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Christian Berg

Publications and source records attributed to Christian Berg.

At least 37 records · Page 2Linked to original sources
arXiv2017

Schoenberg's theorem for real and complex Hilbert spheres revisited

Schoenberg's theorem for the complex Hilbert sphere proved by Christensen and Ressel in 1982 by Choquet theory is extended to the following result: Let L denote a locally compact group and let \overline{\D} denote the closed unit disc in the complex plane. Continuous functions f:\overline{\D}\times L\to \C such that f(ξ\cdot η,u^{-1}v) is a positive definite kernel on the product of the unit sphere in \ell_2(\C) and L are characterized as the functions with a uniformly convergent expansion f(z,u)=\sum_{m,n=0}^\infty φ_{m,n}(u)z^m\overline{z}^n, where φ_{m,n} is a double sequence of continuous positive definite functions on L such that \sumφ_{m,n}(e_L)<\infty (e_L is the neutral element of L). It is shown how the coefficient functions φ_{m,n} are obtained as limits from expansions for positive definite functions on finite dimensional complex spheres via a Rodrigues formula for disc polynomials. Similar results are obtained for the real Hilbert sphere.

math.CA↗
arXiv2016

Orthogonal expansions related to compact Gelfand pairs

Given a compact Gelfand pair (G,K) and a locally compact group L, we characterize the class P_K^\sharp(G,L) of continuous positive definite functions f:G\times L\to \C which are bi-invariant in the G-variable with respect to K. The functions of this class are the functions having a uniformly convergent expansion \sum_{φ\in Z} B(φ)(u)φ(x) for x\in G,u\in L, where the sum is over the space Z of positive definite spherical functions φ:G\to\C for the Gelfand pair, and (B(φ))_{φ\in Z} is a family of continuous positive definite functions on L such that \sum_{φ\in Z}B(φ)(e_L)<\infty. Here e_L is the neutral element of the group L. For a compact abelian group G considered as a Gelfand pair (G,K) with trivial K=\{e_G\}, we obtain a characterization of P(G\times L) in terms of Fourier expansions on the dual group \widehat{G}. The result is described in detail for the case of the Gelfand pairs (O(d+1),O(d)) and (U(q),U(q-1)) as well as for the product of these Gelfand pairs. The result generalizes recent theorems of Berg-Porcu (2016) and Guella-Menegatto (2016)

math.CA↗
arXiv2016

Solvability of the Hankel determinant problem for real sequences

To each nonzero sequence $s:= \{s_{n}\}_{n \geq 0}$ of real numbers we associate the Hankel determinants $D_{n} = \det \mathcal{H}_{n}$ of the Hankel matrices $\mathcal{H}_{n}:= (s_{i + j})_{i, j = 0}^{n}$, $n \geq 0$, and the nonempty set $N_{s}:= \{n \geq 1 \, | \, D_{n-1} \neq 0 \}$. We also define the Hankel determinant polynomials $P_0:=1$, and $P_n$, $n\geq 1$ as the determinant of the Hankel matrix $\mathcal H_n$ modified by replacing the last row by the monomials $1, x, \ldots, x^n$. Clearly $P_n$ is a polynomial of degree at most $n$ and of degree $n$ if and only if $n\in N_s $. Kronecker established in 1881 that if $N_s $ is finite then rank $\mathcal{H}_{n} = r$ for each $n \geq r-1$, where $r := \max N_s $. By using an approach suggested by I.S.Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence $\{t_n\}_{n\geq 0}$ to be of the form $t_n=D_n$, $n\geq 0$ for a real sequence $\{s_n\}_{n\geq 0}$. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial $ P_n $ satisfying deg$P_n = n\geq 1$ is preceded by a nonzero polynomial $P_{n-1}$ whose degree can be strictly less than $n-1$ and which has no common zeros with $ P_n $. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that $D_0 > 0, \ldots, D_{r-1} > 0 $ and $D_n=0$ for all $n\geq r$.

