arXiv subjects

Lutz Klotz

Publications and source records attributed to Lutz Klotz.

5 recordsLinked to original sources

arXiv2026

Extremal representations of functions of matrices and applications to multivariate prediction

Motivated by two seminal results of multivariate prediction theory by Helson and Lowdenslager and by Wiener and Masani we prove extremal representations of functions of matrices and derive their prediction-theoretic consequences. We also sketch a way to obtain matricial inequalities from our results. The main goal of the paper is the computation of the infimum of a set of values of the form $tr(A ΔA^*)$, where $Δ$ is a given non-negative Hermitian $n \times n$ matrix and the choices for $A$ exhauste a certain set of $n \times n$ matrices. In particular, we focus on norm-bounded unit spheres with certain types of properties of unitary invariance, what allows an application of the theory of majorization.

math.FA↗

arXiv2026

Remarks on infimum and maximal lower bounds of a set of bounded self-adjoint operators

The notions of infimum and maximal lower bounds of a set $\mathfrak M$ of bounded self-adjoint operators were mainly studied for a set $\mathfrak M$ of two elements. The present paper deals with more general sets $\mathfrak M$, where it is required that $\mathfrak M$ is nonempty and bounded from below. Kadison's theorem on the existence of the infimum of a two-element set is proved for a countable and weak-operator compact set $\mathfrak M$. Stott's recent results on the structure of the set of maximal lower bounds of a finite set of Hermitian matrices are discussed and partially generalized. We are also concerned with the greatest lower bound and maximal lower bounds under certain restrictions. It is shown that the set of all lower bounds of $\mathfrak M$ commuting with all elements of $\mathfrak M$ possesses the greatest element if $\mathfrak M$ is a set of pairwise commuting operators. The theorem of Moreland and Gudder on the existence of the greatest positive lower bound of a set of two positive matrices is extended to an arbitrary finite set of positive matrices.

math.FA↗

arXiv2021

Stability of trigonometric approximation in $L^p$ and applications to prediction theory

Let $Γ$ be an LCA group and $(μ_n)$ be a sequence of bounded regular Borel measures on $Γ$ tending to a measure $μ_0$. Let $G$ be the dual group of $Γ$, $S$ be a non-empty subset of $G \setminus \{ 0 \}$, and $[{\mathcal T}(S)]_{μ_n,p}$ the subspace of $L^p(μ_n)$, $p \in (0,\infty)$, spanned by the characters of $Γ$ which are generated by the elements of $S$. The limit behaviour of the sequence of metric projections of the function $1$ onto $[{\mathcal T}(S)]_{μ_n,p}$ as well as of the sequence of the corresponding approximation errors are studied. The results are applied to obtain stability theorems for prediction of weakly stationary or harmonizable symmetric $p$-stable stochastic processes. Along with the general problem the particular cases of linear interpolation or extrapolation as well as of a finite or periodic observation set are studied in detail and compared to each other.

math.ST↗

arXiv2017

Duality results for a general trigonometric approximation problem

Let $α\in(1,\infty)$ and $μ$ be a regular finite Borel measure on a locally compact abelian group. The paper deals with a general trigonometric approximation problem in $L^α(μ)$, which arises in prediction theory of harmonizable symmetric $α$-stable processes. To solve it, a duality method is applied, which is due to Nakazi and was generalized by Miamee and Pourahmadi and in the sequel successfully applied by several authors. The novelty of the present paper is that we do not make any additional assumption on $μ$. Moreover, for $α=2$, multivariate extensions are obtained.

math.FA↗

arXiv2016

A necessary condition for certain functions to preserve positive semi-definiteness on partitioned matrices

If $f$ is a symmetric complex-valued function on the $m$-fold Cartesian product of the set of non-negative reals and $A$ is a positive semi-definite $m\times m$ matrix with eigenvalues $λ_j$, we set $f(A):=f(λ_1,\dotsc,λ_m)$. It is shown that if $[f(A_{αβ})]$ is positive semi-definite whenever $[A_{αβ}]$ is a positive semi-definite matrix with positive semi-definite entries $A_{αβ}$, then $f$ has a power series expansion with positive coefficients.

math.FA↗