for the corresponding weighted harmonic Bergman kernels.

     

Comments on some fixed point theorems in hyperconvex metric spaces

In this paper, we show that a number of known fixed point theorems for the Fan-Browder type maps or acyclic maps defined on (subsets of) hyperconvex metric spaces are simple consequences of the previously known theorems for corresponding maps defined on generalized convex spaces.     

Commutator of hypersingular integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator of hypersingular integral T γ from the homogeneous Sobolev space Lpγ (Rn) to the Lebesgue space Lp(Rn) for 1p∞ and 0 γ min{ n/2 , n/p }.     

Commutators of Marcinkiewicz integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator [b, ??_??,?? ] from the homogeneous Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L ^p (? ⁿ ) for 1 < p < ??, where ??_??,?? denotes the Marcinkiewicz integral with rough hypersingular kernel defined by $\mu _{\Omega ,\gamma } f\left( x \right) = \left( {\int_0^\infty {\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega \left( {x - y} \right)}} {{\left| {x - y} \right|^{n - 1} }}f\left( y \right)dy} } \right|^2 \frac{{dt}} {{t^{3 + 2\gamma } }}} } \right)^{\frac{1} {2}} ,$ , with ?? ?? L ¹(S ^n?1) for $0 < \gamma < min\left\{ {\frac{n} {2},\frac{n} {p}} \right\}$ or ?? ?? L(log⁺ L) ^?? (S ^n?1) for $\left| {1 - \frac{2} {p}} \right| < \beta < 1\left( {0 < \gamma < \frac{n} {2}} \right)$ , respectively.     

球面混合插值的逼近性质

球面径向基函数(SBF)和多项式样条函数均为处理球面散乱数据的有效工具. 本文考虑由球面径向基函数与球面多项式函数组成的混合插值模型, 并利用最小二乘法求解该模型. 对于该插值模型, 首先, 给出带Bessel势的Sobolev空间中的Bernstein不等式, 然后利用该不等式建立逼近正定理,并进一步给出该插值工具的误差估计. 最后, 研究该插值方式(即利用最小二乘法求解混合插值模型)的稳定性.     

Asymptotic behaviour of reproducing kernels of weighted Bergman spaces

Let be a domain in , a nonnegative and a positive function on such that is locally bounded, the space of all holomorphic functions on square-integrable with respect to the measure , where is the -dimensional Lebesgue measure, and the reproducing kernel for . It has been known for a long time that in some special situations (such as on bounded symmetric domains with and the Bergman kernel function) the formula

holds true. [This fact even plays a crucial role in Berezin's theory of quantization on curved phase spaces.] In this paper we discuss the validity of this formula in the general case. The answer turns out to depend on, loosely speaking, how well the function can be approximated by certain pluriharmonic functions lying below it. For instance, () holds if is convex (and, hence, can be approximated from below by linear functions), for any function . Counterexamples are also given to show that in general () may fail drastically, or even be true for some and fail for the remaining ones. Finally, we also consider the question of convergence of for , which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of : for instance, if is not real-analytic at some point, then must have zeroes for all sufficiently large.

     

Boundedness of Marcinkiewicz integral related to multilinear commutators with variable kernels on Hardy spaces

Let→b=(b1,b2,…,bm),bi∈∧βi(Rn),1≤I≤m,βi＞0,m∑I=1βi=β,0＜β＜1,μΩ→b(f)(x)=(∫∞0|F→b,t(f)(x)|2dt/t3)1/2,F→b,t(f)(x)=∫|x-y|≤t Ω(x,x-y)/|x-y|n-1 mΠi=1[bi(x)-bi(y)dy.We consider the boundedness of μΩ,→b on Hardy type space Hp→b(Rn).     

Improved stability estimates and a characterization of the native space for matrix-valued RBFs

In this paper we derive several new results involving matrix-valued radial basis functions (RBFs). We begin by introducing a class of matrix-valued RBFs which can be used to construct interpolants that are curl-free. Next, we offer a characterization of the native space for divergence-free and curl-free kernels based on the Fourier transform. Finally, we investigate the stability of the interpolation matrix for both the divergence-free and curl-free cases, and when the kernel has finite smoothness we obtain sharp estimates. An erratum to this article can be found at     

Approximation of solutions to operator equations in spaces of smooth functions and in spaces of distributions

The Galerkin method, in particular, the Galerkin method with finite elements (called finite element method) is widely used for numerical solution of differential equations. The Galerkin method allows us to obtain approximations of weak solutions only. However, there arises in applications a rich variety of problems where approximations of smooth solutions and solutions in the sense of distributions have to be found. This article is devoted to the employment of the Petrov–Galerkin method for solving such problems. The article contains general results on the Petrov–Galerkin approximations of solutions to linear and nonlinear operator equations. The problem on construction of the subspaces, which ensure the convergence of the approximations, is investigated. We apply the general results to two‐dimensional (2D) and 3D problems of the elasticity, to a parabolic problem, and to a nonlinear problem of the plasticity. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 406–450, 2014     

Approximation in Sobolev spaces by piecewise affine interpolation

Functions in a Sobolev space are approximated directly by piecewise affine interpolation in the norm of the space. The proof is based on estimates for interpolations and does not rely on the density of smooth functions.     

Bases in Sobolev weight spaces

In the paper, we construct a system of smooth two-dimensional splines and describe a class of measures for which this system is a basis in the Sobolev weight space on the square. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 343–354, March, 2000.     

The approximation property for spaces of holomorphic functions on infinite dimensional spaces II

Let H(U) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τω and τδ respectively denote the compact-ported topology and the bornological topology on H(U). We show that if E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on E is weakly continuous on bounded sets, then (H(U),τω) and (H(U),τδ) have the approximation property for every open subset U of E. The classical space c₀, the original Tsirelson space T^∗ and the Tsirelson^∗-James space are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains.     

The fractional multilinear singular integral with variable kernels on hardy spaces

The authors discuss Lipschitz boundedness for a class of fractional multilinear operators with variable kernels. It is obtained that these operators are both Lipschitz bounded from Lp to Hq.