相关论文: $\boldsymbol{L}_{\infty}$-approximation in Korobov…

Among other things, we prove that, for a doubling weight $w$, $0< p\leq\infty$, $r\in{\mathbb N}_0$, and $0<\alpha <r+1 - 1/\lambda_p$, we have \[ E_n(f)_{p, w_n} = O(n^{-\alpha}) \iff \omega_\varphi^{r+1}(f, n^{-1})_{p, w_n} =…

This paper deals with some nonlinear problems which exponential and biexponential decays are involved in. A proof of the quasiconvexity of the error function in some of these problems of optimization is presented. This proof is restricted…

We consider $L^2$-approximation on weighted reproducing kernel Hilbert spaces of functions depending on infinitely many variables. We focus on unrestricted linear information, admitting evaluations of arbitrary continuous linear…

In this paper, we study the problem of multivariate $L_2$-approximation of functions belonging to a weighted Korobov space. We propose and analyze a median lattice-based algorithm, inspired by median integration rules, which have attracted…

In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which $s$-variate problems can be approximated to within $\varepsilon$ by using, say, polynomially many in $s$…

Let $E_n(f)_{\alpha,\beta,\gamma}$ denote the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_{\alpha,\beta,\gamma})$ on the triangle $\{(x,y): x, y \ge 0, x+y \le 1\}$, where…

In this paper, we propose a method to approximate the Gaussian function on ${\mathbb R}$ by a short cosine sum. We generalise and extend the differential approximation method proposed in [4, 40] to approximate $\mathrm{e}^{-t^{2}/2\sigma}$…

In this paper, we discuss asymptotic relations for the approximation of $\left\vert x\right\vert ^{\alpha},\alpha>0$ in $L_{\infty}\left[ -1,1\right] $ by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of…

We study multivariate approximation in the average case setting with the error measured in the weighted $L_2$ norm. We consider algorithms that use standard information $\Lambda^{\rm std}$ consisting of function values or general linear…

We study the problem of multivariate $L_2$-approximation of functions in a weighted Korobov space using a median lattice-based algorithm recently proposed by the authors. In the original work, the algorithm requires knowledge of the…

We approximate $d$-variate periodic functions in weighted Korobov spaces with general weight parameters using $n$ function values at lattice points. We do not limit $n$ to be a prime number, as in currently available literature, but allow…

We propose a randomized lattice algorithm for approximating multivariate periodic functions over the $d$-dimensional unit cube from the weighted Korobov space with mixed smoothness $\alpha > 1/2$ and product weights…

We present a lower error bound for approximating linear multivariate operators defined over Hilbert spaces in terms of the error bounds for appropriately constructed linear functionals as long as algorithms use function values. Furthermore,…

In this paper we find extremal one-sided approximations of exponential type for a class of truncated and odd functions with a certain exponential subordination. These approximations optimize the $L^1(\mathbb{R}, |E(x)|^{-2}dx)$-error, where…

An algorithm is given for determining an optimal $b$-step approximation of weighted data, where the error is measured with respect to the $L_\infty$ norm. For data presorted by the independent variable the algorithm takes $\Theta(n + \log n…

We consider binary classification problems with positive definite kernels and square loss, and study the convergence rates of stochastic gradient methods. We show that while the excess testing loss (squared loss) converges slowly to zero as…

Approximation theory has long been concerned with the development of positive linear operators that effectively approximate classes of functions. Among the most well-known results in this area are Korovkin-type approximation theorems, which…

In this paper, we introduce Mellin-Steklov exponential samplingoperators of order $r,r\in\mathbb{N}$, by considering appropriate Mellin-Steklov integrals. We investigate the approximation properties of these operators in continuousbounded…

If $\omega_t > \beta$ for every $t \in \mathbb{N}$ and for some $\beta > 0$, then the sequence $\{\omega_t\}_{t \in \mathbb{N}}$ represents a weighted sequence of real numbers. In this article, we primarily introduce the concepts of rough…

We develop a general framework for estimating the $L_\infty(\mathbb{T}^d)$ error for the approximation of multivariate periodic functions belonging to specific reproducing kernel Hilbert spaces (RHKS) using approximants that are…