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In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

In this paper, we study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. We first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. We propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. We also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fejér Interpolation Problem for matrix rational functions. We then extend the

Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an embedding between the two? A decomposition space We establish readily verifiable criteria which ensure the existence of a continuous inclusion (“an embedding”) In a nutshell, in order to apply the embedding results presented in this article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings. These sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of We also prove a The resulting embedding theory is illustrated by applications to

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness The main tool is the new Hardy–Littlewood–Nikol’skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for the Lipschitz and Besov spaces.

We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin derivatives explicitly. Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of the initial data, in particular upper bounds for their Among other tools, we use methods from harmonic analysis. As a by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the optimality of the bound we established.

The tropical analogue of the lemma on the logarithmic derivative is generalised for noncontinuous tropical meromorphic functions, that is, piecewise linear functions that may have discontinuities. In addition, two Borel type results are generalised for piecewise continuous functions. With the generalisation of the tropical analogue of the lemma on the logarithmic derivative, several tropical analogues of Clunie and Mohon’ko type results are also automatically generalised for noncontinuous tropical meromorphic functions.

The work here studies the communication cost for a multi-server multi-task distributed computation framework, as well as for a broad class of functions and data statistics. Considering the framework where a user seeks the computation of multiple complex (conceivably non-linear) tasks from a set of distributed servers, we establish the communication cost upper bounds for a variety of data statistics, function classes, and data placements across the servers. To do so, we proceed to apply, for the first time here, Körner’s characteristic graph approach—which is known to capture the structural properties of data and functions—to the promising framework of multi-server multi-task distributed computing. Going beyond the general expressions, and in order to offer clearer insight, we also consider the well-known scenario of cyclic dataset placement and linearly separable functions over the binary field, in which case, our approach exhibits considerable gains over the state of the art. Similar gains are identified for the case of multi-linear functions.

Dynamic linear functions on the boolean hypercube are functions which assign to each bit a positive weight, but the weights change over time. Throughout optimization, these functions maintain the same global optimum, and never have defecting local optima. Nevertheless, it was recently shown [Lengler, Schaller, FOCI 2019] that the ( 1 + 1 ) -Evolutionary Algorithm needs exponential time to find or approximate the optimum for some algorithm configurations. In this experimental paper, we study the effect of larger population sizes for dynamic binval , the extreme form of dynamic linear functions. We find that moderately increased population sizes extend the range of efficient algorithm configurations, and that crossover boosts this positive effect substantially. Remarkably, similar to the static setting of monotone functions in [Lengler, Zou, FOGA 2019], the hardest region of optimization for ( μ + 1 ) -EA is not close the optimum, but far away from it. In contrast, for the ( μ + 1 ) -GA, the region around the optimum is the hardest region in all studied cases.Kindly check and confirm the inserted city name is correctly identified.Correct.

This volume is dedicated to the memory of Björn Jawerth. It contains original research contributions and surveys in several of the areas of mathematics to which Björn made important contributions. Those areas include harmonic analysis, image processing, and functional analysis, which are of course interrelated in many significant and productive ways.Among the contributors are some of the world's leading experts in these areas. With its combination of research papers and surveys, this book may become an important reference and research tool.This book should be of interest to advanced graduate students and professional researchers in the areas of functional analysis, harmonic analysis, image processing, and approximation theory. It combines articles presenting new research with insightful surveys written by foremost experts.

The problem of multi-function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node is required to compute with zero error multiple vector-linear functions, where each vector-linear function has distinct inputs generated by multiple source nodes. The of an admissible code is defined as a tuple consisting of the average number of zero-error computations for each vector-linear function when the network is used once jointly. From the information theoretic point of view, we are interested in characterizing the , which is defined as the closed set of all achievable computing rate tuples. In particular, when the sink node is required to compute a single vector-linear function, the problem degenerates to the problem. We prove an outer bound on the rate region by developing the approach of the cut-set strong partition. We also illustrate that the obtained outer bound is tight for a typical model of computing two vector-linear functions over the diamond network. Furthermore, we establish the relationship between the network multi-function computation rate region and the network function computation rate region. Also, we show that the best known outer bound on the rate region for computing an arbitrary vector-linear function over an arbitrary network is a straightforward consequence of our outer bound.