Analysis Mathematica最新文献

We use the characterization of the case of equality in Barthe's geometric reverse Brascamp-Lieb inequality to characterize equality in Liakopoulos's volume estimate in terms of sections by certain lower-dimensional linear subspaces.

This paper is a survey on the theory of finely holomorphicfunctions in one variable as developed by Fuglede in the 1980s andin several variables as developed recently by El Kadiri, Fuglede and Wiegerinck. Some applications to pluripotential theory are also given.

A new approach to normal operators in real Hilbert spaces isdiscussed, and a spectral representation is obtained, derived directly from thecomplex case. The results are then applied to quaternionic normal operators,regarded as a special class of real normal operators. This point of view allows usto consider their spectrum and associated measures to be defined on subsets ofthe complex plane, in a classical manner.

Bounded holomorphic interpolation problems associated to finitely many data have, in general, distinct solutions. Uniqueness arises only in some convex extreme configurations. Rational inner functions in a polydisk are the best understood examples in this sense. We analyze the continuity of global solutions as functions of the finite interpolation data in neighbourhoods of elements distinguished by this uniqueness property. Our study covers rational inner or Cayley rational inner functions in the polydisk and automorphisms of the Euclidean ball. The proof of the main superresolution result is derived from optimization theory techniques and volume estimates of sublevel sets of real polynomials, both emerging from Markov's multivariable moment problem.

We prove that every plurifinely continuous function on an open set (Omegasubset mathbb{C}^n (ngeq 1)) is quasi-continuous relatively to the Bedford-Taylor capacity on (Omega).

For the (alpha)-Green kernel (g^alpha_D) on a domain (Dsubsetmathbb R^n), (n geq 2), associated with the (alpha)-Riesz kernel (|x-y|^{alpha-n}), where (alphain(0,n)) and (alpha leq 2), and a relatively closed set (Fsubset D), we investigate the problem on minimizing the Gauss functional

(vartheta) being a given positive (Radon) measure concentrated on (Dsetminus F), and (mu) ranging over all probability measures of finite energy, supported in (D) by (F). For suitable (vartheta), we find necessary and/or sufficient conditions for the existence of the solution to the problem, give a description of its support, provide various alternative characterizations, and prove convergence theorems when (F) is approximated by partially ordered families of sets. The analysis performed is substantially based on the perfectness of the (alpha)-Green kernel, discovered by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018).

We study the C(^*) algebra generated by the composition operator (C_a) acting on the Hardy space (H^2) of the unit disk, given by (C_af=fcircvarphi_a) , where

for (|a|<1). Also several operators related to (C_a) are examined.

The notion of weak tiling played a key role in the proof of Fuglede's spectral set conjecture for convex domains, due to the fact that every spectral set must weakly tile its complement. In this paper, we revisit the notion of weak tiling and establish some geometric properties of sets that weakly tile their complement. If (A subset mathbb{R}^d) is a convex polytope, we give a direct and self-contained proof that A must be symmetric and have symmetric facets. If (A subset mathbb{R}) is a finite union of intervals, we give a necessary condition on the lengths of the gaps between the intervals.

It is known that the inequality

between the quadratic spectral concentration of a function and that of its decreasing rearrangement holds for any function (fin L^2 ), ( lvert {textrm{supp}} f|=T ), if and only if the product (WT) does not exceed the critical value (approx 0.81). We show that by restricting ourselves to characteristic functions we can enlarge this range up to (WTleq 4/3). Besides, we establish various properties of minimizers of the difference (int_{-W/2}^{W/2} |widehat{{mathbb{1}}_A^*}(xi) | ^2 , dxi -int_{-W/2}^{W/2} |widehat{{mathbb{1}}_A}(xi) | ^2 , dxi ) over sets (A) of finite measure and prove that this difference is non-negative for all (W,T>0) if (A) is the union of two intervals. As a corollary, we obtain a sharp (up to a constant) estimate for the (L^2 )-norms of non-harmonic trigonometric polynomials with alternating coefficients (pm 1).