Journal of Approximation Theory最新文献

We correct an error in the statement of Levis et al. (2023, Theorem 4.5).

We prove lower bounds for the randomized approximation of the embedding ${\ell }_1^m\hookrightarrow {\ell }_{\infty }^m$ based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix $N\in R^{n\times m}$. These lower bounds reflect the increasing difficulty of the problem for $m\to \infty$, namely, a term $\sqrt{logm}$ in the complexity $n$. This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity $n$ only exhibits a $(loglogm)$-dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order $n^{1/2}{(logn)}^{-1/2}$.

Motivated by some properties of the geometric measures for compact convex sets in the Brunn–Minkowski theory, such as the properties of the volume, the $p$-capacity $(1<p<n)$ and the torsional rigidity for compact convex sets, we introduce a more general geometric invariant, called the compatible functional $F$. Inspired also by the $L_p$ Minkowski problem associated with the volume, the $p$-capacity and the torsional rigidity for compact convex sets, we pose the $L_p$ Minkowski problem associated with the compatible functional $F$ and prove the existence of the solutions to this problem for $p>0$. We will show that the volume, the $p$-capacity $(1<p<2)$ and the torsional rigidity for compact convex sets are the compatible functionals. Thus, as an application, we provide the solution to the $L_p$ Minkowski problem $(0<p<1)$ for arbitrary measure associated with $p$-capacity $(1<p<2)$.

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of ${ℒ}_2\left(R\right)$. This allows us to derive explicit expansions on the real line for (i) Matérn kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.

We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the form ${(z-1)}^s$ where $s>0$. For integer values of $s$ this corresponds to prescribing a zero of the polynomial on the boundary. As such, we extend findings of Lachance et al. (1979), to non-integer $s$. Using this generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established, categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the Erdős–Lax inequality to encompass powers of polynomials. We believe that this particular result holds significance in its own right.

In this paper, order estimates for the Kolmogorov $n$-widths of an intersection of a family of balls in a mixed norm in the space $l_{q,\sigma }^{m,k}$ with $2⩽q,\sigma <\infty$, $n⩽mk/2$ are obtained.

For $\lambda \ge 0$, a $C^2$ function $f$ defined on the unit disk $D$ is said to be $\lambda$-analytic if $D_{\bar{z}}f=0$, where $D_{\bar{z}}$ is the (complex) Dunkl operator given by $D_{\bar{z}}f={\partial }_{\bar{z}}f-\lambda (f\left(z\right)-f\left(\bar{z}\right))/(z-\bar{z})$. The aim of the paper is to study several problems on the associated Bergman spaces $A_{\lambda }^p\left(D\right)$ and Hardy spaces $H_{\lambda }^p\left(D\right)$ for $p\ge 2\lambda /(2\lambda +1)$, such as boundedness of the Bergman projection, growth of functions, density, completeness, and the dual spaces of $A_{\lambda }^p\left(D\right)$ and $H_{\lambda }^p\left(D\right)$, and characterization and interpolation of $A_{\lambda }^p\left(D\right)$.

Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process is inspired by the Halpern–Lions–Wittmann–Bauschke algorithm and the classical alternating process of Cheney and Goldstein, and its advantage is that there is no need to project onto the intersections themselves, a task which can be rather demanding. We prove that under certain conditions the two interlaced subsequences converge to a best approximation pair. These conditions hold, in particular, when the space is Euclidean and the subsets which generate the intersections are compact and strictly convex. Our result extends the one of Aharoni, Censor and Jiang [“Finding a best approximation pair of points for two polyhedra”, Computational Optimization and Applications 71 (2018), 509–23] who considered the case of finite-dimensional polyhedra.

We consider the general question of when all orbits under the unitary action of a finite group give a complex spherical design. Those orbits which have large stabilisers are then good candidates for being optimal complex spherical designs. This is done by developing the general theory of complex designs and associated (harmonic) Molien series for group actions. As an application, we give explicit constructions of some putatively optimal real and complex spherical $t$-designs.

Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form ${\int }_a^{b}f\left(x\right)d\mu \left(x\right)={\sum }_{i=1}^n{w}_{i}f\left(x_i\right)$ where $f$ belongs to $H^1\left(\mu \right)$. Here, $\mu$ belongs to a class of continuous probability distributions on $[a,b]\subset R$ and ${\sum }_{i=1}^n{w}_i{\delta }_{x_i}$ is a discrete probability distribution on $[a,b]$. We show that $H^1\left(\mu \right)$ is a reproducing kernel Hilbert space with a continuous kernel $K$, which allows to reformulate the quadrature question as a kernel (or Bayesian) quadrature problem. Although $K$ has not an easy closed form in general, we establish a correspondence between its spectral decomposition and the one associated to Poincaré inequalities, whose common eigenfunctions form a $T$-system (Karlin and Studden, 1966). The quadrature problem can then be solved in the finite-dimensional proxy space spanned by the first eigenfunctions. The solution is given by a generalized Gaussian quadrature, which we call Poincaré quadrature.

We derive several results for the Poincaré quadrature weights and the associated worst-case error. When $\mu$ is the uniform distribution, the results are explicit: the Poincaré quadrature is equivalent to the midpoint (rectangle) quadrature rule. Its nodes coincide with the zeros of an eigenfunction and the worst-case error scales as $\frac{b-a}{2\sqrt{3}}{n}^{-1}$ for lar