Journal of Approximation Theory最新文献

Let $L_{p,w},1\le p<\infty ,$ denote the weighted $L_p$ space of functions on the unit ball $B^d$ with a doubling weight $w$ on $B^d$. The Markov factor for $L_{p,w}$ of a polynomial $P$ is defined by $\frac{{\Vert |\nabla P|\Vert }_{p,w}}{{\Vert P\Vert }_{p,w}},$ where $\nabla P$ is the gradient of $P$. We investigate the worst case Markov factors for $L_{p,w}$ and prove that the degree of these factors is at most 2. In particular, for the Gegenbauer weight $w_{\mu }\left(x\right)={(1-{\left|x\right|}^2)}^{\mu -1/2},\mu \ge 0,$ the exponent 2 is sharp. We also study the average case Markov factor for $L_{2,w}$ on random polynomials with independent $N(0,{\sigma }^2)$ coefficients and obtain that the upper bound of the average (expected) Markov factor is order degree to the $3/2$, as compared to the degree squared worst case upper bound.

For any $p\in [1,\infty ]$, we prove that the set of simple functions taking at most $k$ different values is proximinal in $L^p$ for all $k\ge 1$. Moreover, if $1<p<\infty$, we prove that these sets are approximatively norm-compact. We introduce the class of uniformly approximable subsets of $L^p$, which is larger than the class of uniformly integrable sets. This new class is characterized in terms of the $p$-variation if $p\in [1,\infty )$ and in terms of covering numbers if $p=\infty$. We study properties of uniformly approximable sets. In particular, we prove that the convex hull of a uniformly approximable bounded set is also uniformly approximable and that this class is stable under Hölder transformations.

We use methods of approximation theory to find the absolute minima on the sphere of the potential of spherical $(2m-3)$-designs with a non-trivial index $2m$ that are contained in a union of $m$ parallel hyperplanes, $m\ge 2$, whose locations satisfy certain additional assumptions. The interaction between points is described by a function of the dot product, which has positive derivatives of orders $2m-2$, $2m-1$, and $2m$. This includes the case of the classical Coulomb, Riesz, and logarithmic potentials as well as a completely monotone potential of the distance squared. We illustrate this result by showing that the absolute minimum of the potential of the set of vertices of the icosahedron on the unit sphere $S^2$ in $R^3$ is attained at the vertices of the dual dodecahedron and the one for the set of vertices of the dodecahedron is attained at the vertices of the dual icosahedron. The absolute minimum of the potential of the configuration of 240 minimal vectors of $E_8$ root lattice normalized to lie on the unit sphere $S^7$ in $R^8$ is attained at a set of 2160 points on $S^7$ which we describe.

A design is a collection of distinct points in a given set $X$, which is assumed to be a compact subset of $R^d$, and the mesh-ratio of a design is the ratio of its fill distance to its separation radius. The uniformity constant of a sequence of nested designs is the smallest upper bound for the mesh-ratios of the designs. We derive a lower bound on this uniformity constant and show that a simple greedy construction achieves this lower bound. We then extend this scheme to allow more flexibility in the design construction.

The concept of $H$-sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and $H$-sets exist, but are only rarely accessible by mathematical arguments. However, in Reproducing Kernel Hilbert spaces, $H$-sets are shown here to have a rather simple and complete characterization. As a byproduct, the strong connection of $H$-sets to Linear Programming is studied. But on the downside, it is explained why $H$-sets have a very limited range of applicability in the times of large-scale computing.

We study approximation properties of linear sampling operators in the spaces $L_p$ for $1\le p<\infty$. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of a function in $L_p$ and discrete information on the behaviour of a function at sampling points. The new measure of smoothness enables us to improve and extend several classical results of approximation theory to the case of linear sampling operators. In particular, we obtain matching direct and inverse approximation inequalities for sampling operators in $L_p$, find the exact order of decay of the corresponding $L_p$-errors for particular classes of functions, and introduce a special $K$-functional and its realization suitable for studying smoothness properties of sampling operators.

This note conducts a comparative study of some approximating properties of the metric projection, generalized projection, and generalized metric projection in uniformly convex and uniformly smooth Banach spaces. We prove that the inverse images of the metric projections are closed and convex cones, but they are not necessarily convex. In contrast, inverse images of the generalized projection are closed and convex cones. Furthermore, the inverse images of the generalized metric projection are neither a convex set nor a cone. We also prove that the distance from a point to its projection on a convex set is a weakly lower semicontinuous function for all three notions of projections. We provide illustrating examples to highlight the different behavior of the three projections in Banach spaces.