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Register a new userProducts of generalized Nevanlinna functions with symmetric rational functions
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New classes of generalized Nevanlinna functions, which under multiplication with an arbitrary fixed symmetric rational function remain generalized Nevanlinna functions, are introduced. Characterizations for these classes of functions are established by connecting the canonical factorizations of the product function and the original generalized Nevanlinna function in a constructive manner. Also a detailed functional analytic treatment of these classes of functions is carried out by investigating the connection between the realizations of the product function and the original function. The operator theoretic treatment of these realizations is based on the notions of rigged spaces, boundary triplets, and associated Weyl functions.
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We analyze the fine structure of Clark measures and Clark isometries associated with two-variable rational inner functions on the bidisk. In the degree (n,1) case, we give a complete description of supports and weights for both generic and exceptional Clark measures, characterize when the associated embedding operators are unitary, and give a formula for those embedding operators. We also highlight connections between our results and both the structure of Agler decompositions and study of extreme points for the set of positive pluriharmonic measures on 2-torus.
For each integrability parameter $p in (0,infty]$, the critical smoothness of a periodic generalized function $f$, denoted by $s_f(p)$ is the supremum over the smoothness parameters $s$ for which $f$ belongs to the Besov space $B_{p,p}^s$ (or other similar function spaces). This paper investigates the evolution of the critical smoothness with respect to the integrability parameter $p$. Our main result is a simple characterization of all the possible critical smoothness functions $pmapsto s_f(p)$ when $f$ describes the space of generalized periodic functions. We moreover characterize the compressibility of generalized periodic functions in wavelet bases from the knowledge of their critical smoothness function.
We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Frolicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Frolicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Frolicher spaces and diffeological spaces, and they form a complete and cocomplete Cartesian closed category. We deeply study functionally generated spaces (and Frolicher spaces) as a framework for Schwartz distributions, and prove that in the category of diffeological spaces, both the special and the full Colombeau algebras are smooth differential algebras, with a smooth embedding of Schwartz distributions and smooth pointwise evaluations of Colombeau generalized functions.
We prove that generalized matrix functions satisfy a block-matrix strong superadditivity inequality over the cone of positive semidefinite matrices. Our result extends a recent result of Paksoy-Turkmen-Zhang (V. Paksoy, R. Turkmen, F. Zhang, Inequalities of generalized matrix functions via tensor products, Electron. J. Linear Algebra 27 (2014) 332-341.). As an application, we obtain a short proof of a classical inequality of Thompson (1961) on block matrix determinants.
The properties of the compactness of interpolation sets of algebras of generalized analytic functions are investigated and convenient sufficient conditions for interpolation are given.
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