Let \({\mathcal T}_n\) be the set of all trigonometric polynomials of degree at most n, and let \(B_1\) be the set of all entire functions of exponential type at most 1. We discuss limit relations between the sharp constants in the Bernstein–Nikolskii inequalities defined by $$\begin{aligned}&A_{p,q,n}^{(s)}:=n^{-s-1/p+1/q}\sup _{T\in {\mathcal T}_n{\setminus }\{0\}}\frac{\Vert T^{(s)}\Vert _{L_q([-\pi ,\pi ])}}{\Vert T\Vert _{L_p([-\pi ,\pi ])}},\quad \\&D_{p,q}^{(s)}:= \sup _{f\in (B_1\cap L_p({\mathbb R})){\setminus }\{0\}}\frac{\Vert f^{(s)}\Vert _{L_q({\mathbb R})}}{\Vert f\Vert _{L_p({\mathbb R})}}, \end{aligned}$$ where \(0<p<q\le {\infty }\) and \(s=0,\,1,\ldots .\) We prove that $$\begin{aligned} D_{p,q}^{(s)}\le \liminf _{n\rightarrow {\infty }}A_{p,q,n}^{(s)},\qquad D_{p,{\infty }}^{(s)}= \lim _{n\rightarrow {\infty }}A_{p,{\infty },n}^{(s)}. \end{aligned}$$

An effective algorithm is presented for solving the Beltrami equation $$\partial f / \partial \overline{z } =\mu \,\partial f/\partial z$$ in a planar disk. The disk is triangulated in a simple way, and f is approximated by piecewise linear mappings; the images of the vertices of the triangles are defined by an overdetermined system of linear equations. (Certain apparently nonlinear conditions on the boundary are eliminated by means of a symmetry construction.) The linear system is sparse, and its solution is obtained by standard least-squares, so the algorithm involves no evaluation of singular integrals nor any iterative procedure for obtaining a single approximation of f. Numerical examples are provided, including a deformation in a Teichmuller space of a Fuchsian group.

We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense—they cannot be improved by employing polynomials of the same or lower degrees in the Delsarte–Yudin method. However, improvements are sometimes possible, and we provide a necessary and sufficient condition for the existence of such better bounds. All our bounds can be obtained in a unified manner that does not depend on the potential function, provided the potential is given by an absolutely monotone function of the inner product between pairs of points, and this is the reason we call them universal. We also establish a criterion for a given code of dimension n and cardinality N not to be LP-universally optimal; e.g., we show that two codes conjectured by Ballinger et al. to be universally optimal are not LP-universally optimal.

In this paper, we show that Besov and Triebel–Lizorkin functions can be approximated by a Holder continuous function both in the Lusin sense and in norm. The results are proved in metric measure spaces for Hajlasz–Besov and Hajlasz–Triebel–Lizorkin functions defined by a pointwise inequality. We also prove new inequalities for medians, including a Poincare type inequality, which we use in the proof of the main result.

We prove the following higher-order Szegő theorem: If a measure on the unit circle has absolutely continuous part \(w(\theta )\) and Verblunsky coefficients \(\alpha \) with square-summable variation, then for any positive integer m, Open image in new window is finite if and only if \(\alpha \in \ell ^{2m+2}\). This is the first known equivalence result of this kind in the regime of very slow decay, i.e., with \(\ell ^p\) conditions with arbitrarily large p. The usual difficulty of controlling higher-order sum rules is avoided by a new test sequence approach.

We conjecture that every ideal projector on \({\mathbb {C}}\left[ x_1,\ldots ,x_d\right] \) whose kernel is generated by precisely d polynomials is Hermite (i.e., the limit of Lagrange interpolation projectors). We validate this conjecture in case the d generators of the kernel have no roots at infinity.

We call a set $$K \subset {\mathbb {R}}^s$$ with positive Lebesgue measure a spectral set if $$L^2(K)$$ admits an exponential orthonormal basis. It was conjectured that K is a spectral set if and only if K is a tile (Fuglede’s conjecture). Although the conjecture was proved to be false in $${\mathbb {R}}^s$$ , $$s\ge 3$$ (Kolountzakis in Forum Math 18:519–528, 2006; Tao in Math Res Lett 11:251–258, 2004), it still poses challenging questions with additional assumptions. In this paper, our additional assumption is self-similarity. We study the spectral properties of a class of self-similar tiles K in $${\mathbb {R}}$$ that has a product structure on the associated digit sets. We show that strict product-form tiles and the associated modulo product-form tiles are spectral sets. For the converse question, we give a pilot study for the self-similar set K generated by arbitrary digit sets with four elements. We investigate the zeros of its Fourier transform due to the orthogonality and verify Fuglede’s conjecture for this special case. We also make use of this case to illustrate the theorems and discuss some questions that arise in the general situation.

Consider a set of points $$(x_1,y_1),\ldots ,(x_n,y_n)$$ with distinct $$0 \le x_i \le 1$$ and with $$-1 < y_i < 1$$ . The question of whether the function $$y = \sin (\omega x)$$ can approximate these points arbitrarily closely for a suitable choice of $$\omega $$ is considered. It is shown that such approximation is possible if and only if the set $$\{x_1,\ldots ,x_n\}$$ is linearly independent over the rationals. Furthermore, a constructive sufficient condition for such approximation is provided. The results provide a sort of counterpoint to the classical sampling theorem for bandlimited signals. They also provide a stronger statement than the well-known result that the collection of functions $$\{\sin (\omega x) : \omega < \infty \}$$ has infinite pseudo-dimension.

A Bernstein–Bezier basis is developed for \({\varvec{H}}({{\mathrm{div}}})\)-conforming finite elements that gives a clear separation between the curls of the Bernstein basis for the polynomial discretization of the space \(H^1\), and the noncurls that characterize the specific \({\varvec{H}}({{\mathrm{div}}})\) finite element space (Raviart–Thomas in our case). The resulting basis has two distinct components reflecting this separation with the basis functions in each component having a natural identification with a domain point, or node, on the element. It is shown that the basis retains the favorable properties of the Bernstein basis that were used in Ainsworth et al. (SIAM J Sci Comput 3087–3109, 2011) to develop efficient computational procedures for the application of the elements.