Journal of Approximation Theory最新文献

We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold $M$ when given a sample on a finite point set. We prove that the quality of the sample is given by the $L_{\gamma }\left(M\right)$-average of the geodesic distance to the point set and determine the value of $\gamma \in (0,\infty ]$. This extends our findings on bounded convex domains [IMA J. Numer. Anal., 2024]. As a byproduct, we prove the optimal rate of convergence of the $n$th minimal worst case error for $L_q\left(M\right)$-approximation for all $1\le q\le \infty$.

Further, a limit theorem for moments of the average distance to a set consisting of i.i.d. uniform points is proven. This yields that a random sample is asymptotically as good as an optimal sample in precisely those cases with $\gamma <\infty$. In particular, we obtain that cubature formulas with random nodes are asymptotically as good as optimal cubature formulas if the weights are chosen correctly. This closes a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput., 29:1203-1214, 2019].

Let ${\Pi }_n$ be the class of algebraic polynomials $P$ of degree $n$, all of whose zeros lie on the segment $[-1,1]$. In 1995, S. P. Zhou has proved the following Turán type reverse Markov–Nikol’skii inequality: ${\Vert P^{\prime }\Vert }_{L_p[-1,1]}>c{\left(\sqrt{n}\right)}^{1-1/p+1/q}{\Vert P\Vert }_{L_q[-1,1]}$, $P\in {\Pi }_n$, where $0<p\le q\le \infty$, $1-1/p+1/q\ge 0$($c>0$ is a constant independent of $P$ and $n$). We show that Zhou’s estimate remains true in the case $p=\infty$, $q>1$. Some of related Turán type inequalities are also discussed.

We estimate the number of zeros of a polynomial in $ℂ\left[z\right]$ within any small circular disk centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erdélyi, and Littmann in 2008. Furthermore, by combining this result with Euclidean geometry, we derive an upper bound on the number of zeros of such a polynomial within a region resembling a gear wheel. Additionally, we obtain a sharp upper bound on the annular discrepancy of such zeros near the unit circle. Our approach builds upon a modified version of the method described in Borwein et al. (2008), combined with the refined version of the best-known upper bound for angular discrepancy of zeros of polynomials.

We study the convergence of Bernstein type operators leading to two results. The first: The kernel $K_n$ of the Bernstein–Durrmeyer operator at each point $x\in (0,1)$ — that is $K_n(x,t)dt$ — once standardised converges to the normal distribution. The second result computes the pointwise limit of a generalised Bernstein–Durrmeyer operator applied to — possibly discontinuous — functions $f$ of bounded variation.

The Matrix Bochner Problem aims to classify which weight matrices have their sequence of orthogonal polynomials as eigenfunctions of a second-order differential operator. Casper and Yakimov, in [4], demonstrated that, under certain hypotheses, all solutions to the Matrix Bochner Problem are noncommutative bispectral Darboux transformations of a direct sum of classical scalar weights. This paper aims to provide the first proof that there are solutions to the Matrix Bochner Problem that do not arise through a noncommutative bispectral Darboux transformation of any direct sum of classical scalar weights. This initial example could contribute to a more comprehensive understanding of the general solution to the Matrix Bochner Problem.

The aim of this paper is to calculate relative and absolute projection constants of certain subspaces of $l_1^{\left(n\right)}$ and $l_{\infty }^{\left(n\right)}$ generated by eigenvectors of sign matrices. The main tool in our considerations is so called Chalmers–Metcalf operator (see Chalmers & Metcalf (1994) and Lewicki & Prophet (2021)). Also, some results from Castejon & Lewicki (2014) and Castejon & Lewicki (2019) will be applied.

We consider best approximation by rational functions in Musielak–Orlicz spaces of real-valued measurable functions over the unit interval equipped with the Lebesgue measure. We prove several properties of the respective multi-value projection operator, including its semi-continuity. Our results generalise known results for Lebesgue and variable Lebesgues spaces, and can be applied to special cases including Orlicz spaces and variable Lebesgue spaces with weights. We touch upon applications to image processing.

The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of $C(I\times J,X\otimes Y)$ as the closure of $C(I,X)\otimes C(J,Y)$ in different senses and the extension of continuous vector-valued functions are studied.