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Carleson's Theorem asserts the pointwise convergence of Fourier series of square integrable functions. We give a complete proof, following joint work of the author and C. Thiele. Over 20 exercises are also detailed. We also discuss the…

Classical Analysis and ODEs · Mathematics 2007-05-23 Michael Lacey

Starting with univariate polynomial interpolation we arrive to a natural generalization of fundamental theorem of algebra for certain systems of multivariate algebraic equations.

Numerical Analysis · Mathematics 2025-10-20 H. Hakopian , M. Tonoyan

We give necessary and sufficient conditions under which a density matrix acting on a two-fold tensor product space is separable. Our conditions are given in terms of quantum conditional information transmission.

Quantum Physics · Physics 2016-09-08 Robert R. Tucci

In this work we study Hardy Sobolev spaces in the ball of $C^n$ with respect to interpolating sequences and Carleson measures. We compare them with the classical Hardy spaces of the ball and we stress analogies and differences.

Complex Variables · Mathematics 2015-09-08 Eric Amar

This paper is meant to be a gentle introduction to Carleson's Theorem on pointwise convergence of Fourier series.

Classical Analysis and ODEs · Mathematics 2016-12-09 Ciprian Demeter

We characterize simply interpolating sequences (also known as onto interpolating sequences) for complete Pick spaces. We show that a sequence is simply interpolating if and only if it is strongly separated. This answers a question of Agler…

Functional Analysis · Mathematics 2023-06-27 Nikolaos Chalmoukis , Alberto Dayan , Michael Hartz

In this article, we define a matrix multinomial distribution. We prove some properties of the matrix multinomial distribution. We prove that the matrix Poisson distribution can be used as an approximation to the matrix multinomial…

Probability · Mathematics 2021-04-30 Yuriy Yurchenko

In this paper we develop a general method for improving Jensen-type inequalities for convex and, even more generally, for piecewise convex functions. Our main result relies on the linear interpolation of a convex function. As a consequence,…

Classical Analysis and ODEs · Mathematics 2016-10-06 Daeshik Choi , Mario Krnić , Josip Pecarić

Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an $H$-polar decomposition are found. In the process an equivalent to Witt's…

Functional Analysis · Mathematics 2021-06-22 G. J. Groenewald , D. B. Janse van Rensburg , A. C. M. Ran , F. Theron , M. van Straaten

We study simply interpolating sequences for the Dirichlet space in the unit disc. In particular we are interested in comparing three different sufficient conditions for simply interpolating sequences. The first one is the the so called one…

Complex Variables · Mathematics 2022-10-25 Nikolaos Chalmoukis

Multilinear interpolation is a powerful tool used in obtaining strong type boundedness for a variety of operators assuming only a finite set of restricted weak-type estimates. A typical situation occurs when one knows that a multilinear…

Functional Analysis · Mathematics 2007-05-23 Loukas Grafakos , Terence Tao

The concept of a fully interlacing matrix of formal power series with real coefficients is introduced. This concept extends and strengthens that of an interlacing sequence of real-rooted polynomials with nonnegative coefficients, in the…

Combinatorics · Mathematics 2024-04-22 Christos A. Athanasiadis , David G. Wagner

We construct automorphisms of $\C^n$ which map certain discrete sequences one onto another with prescribed finite jet at each point, thus solving a general Mittag-Leffler interpolation problem for automorphisms. Under certain circumstances,…

Complex Variables · Mathematics 2016-09-06 Gregery T. Buzzard , Franc Forstneric

Various algebraic multigrid algorithms have been developed for solving problems in scientific and engineering computation over the past decades. They have been shown to be well-suited for solving discretized partial differential equations…

Numerical Analysis · Mathematics 2018-05-08 Xuefeng Xu , Chen-Song Zhang

In the space of all entire functions it is solved the problem of interpolation taking into account multiplicities by sums of the series of exponentials with the exponents from a given set. It is found a criterion of solubility of the…

Complex Variables · Mathematics 2016-12-20 S. G. Merzlyakov , S. V. Popenov

Some inequalities for positive linear maps on matrix algebras are given, especially asymmetric extensions of Kadison's inequality and several operator versions of Chebyshev's inequality. We also discuss well-known results around the matrix…

Functional Analysis · Mathematics 2010-03-12 Jean-Christophe Bourin , Éric Ricard

Interpolation Theory gives techniques for constructing spaces from two initial Banach spaces. We provide several conditions under which the restriction of a holomorphic map $f:X_0+X_1 \rightarrow Y_0+Y_1$ to the interpolated spaces (using…

Functional Analysis · Mathematics 2017-06-21 Pablo Jiménez-Rodíguez

We give a description, in analytic and geometric terms, of the interpolation sequences for the algebra of entire functions of exponential type which are bounded on the real line.

Complex Variables · Mathematics 2009-09-03 Xavier Massaneda , Joaquim Ortega-Cerdà

We prove analogues of the Craig interpolation theorem for the continuous model theory of metric structures.

Logic · Mathematics 2025-01-17 H. Jerome Keisler

Given a function on diagonal matrices, there is a unique way to extend this to an invariant (by conjugation) function on symmetric matrices. We show that the extension preserves regularity -- that is, if the original function is k times…

Functional Analysis · Mathematics 2007-05-23 Yury Grabovsky , Omar Hijab , Igor Rivin