18,581 research outputs found
Difference Balanced Functions and Their Generalized Difference Sets
Difference balanced functions from to are closely related
to combinatorial designs and naturally define -ary sequences with the ideal
two-level autocorrelation. In the literature, all existing such functions are
associated with the -homogeneous property, and it was conjectured by Gong
and Song that difference balanced functions must be -homogeneous. First we
characterize difference balanced functions by generalized difference sets with
respect to two exceptional subgroups. We then derive several necessary and
sufficient conditions for -homogeneous difference balanced functions. In
particular, we reveal an unexpected equivalence between the -homogeneous
property and multipliers of generalized difference sets. By determining these
multipliers, we prove the Gong-Song conjecture for prime. Furthermore, we
show that every difference balanced function must be balanced or an affine
shift of a balanced function.Comment: 17 page
Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group
We inspect the relationship between relative Fourier multipliers on
noncommutative Lebesgue-Orlicz spaces of a discrete group and relative
Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four
applications are given: lacunary sets; unconditional Schauder bases for the
subspace of a Lebesgue space determined by a given spectrum, that is, by a
subset of the group; the norm of the Hilbert transform and the Riesz projection
on Schatten-von-Neumann classes with exponent a power of 2; the norm of
Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less
than 1.Comment: Corresponds to the version published in the Canadian Journal of
Mathematics 63(5):1161-1187 (2011
The Dirichlet space: A Survey
In this paper we survey many results on the Dirichlet space of analytic
functions. Our focus is more on the classical Dirichlet space on the disc and
not the potential generalizations to other domains or several variables.
Additionally, we focus mainly on certain function theoretic properties of the
Dirichlet space and omit covering the interesting connections between this
space and operator theory. The results discussed in this survey show what is
known about the Dirichlet space and compares it with the related results for
the Hardy space.Comment: 35 pages, typoes corrected, some open problems adde
Closable Multipliers
Let (X,m) and (Y,n) be standard measure spaces. A function f in
is called a (measurable) Schur multiplier if
the map , defined on the space of Hilbert-Schmidt operators from
to by multiplying their integral kernels by f, is bounded
in the operator norm.
The paper studies measurable functions f for which is closable in the
norm topology or in the weak* topology. We obtain a characterisation of
w*-closable multipliers and relate the question about norm closability to the
theory of operator synthesis. We also study multipliers of two special types:
if f is of Toeplitz type, that is, if f(x,y)=h(x-y), x,y in G, where G is a
locally compact abelian group, then the closability of f is related to the
local inclusion of h in the Fourier algebra A(G) of G. If f is a divided
difference, that is, a function of the form (h(x)-h(y))/(x-y), then its
closability is related to the "operator smoothness" of the function h. A number
of examples of non-closable, norm closable and w*-closable multipliers are
presented.Comment: 35 page
Infinite Dimensional Multipliers and Pontryagin Principles for Discrete-Time Problems
The aim of this paper is to provide improvments to Pontryagin principles in
infinite-horizon discrete-time framework when the space of states and of space
of controls are infinite-dimensional. We use the method of reduction to finite
horizon and several functional-analytic lemmas to realize our aim
Cycles and 1-unconditional matrices
We characterize the 1-unconditional subsequences of the canonical basis
(e_rc) of elementary matrices in the Schatten-von-Neumann class S^p . The set I
of couples (r,c) must be the set of edges of a bipartite graph without cycles
of even length 4<=l<=p if p is an even integer, and without cycles at all if p
is a positive real number that is not an even integer. In the latter case, I is
even a Varopoulos set of V-interpolation of constant 1. We also study the
metric unconditional approximation property for the space S^p_I spanned by
(e_rc)_{(r,c)\in I} in S^p .Comment: 29 pages. This new version computes explicitly certain
unconditionality constants, shows how our results generalize Varopoulos' work
on V-Sidon sets, investigates the metric unconditional approximation property
in the same contex
Local Operator Multipliers and Positivity
We establish an unbounded version of Stinespring's Theorem and a lifting
result for Stinespring representations of completely positive modular maps
defined on the space of all compact operators. We apply these results to study
positivity for Schur multipliers. We characterise positive local Schur
multipliers, and provide a description of positive local Schur multipliers of
Toeplitz type. We introduce local operator multipliers as a non-commutative
analogue of local Schur multipliers, and obtain a characterisation that extends
earlier results concerning operator multipliers and local Schur multipliers. We
provide a description of the positive local operator multipliers in terms of
approximation by elements of canonical positive cones.Comment: 31 page
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