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Peter Grabner

Publications and source records attributed to Peter Grabner.

9 recordsLinked to original sources
arXiv2026

Levenshtein-type bounds for $(a,b)$-avoiding spherical codes

We develop a Levenshtein-type linear programming framework (for maximal $s$-codes and for minimal energy) for $(a,b)$-avoiding spherical codes, that is, codes whose inner products between distinct points avoid $(a,b)$. We introduce and investigate signed measures that are positive definite up to an appropriate degree and yield stationary auxiliary linear programming polynomials and exact quadrature formulas. We give explicit conditions for positivity of the quadrature weights and sufficient Krein-type conditions for nonnegative Gegenbauer coefficients (positive definiteness of the candidate optimal linear programming polynomials). Under these hypotheses and suitable node-location assumptions, the resulting cardinality bounds are optimal within the corresponding degree restriction and unique under natural nondegeneracy conditions. Under proper conditions on the quadrature nodes, we obtain universal lower energy bounds by utilizing Hermite interpolants as polynomial solutions of the respective linear program. Common test functions characterize optimality in the full polynomial linear programs under mild additional assumptions.

cs.IT↗
arXiv2024

Positive definite singular kernels on two-point homogeneous spaces

We study positive definiteness of kernels $K(x,y)$ on two-point homogeneous spaces. As opposed to the classical case, which has been developed and studied in the existing literature, we allow the kernel to have an (integrable) singularity for $x=y$. Specifically, the Riesz kernel $d(x,y)^{-s}$ (where $d$ denotes some distance on the space) is a prominent example. We derive results analogous to Schoenberg's characterization of positive definite functions on the sphere, Schur's lemma on the positive definiteness of the product of positive definite functions, and Schoenberg's characterization of functions positive definite on all spheres. We use these results to better understand the behavior of the Riesz kernels for the geodesic and chordal distances on projective spaces.

math.CA↗
arXiv2024

Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus

Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so-called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere $\mathbb{S}^d$ and the flat torus $\mathbb{T}^d$, and the so-called spherical ensemble on $\mathbb{S}^2$, which originates in random matrix theory. We extend results of Beltrán, Marzo and Ortega-Cerdà on the Riesz $s$-energy of the harmonic ensemble to the nonsingular regime $s<0$, and as a corollary find the expected value of the spherical cap $L^2$ discrepancy via the Stolarsky invariance principle. We find the expected value of the $L^2$ discrepancy with respect to axis-parallel boxes and Euclidean balls of the harmonic ensemble on $\mathbb{T}^d$. We also show that the spherical ensemble and the harmonic ensemble on $\mathbb{S}^2$ and $\mathbb{T}^2$ with $N$ points attain the optimal rate $N^{-1/2}$ in expectation in the Wasserstein metric $W_2$, in contrast to i.i.d. random points, which are known to lose a factor of $(\log N)^{1/2}$.

math.CA↗
arXiv2021

How well-conditioned can the eigenvalue problem be?

The condition number for eigenvalue computations is a well--studied quantity. But how small can we expect it to be? Namely, which is a perfectly conditioned matrix w.r.t. eigenvalue computations? In this note we answer this question with exact first order asymptotic.

math.NA↗
arXiv2018

Comparison of probabilistic and deterministic point sets

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (spherical $t$-designs) are better or as good as probabilistic ones. We find asymptotic equalities for the discrete Riesz $s$-energy of sequences of well separated $t$-designs on the unit sphere $\mathbb{S}^d \subset \mathbb{R}^{d+1}$, $d\geq2$. The case $d=2$ was studied Hesse and Leopardi. Bondarenko, Radchenko, and Viazovska established, that for $d\geq 2$, there exists a constant $c_{d}$, such that for every $N> c_{d}t^{d}$ there exists a well-separated spherical $t$-design on $\mathbb{S}^{d}$ with $N$ points. For this reason, in our paper we assume, that the sequence of well separated spherical $t$-designs is such that $t$ and $N$ are related by $N\asymp t^{d}$.

math.CA↗
arXiv2018

Upper and lower estimates for numerical integration errors on spheres of arbitrary dimension

In this paper we study the worst-case error of numerical integration on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, for certain spaces of continuous functions on $\mathbb{S}^{d}$. For the classical Sobolev spaces $\mathbb{H}^s(\mathbb{S}^d)$ ($s>\frac d2$) upper and lower bounds for the worst case integration error have been obtained By Brauchart, Hesse, and Sloan earlier in papers. We investigate the behaviour for $s\to\frac d2$ by introducing spaces $\mathbb{H}^{\frac d2,γ}(\mathbb{S}^d)$ with an extra logarithmic weight. For these spaces we obtain similar upper and lower bounds for the worst case integration error.

math.CA↗
arXiv2012

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincaré and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpiński gasket.

math.SP↗
arXiv2005

The Zeta Function of the Laplacian on Certain Fractals

We prove that the zeta-function $ζ_Δ$ of the Laplacian $Δ$ on a self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta-function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function.

math.SP↗