math.CA↗
arXiv2015

From Schoenberg coefficients to Schoenberg functions

In his seminal paper, Schoenberg (1942) characterized the class P(S^d) of continuous functions f:[-1,1] \to \R such that f(\cos θ) is positive definite over the product space S^d \times S^d, with S^d being the unit sphere of \R^{d+1} and θbeing the great circle distance. In this paper, we consider the product space S^d \times G, for G a locally compact group, and define the class P(S^d, G) of continuous functions f:[-1,1]\times G \to \C such that f(\cos θ, u^{-1}\cdot v) is positive definite on S^d \times S^d \times G \times G. This offers a natural extension of Schoenberg's Theorem. Schoenberg's second theorem corresponding to the Hilbert sphere S^\infty is also extended to this context. The case G=\R is of special importance for probability theory and stochastic processes, because it characterizes completely the class of space-time covariance functions where the space is the sphere, being an approximation of Planet Earth.

math.CA↗
arXiv2014

A determinant characterization of moment sequences with finitely many mass-points

To a sequence (s_n)_{n\ge 0} of real numbers we associate the sequence of Hankel matrices \mathcal H_n=(s_{i+j}),0\le i,j \le n. We prove that if the corresponding sequence of Hankel determinants D_n=\det\mathcal H_n satisfy D_n>0 for n<n_0 while D_n=0 for n\ge n_0, then all Hankel matrices are positive semi-definite, and in particular (s_n) is the sequence of moments of a discrete measure concentrated in n_0 points on the real line. We stress that the conditions D_n\ge 0 for all n do not imply the positive semi-definiteness of the Hankel matrices.

math.CA↗
arXiv2013

On the order of indeterminate moment problems

For an indeterminate moment problem we denote the orthonormal polynomials by P_n. We study the relation between the growth of the function P(z)=(\sum_{n=0}^\infty|P_n(z)|^2)^{1/2} and summability properties of the sequence (P_n(z)). Under certain assumptions on the recurrence coefficients from the three term recurrence relation zP_n(z)=b_nP_{n+1}(z)+a_nP_n(z)+b_{n-1}P_{n-1}(z), we show that the function P is of order αwith 0<α<1, if and only if the sequence (P_n(z)) is absolutely summable to any power greater than 2α. Furthermore, the order αis equal to the exponent of convergence of the sequence (b_n). Similar results are obtained for logarithmic order and for more general types of slow growth. To prove these results we introduce a concept of an order function and its dual. We also relate the order of P with the order of certain entire functions defined in terms of the moments or the leading coefficient of P_n

math.CA↗
arXiv2013

Asymptotic behaviour of the Urbanik semigroup

We revisit the product convolution semigroup of probability densities e_c(t),c>0 on the positive half-line with moments (n!)^c and determine the asymptotic behaviour of e_c(t) for large and small t>0. This shows that (n!)^c is indeterminate as Stieltjes moment sequence if and only if c>2

math.CV↗
arXiv2012

Exact Gap Computation for Code Coverage Metrics in ISO-C

Test generation and test data selection are difficult tasks for model based testing. Tests for a program can be meld to a test suite. A lot of research is done to quantify the quality and improve a test suite. Code coverage metrics estimate the quality of a test suite. This quality is fine, if the code coverage value is high or 100%. Unfortunately it might be impossible to achieve 100% code coverage because of dead code for example. There is a gap between the feasible and theoretical maximal possible code coverage value. Our review of the research indicates, none of current research is concerned with exact gap computation. This paper presents a framework to compute such gaps exactly in an ISO-C compatible semantic and similar languages. We describe an efficient approximation of the gap in all the other cases. Thus, a tester can decide if more tests might be able or necessary to achieve better coverage.

cs.SE↗
arXiv2011

On an iteration leading to a q-analogue of the Digamma function

We show that the q-Digamma function psi_q for 0<q<1 appears in an iteration studied by Berg and Durán. In addition we determine the probability measure ν_q with moments 1/\sum_{k=1}^{n+1} (1-q)/(1-q^k), which are q-analogues of the reciprocals of the harmonic numbers.

math.CA↗
arXiv2011

A completely monotonic function used in an inequality of Alzer

The function $G(x)=(1-\ln x /\ln(1+x))x\ln x$ has been considered by Alzer, Qi and Guo. We prove that $G'$ is completely monotonic by finding an integral representation of the holomorphic extension of $G$ to the cut plane. A main difficulty is caused by the fact that $G'$ is not a Stieltjes function.

