18,581 research outputs found

    Difference Balanced Functions and Their Generalized Difference Sets

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    Difference balanced functions from Fqn∗F_{q^n}^* to FqF_q are closely related to combinatorial designs and naturally define pp-ary sequences with the ideal two-level autocorrelation. In the literature, all existing such functions are associated with the dd-homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be dd-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 dd-homogeneous difference balanced functions. In particular, we reveal an unexpected equivalence between the dd-homogeneous property and multipliers of generalized difference sets. By determining these multipliers, we prove the Gong-Song conjecture for qq 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

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    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

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    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

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    Let (X,m) and (Y,n) be standard measure spaces. A function f in L∞(X×Y,m×n)L^\infty(X\times Y,m\times n) is called a (measurable) Schur multiplier if the map SfS_f, defined on the space of Hilbert-Schmidt operators from L2(X,m)L_2(X,m) to L2(Y,n)L_2(Y,n) by multiplying their integral kernels by f, is bounded in the operator norm. The paper studies measurable functions f for which SfS_f 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

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    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

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    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

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    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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