Journal of Approximation Theory最新文献

We prove mean convergence of the Fourier series in Akhiezer–Chebyshev polynomials in $L^p$, $p>1$, using a weighted inequality for the Hilbert transform in an arc of the unit circle.

The research about harmonic analysis associated with Jacobi expansions carried out in Arenas et al. (2020) and Arenas et al. (2022) is continued in this paper. Given the operator $J^{(\alpha ,\beta )}=J^{(\alpha ,\beta )}-I$, where $J^{(\alpha ,\beta )}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we define the corresponding Littlewood–Paley–Stein $g_k^{(\alpha ,\beta )}$-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.

Explicit expressions and computational approaches are given for the Fortet–Mourier distance between a positively weighted sum of Dirac measures on a metric space and a positive finite Borel measure. Explicit expressions are given for the distance to a single Dirac measure. For the case of a sum of several Dirac measures one needs to resort to a computational approach. In particular, two algorithms are given to compute the Fortet–Mourier norm of a molecular measure, i.e. a finite weighted sum of Dirac measures. It is discussed how one of these can be modified to allow computation of the dual bounded Lipschitz (or Dudley) norm of such measures.

In the present paper we establish estimates for the error of approximation (in the $L^p$-norm) achieved by neural network (NN) operators. The above estimates have been given by means of an averaged modulus of smoothness introduced by Sendov and Popov, also known with the name of $\tau$-modulus, in case of bounded and measurable functions on the interval $[-1,1]$. As a consequence of the above estimates, we can deduce an $L^p$ convergence theorem for the above family of NN operators in case of functions which are bounded, measurable, and Riemann integrable on the above interval. In order to reach the above aims, we preliminarily establish a number of results; among them we can mention an estimate for the $p$-norm of the operators, and an asymptotic type theorem for the NN operators in case of functions belonging to Sobolev spaces.

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.