Given positive integers$n$_{1}<$n$_{2}<... we show that the Hardy-type inequality $$\sum\limits_{k = 1}^\infty {\frac{{\left| {\hat f(n_k )} \right|}}{k}} \leqslant const\left\| f \right\|1$$ holds true for all$f$∈$H$^{1}, provided that the$n$_{$k$}'s, satisfy an appropriate (and indispensable) regularity condition. On the other hand, we exhibit inifinite-dimensional subspaces of$H$^{1}for whose elements the above inequality is always valid, no additional hypotheses being imposed. In conclusion, we extend a result of Douglas, Shapiro and Shields on the cyclicity of lacunary series for the backward shift operator.

Rerandomization is a strategy for improving balance on observed covariates in randomized controlled trials. It has been both advocated and advised against by renowned scholars of experimental design. However, the relationship and differences between stratification, rerandomization, and the combination of the two have not been previously investigated. In this paper, we show that stratified designs can be recreated by rerandomization and explain why, in most cases, stratification on binary covariates followed by rerandomization on continuous covariates is more efficient than rerandomization on all covariates at the same time.

Let$M$be a smooth manifold and$V$a Euclidean space. Let $ \overline{{{\text{Emb}}}} $ ($M$,$V$) be the homotopy fiber of the map Emb($M$,$V$) → Imm($M$,$V$). This paper is about the rational homology of $ \overline{{{\text{Emb}}}} $ ($M$,$V$). We study it by applying embedding calculus and orthogonal calculus to the bifunctor ($M$,$V$)↦$HQ$∧ $ \overline{{{\text{Emb}}}} $ ($M$,$V$)_{+}. Our main theorem states that if $$ \dim V \geqslant 2{\text{ED}}{\left( M \right)} + 1 $$ (where ED($M$) is the embedding dimension of$M$), the Taylor tower in the sense of orthogonal calculus (henceforward called “the orthogonal tower”) of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at$E$^{1}. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor $$ HQ \wedge \overline{{{\text{Emb}}}} {\left( {M,V} \right)}_{ + }. $$ The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of$M$. This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of $ \overline{{{\text{Emb}}}} $ (–,V).

Let$D$⊂$C$be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. For$R$>0, let$ω$_{$D$}($R$) denote the harmonic measure at 0 of the set {$z$:|$z$|≽$R$}⋔∂$D$. Then it is shown that$there exist$β>0$and C$>0$such that for each such D$,$ω$_{$D$}($R$)≤$Ce$^{−$βR$},$for every R$>0. Thus a natural question is: What is the supremum of all β′s , call it β_{0}, for which the above inequality holds for every such$D$? Another formulation of the problem involves hyperbolic metric instead of harmonic measure. Using this formulation a lower bound for β_{0}is found. Upper bounds for β_{0}can be obtained by constructing examples of domains$D$. It is shown that a certain domain whose boundary consists of an infinite number of vertical half-lines, i.e. a comb domain, gives a good upper bound. This bound disproves a conjecture of C. Bishop which asserted that the strips of width 2 are extremal domains. Harmonic measures on comb domains are also studied.

This paper presents an elementary proof of the following theorem: Given {$r$_{$j$}}_{$j$}^{$m$}=1 with m=d+1, fix $$fix R \geqslant \sum\nolimits_{j = 1}^m {r_j } $$ and let Q=[−R, R]^{d}. Then any f∈ L^{2}(Q) is completely determined by its averages over cubes of side r_{j}that are completely contained in Q and have edges parallel to the coordinate axes if and only if r_{j}/r_{k}is irrational for j≠k. When$d$=2 this theorem is known as the local three squares theorem and is an example of a Pompeiu-type theorem. The proof of the theorem combines ideas in multisensor deconvolution and the theory of sampling on unions of rectangular lattices having incommensurate densities with a theorem of Young on sequences biorthogonal to exact sequences of exponentials.