Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
31,319
result(s) for
"Linear function"
Sort by:
Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations
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.
eBook
Embeddings of Decomposition Spaces
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
eBook
Matrix Functions of Bounded Type: An Interplay Between Function Theory and Operator Theory
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
eBook
Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs
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.
eBook
Hardy–Littlewood and Ulyanov inequalities
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.
eBook
Multi-Function Computation over a Directed Acyclic Network
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.
Journal Article
Functional Analysis, Harmonic Analysis, and Image Processing
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.
eBook
Security, Privacy, and Linear Function Retrieval in Combinatorial Multi-Access Coded Caching with Private Caches
We consider combinatorial multi-access coded caching with private caches, where users are connected to two types of caches: private caches and multi-access caches. Each user has its own private cache, while multi-access caches are connected in the same way as caches are connected in a combinatorial topology. A scheme is proposed that satisfies the following three requirements simultaneously: (a) Linear Function Retrieval (LFR), (b) content security against an eavesdropper, and (c) demand privacy against a colluding set of users. It is shown that the private caches included in this work enable the proposed scheme to provide privacy against colluding users. For the same rate, our scheme requires less total memory accessed by each user and less total system memory than the existing scheme for multi-access combinatorial topology (no private caches) in the literature. We derive a cut-set lower bound and prove optimality when r≥C−1. For r
Journal Article
Hierarchical Cache-Aided Networks for Linear Function Retrieval
In a hierarchical caching system, a server is connected to multiple mirrors, each of which is connected to a different set of users, and both the mirrors and the users are equipped with caching memories. All the existing schemes focus on single file retrieval, i.e., each user requests one file. In this paper, we consider the linear function retrieval problem, i.e., each user requests a linear combination of files, which includes single file retrieval as a special case. We propose a new scheme that reduces the transmission load of the first hop by jointly utilizing the two layers’ cache memories, and we show that our scheme achieves the optimal load for the second hop in some cases.
Journal Article