math.CA↗
arXiv2011

A question by Chihara about shell polynomials and indeterminate moment problems

The generalized Stieltjes--Wigert polynomials depending on parameters 0\le p<1 and 0<q<1 are discussed. By removing the mass at zero of the N-extremal solution concentrated in the zeros of the D-function from the Nevanlinna parametrization, we obtain a discrete measure μ^M which is uniquely determined by its moments. We calculate the coefficients of the corresponding orthonormal polynomials (P^M_n). As noticed by Chihara, these polynomials are the shell polynomials corresponding to the maximal parameter sequence for a certain chain sequence. We also find the minimal parameter sequence, as well as the parameter sequence corresponding to the generalized Stieltjes--Wigert polynomials, and compute the value of related continued fractions. The mass points of μ^M have been studied in recent papers of Hayman, Ismail--Zhang and Huber. In the special case of p=q, the maximal parameter sequence is constant and the determination of μ^M and (P^M_n) gives an answer to a question posed by Chihara in 2001

math.CA↗
arXiv2010

Fibonacci numbers and orthogonal polynomials

We prove that the sequence $(1/F_{n+2})_{n\ge 0}$ of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability, and we identify the orthogonal polynomials as little $q$-Jacobi polynomials with $q=(1-\sqrt{5})/(1+\sqrt{5})$. We prove that the corresponding kernel polynomials have integer coefficients, and from this we deduce that the inverse of the corresponding Hankel matrices $(1/F_{i+j+2})$ have integer entries. We prove analogous results for the Hilbert matrices.

math.NT↗
arXiv2010

On a fixed point in the metric space of normalized Hausdorff moment sequences

We show that the transformation (x_n)_{n\ge 1}\to (1/(1+x_1+...+x_n))_{n\ge 1} of the compact set of sequences (x_n)_{n\ge 1} of numbers from the unit interval [0,1] has a unique fixed point, which is attractive. The fixed point turns out to be a Hausdorff moment sequence studied in papers by Berg and Durán in 2008.

math.CA↗
arXiv2009

A Pick function related to the sequence of volumes of the unit ball in n-space

We show that F_a(x)=\frac{\ln Γ(x+1)}{x\ln(ax)} is a Pick function for a\ge 1 and find its integral representation. We also consider the function f(x)=(\frac{π^{x/2}}{Γ(1+x/2)})^{1/(x\ln x)} and show that \ln f(x+1) is a Stieltjes function and that f(x+1) is completely monotonic on (0,\infty). In particular f(n)=Ω_n^{1/(n\ln n)},n\ge 2 is a Hausdorff moment sequence. Here Ω_n is the volume of the unit ball in Euclidean n-space

math.CA↗
arXiv2009

The smallest eigenvalue of Hankel matrices

Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of lambda_N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where lambda_N is known to be bounded below by a positive constant, we prove that the limit of the n'th smallest eigenvalue of H_N for N tending to infinity tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.

math.CA↗
arXiv2009

Fibonacci numbers, Euler's 2-periodic continued fractions and moment sequences

We prove that certain sequences of finite continued fractions associated with a 2-periodic continued fraction with period a,b>0 are moment sequences of discrete signed measures supported in the interval [-1,1], and we give necessary and sufficient conditions in order that these measures are positive. For a=b=1 this proves that the sequence of ratios F_{n+1}/F_{n+2}, n\ge 0 of consecutive Fibonacci numbers is a moment sequence.

math.CA↗
arXiv2008

The Dagum family of isotropic correlation functions

A function $ρ:[0,\infty)\to(0,1]$ is a completely monotonic function if and only if $ρ(\Vert\mathbf{x}\Vert^2)$ is positive definite on $\mathbb{R}^d$ for all $d$ and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function \[ρ(β,γ)(x)=1-\biggl(\frac{x^β}{1+x^β}\biggr )^γ,\qquad x\ge 0, β,γ>0,\] called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any $d$-dimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.

math.ST↗