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We study the minimality properties of a new type of “soft” theta functions. For a lattice $$L\subset {\mathbb {R}}^d$$ , an L-periodic distribution of mass $$\mu _L$$ , and another mass $$\nu _z$$ centered at $$z\in {\mathbb {R}}^d$$ , we define, for all scaling parameters $$\alpha >0$$ , the translated lattice theta function $$\theta _{\mu _L+\nu _z}(\alpha )$$ as the Gaussian interaction energy between $$\nu _z$$ and $$\mu _L$$ . We show that any strict local or global minimality result that is true in the point case $$\mu =\nu =\delta _0$$ also holds for $$L\mapsto \theta _{\mu _L+\nu _0}(\alpha )$$ and $$z\mapsto \theta _{\mu _L+\nu _z}(\alpha )$$ when the measures are radially symmetric with respect to the points of $$L\cup \{z\}$$ and sufficiently rescaled around them (i.e., at a low scale). The minimality at all scales is also proved when the radially symmetric measures are generated by a completely monotone kernel. The method is based on a generalized Jacobi transformation formula, some standard integral representations for lattice energies, and an approximation argument. Furthermore, for the honeycomb lattice $${\mathsf {H}}$$ , the center of any primitive honeycomb is shown to minimize $$z\mapsto \theta _{\mu _{{\mathsf {H}}}+\nu _z}(\alpha )$$ , and many applications are stated for other particular physically relevant lattices including the triangular, square, cubic, orthorhombic, body-centered-cubic, and face-centered-cubic lattices.

In this paper, we focus on Strichartz’s derivatives, a family of derivatives including the normal derivative, on post critically finite fractals, which are defined at vertices in the graphs that approximate the fractal. We obtain a weak continuity property of the derivatives for functions in the domain of the Laplacian. For a function with zero normal derivative at any fixed vertex, the derivatives, including the normal derivatives, of the neighboring vertices will decay to zero. The rates of approximations are described, and several nontrivial examples are provided to illustrate that our estimates are optimal. We also study the boundedness property of derivatives for functions in the domain of the Laplacian. A necessary condition for a function having a weak tangent of order one at a vertex is provided. Furthermore, we give a counterexample of a conjecture of Strichartz on the existence of higher-order weak tangents.

WithE 2n (|x|) denoting the error of best uniform approximation to |x| by polynomials of degree at most 2n on the interval [−1, +1], the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constantβ for which lim 2nE 2n (|x|)=β.n→∞ Moreover, by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds forβ: 0.278<β<0.286. Now, the average of these bounds is 0.282, which, as Bernstein noted as a “curious coincidence,” is very close to 1/(2√π)=0.2820947917... This observation has over the years become known as the Bernstein Conjecture: Isβ=1/(2√π)? We show here that the Bernstein conjecture isfalse. In addition, we determine rigorous upper and lower bounds forβ, and by means of the Richardson extrapolation procedure, estimateβ to approximately 50 decimal places.

Some new conditions that arise naturally in the study of the Thresholding Greedy Algorithm are introduced for bases of Banach spaces. We relate these conditions to best n-term approximation and we study their duality theory. In particular, we obtain a complete duality theory for greedy bases.

Morrey (function) spaces and, in particular, smoothness spaces of Besov–Morrey or Triebel–Lizorkin–Morrey type have enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces $$m_{u,p}=m_{u,p}(\mathbb {Z}^d)$$, $$0