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Exact estimates with respect to the order of magnitude are obtained for the ortho-projective and linear diameters of the classes B _p,?? ^r periodic functions of several variables in the spaces L _q , 1 ?? p, q ?? ??. The order of magnitude of the best approximation is established in the space Leo of the classes B _??,?? ^r of periodic functions of two variables with trigonometric polynomials with harmonics from a hyperbolic cross.     

用指数型整函数的最佳限制逼近

我们研究了由仅有实零点的代数多项式导出的微分算子确定的广义Sobolev类利用指数型整函数作为逼近工具的最佳限制逼近问题.利用Fourier变换和周期化等方法,得到在L_2(R)范数下的广义Sobolev光滑函数类的相对平均宽度和最佳限制逼近的精确常数,以及当0是这个代数多项式的一个至多2重的零点时,得到最佳限制逼近在L_1(R)范数和一致范数下的广义Sobolev类的精确到阶的结果.     

Some extremal properties of multivariate polynomial splines in the metric Lp(ℝd)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (?^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (?^d in the metric L_p((?^d).     

Bernstein Fractal Trigonometric Approximation

Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein \(\alpha \)-fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein \(\alpha \)-fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function \(f\) defined on \([-l,l]\). The Bernstein fractal Fourier series converges to \(f\) even if the magnitude of the scaling factors does not approach zero. Existence of the \(\mathcal{C}^{r}\)-Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of \(\mathcal{C}^{r}\)-Bernstein fractal functions.

     

A note on orthonormal polynomial bases and wavelets

A simple and explicit construction of an orthnormal trigonometric polynomial basis in the spaceC of continuous periodic functions is presented. It consists simply of periodizing a well-known wavelet on the real line which is orthonormal and has compactly supported Fourier transform. Trigonometric polynomials resulting from this approach have optimal order of growth of their degrees if their indices are powers of 2. Also, Fourier sums with respect to this polynomial basis are projectors onto subspaces of trigonometric polynomials of high degree, which implies almost best approximation properties.     

Tractability through increasing smoothness

We prove that some multivariate linear tensor product problems are tractable in the worst case setting if they are defined as tensor products of univariate problems with logarithmically increasing smoothness. This is demonstrated for the approximation problem defined over Korobov spaces and for the approximation problem of certain diagonal operators. For these two problems we show necessary and sufficient conditions on the smoothness parameters of the univariate problems to obtain strong polynomial tractability. We prove that polynomial tractability is equivalent to strong polynomial tractability, and that weak tractability always holds for these problems. Under a mild assumption, the Korobov space consists of periodic functions. Periodicity is crucial since the approximation problem defined over Sobolev spaces of non-periodic functions with a special choice of the norm is not polynomially tractable for all smoothness parameters no matter how fast they go to infinity. Furthermore, depending on the choice of the norm we can even lose weak tractability.     

Some extremal properties of multivariate polynomial splines in the metric Lp(?d)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (ℝ^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (ℝ^d in the metric L_p((ℝ^d).     

Greedy Algorithm and m -Term Trigonometric Approximation

We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form , where is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients of function f . We compare the efficiency of this method with the best m -term trigonometric approximation both for individual functions and for some function classes. It turns out that the operator G _m provides the optimal (in the sense of order) error of m -term trigonometric approximation in the L _p -norm for many classes. September 23, 1996. Date revised: February 3, 1997.     

On the unimprovability of full-memory strategies in the risk minimization problem

We continue the study of approximation properties of local exponential splines on a uniform grid with step h > 0 corresponding to a linear differential operator L with constant coefficients and real pairwise different roots of the characteristic polynomial (such splines were constructed by E.V. Strelkova and V.T. Shevaldin). We find order estimates as h → 0 for the error of approximation of certain Sobolev classes of functions by splines of the described type that are exact on the kernel of the operator L.     

Approximation of classes B_{p,\theta }^r of periodic functions of one and several variables

We obtain order-sharp estimates of best approximations to the classes $B_{p,\theta }^r$ of periodic functions of several variables in the space L _q , 1 ≤ p, q ≤ ∞ by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses. In the one-dimensional case, we establish the order of deviation of Fourier partial sums of functions from the classes $ B_{1,\theta }^{r_1 } $ in the space L ₁.     

