arXiv subjects

Patricia Szokol

Publications and source records attributed to Patricia Szokol.

5 recordsLinked to original sources

arXiv2026

Conjectures of Bernstein and Erd\H os for weighted Lagrange interpolation on the halfline with exponential weights

Let I=[a,b] and consider the degree n Lagrange interpolation at the nodes x, where x\in S:={x=(x_0,x_1,...,x_n):a=x_0 m_j(x). The conjectures were proved by Kilgore and de Boor--Pinkus in 1978. Since then, analogous results were obtained only for a few cases when interpolation is made to certain very special spaces of polynomials, or when we apply weighted interpolation with rather special weights. Worse than that, it turned out that published proofs of results on infinite intervals and weighted interpolation were seriously flawed. Here we prove the Bernstein and Erd\H os Conjectures for the case of exponentially weighted polynomials on the halfline. This is the first proof of these conjectures in a situation where, contrary to all existing successful proofs, we encounter singularity of certain derivative matrices.

math.CA↗

arXiv2026

On some examples and counterexamples about weighted Lagrange interpolation with Exponential and Hermite weights

The famous Bernstein conjecture about optimal node systems in classical polynomial Lagrange interpolation, standing unresolved for about half a century, was solved by T. Kilgore in 1978. Immediately following him, also the additional conjecture of Erdős was solved by de Boor and Pinkus. These breakthrough achievements were built on a fundamental auxiliary result on nonsingularity of derivative (Jacobian) matrices of certain interval maxima in function of the nodes. After the above breakthrough, a considerable effort was made to extend the results to the case of at least certain Chebyshev-Haar spaces of functions. Here, we analyse, in what extent the key nonsingularity statement remains true in case of exponentially weighted interpolation on the halfline, or with Hermite weights on the full real line. In these settings counterexamples demonstrate that the respective derivative matrices may as well be singular. It remains to further study if the Bernstein- and Erdős characterizations remain valid. The ``hybrid'' Chebyshev-Haar system of exponentially weighted polynomials adjoined with constant functions and the corresponding interpolation were previously studied, as well. Some hints were also given for the proof of the respective Bernstein and Erdős conjectures. We present in detail the full proof together with all the auxiliary results needed in this setting.

math.CA↗

arXiv2022

A dichotomy result for strictly increasing bisymmetric maps

In this paper we show some remarkable consequences of the method which proves that every bisymmetric, symmetric, reflexive, strictly monotonic binary map on a proper interval is continuous, in particular it is a quasi-arithmetic mean. Now we demonstrate that this result can be refined in the way that the symmetry condition can be weakened by assuming symmetry only for a pair of distinct points of an interval.

math.CA↗

arXiv2021

Characterization of quasi-arithmetic means without regularity condition

In this paper we show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function $F:I^2\to I$ is continuous. As a consequence, we obtain a finer characterization of quasi-arithmetic means than the classical results of Aczél, Kolmogoroff, Nagumo and de Finetti.

math.CA↗

arXiv2016

Trace and determinant preserving maps of matrices

Suppose a map $ϕ$ on the set of positive definite matrices satisfies $\det(A+B)=\det(ϕ(A)+ϕ(B))$. Then we have $${\rm tr}(AB^{-1}) = {\rm tr}(ϕ(A){ϕ(B)}^{-1}).$$ Through this viewpoint, we show that $ϕ$ is of the form $ϕ(A)= M^*AM$ or $ϕ(A)= M^*A^tM$ for some invertible matrix $M$ with $\det (M^*M)=1$. We also characterize the map $ϕ: \mathcal{S} \rightarrow \mathcal{S}$ preserving the determinant of convex combinations in $\mathcal{S}$ by using similar method. Here $\mathcal{S}$ can be the set of complex matrices, positive definite matrices, symmetric matrices, and upper triangular matrices.

math.RA↗