Some extremal properties of multivariate polynomial splines in the metric Lp (Rd )

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the Bd instead of the usual multivariate cardinal interpolation oper-ators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asymptoti-cally optimal for the Kolmogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on Bd in the metric Lp(Bd).     

Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

In the spaceL _q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB _1,θ ^r andB _p,α ^r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.     

Maharam-type kernel representation for operators with a trigonometric domination

Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in \(L^{2}[0,1]\) by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.     

Approximation of Sobolev Classes by Their Finite-Dimensional Sections

We consider relative widths characterizing the best approximation of a fixed set by its sections of given dimension. For Sobolev classes of periodic functions of a single variable with constraints inL _∞ orL ₁ on higher-order derivatives, we present the exact orders of such widths in the spaces L _q.     

On Trigonometric Blending Interpolation and Cubature Formulae

We discuss error representations for Hermite-Lagrange trigonometric interpolation introduced in Dryanov and Petrov (Interpolation and L ₁-approximation by trigonometric polynomials and blending functions, J. Approx. Theory 164, 1049–1064 (2012)) and obtain one-sided trigonometric quadratures for approximate integration of one-dimensional integrals. Next, we study error representations of multivariate Hermite-Lagrange transfinite trigonometric interpolation and derive one-sided trigonometric blending interpolants to multivariate functions, under some restrictions. Then, we construct one-sided transfinite cubature formulae for approximate integration of multivariate integrals. We construct also cubature formulae with positive coefficients, based on line integrals and exact in a vector space of trigonometric blending functions with prescribed order.     

Spline fitting discontinuous functions given just a few Fourier coefficients

This paper considers the use of polynomial splines to approximate periodic functions with jump discontinuities of themselves and their derivatives when the information consists only of the first few Fourier coefficients and the location of the discontinuities. Spaces of splines are introduced which include, members with discontinuities at those locations. The main results deal with the orthogonal projection of such a spline space on spaces of trigonometric polynomials corresponding to the known coefficients. An approximation is defined based on inverting this projection. It is shown that when discontinuities are sufficiently far apart, the projection is invertible, its inverse has norm close to 1, and the approximation is nearly as good as directL ₂ approximation by members of the spline space. An example is given which illustrates the results and which is extended to indicate how the approximation technique may be used to provide smoothing which which accurately represents discontinuities.     

Best approximations of classes of functions defined by means of the modulus of continuity

This paper is devoted to an exact solution of problems of best approximation in the uniform and integral metrics of classes of periodic functions representable as a convolution of a kernel not increasing the oscillation with functions having a given convex upwards majorant of the modulus of continuity. The approximating sets are taken to be the trigonometric polynomials in the case of the uniform and integral metrics, and convolutions of the kernel defining the class with polynomial splines in the case of the integral metric.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 579–589, May, 1992.     

Jacobi Polynomials,Weighted Sobolev Spaces and Approximation Results of Some Singularities

In this paper, we give some polynomial approximation results in a class of weighted Sobolev spaces, which are related to the Jacobi operator. We further give some embeddings of those weighted Sobolev spaces into usual ones and into spaces of continuous functions, in order to use the above approximation results in the p‐version (or the spectral method) of some finite or boundary element methods. Finally, two typical examples of the polynomial approximation of some singularities of boundary value problems in polygonal or polyhedral domains are presented.     

Analysis of Tensor Approximation Schemes for Continuous Functions

In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate dimension weights.

     

On the Dirichlet–Riquier problem for biharmonic equations

In the paper, it is proved that the distribution of a measurable polynomial on an infinite-dimensional space with log-concave measure is absolutely continuous if the polynomial is not equal to a constant almost everywhere. A similar assertion is proved for analytic functions and for some other classes of functions. Properties of distributions of norms of polynomial mappings are also studied. For the space of measurable polynomial mappings of a chosen degree, it is proved that the L ¹-norm with respect to a log-concave measure is equivalent to the L ¹-norm with respect to the restriction of the measure to an arbitrarily chosen set of positive measure.