arXiv subjects
Publications and source records attributed to Christian Berg.
In a paper by Novi Inverardi and Tagliani from 2024 it is claimed that the log-normal density has maximal entropy in the class of densities having the same moments as the log-normal density. We disprove the claim by finding a family of densities with larger entropy.
For an N-extremal solution $\mu$ to an indeterminate moment problem it is known by a theorem of M. Riesz that the measure $(1+x^2)^{-1}d\mu(x)$ is determinate. For $0<\alpha<1$ we show by contradiction that there exist indeterminate N-extremal solutions $\mu$ such that $(1+x^2)^{-\alpha}d\mu(x)$ is determinate, and there exist also indeterminate N-extremal solutions $\mu$ such that $(1+x^2)^{-\alpha}d\mu(x)$ is indeterminate. Explicit examples of such measures are so far only known when $\alpha=1/2$. For indeterminate Stieltjes moment problems and for N-extremal solutions $\mu$, we show that $(1+x^2)^{-1/2}d\mu(x)$ is indeterminate except when $\mu=\mu_F$ is the Friedrichs solution in case of which $(1+x^2)^{-1/2}d\mu_F(x)$ is determinate. We identify the Friedrichs and Krein solutions for some indeterminate Stieltjes moment problems.
For an indeterminate Hamburger moment problem we consider an infinite family of analytic densities solving the moment problem and we prove that they all have finite (Shannon) entropy. These densities are either all bounded or all unbounded. The result is illustrated by the Al-Salam--Carlitz moment problem, where all the densities in the family are bounded.
We consider the Jacobi operator (T,D(T)) associated with an indeterminate Hamburger moment problem, and present countable subsets S of the domain D(T) such that span(S) is dense in \ell^2. As an example we have S={(p_n(u))+B(u)(p_n(0)):D(u)=0}, where (p_n) denotes the orthonormal polynomials of the moment problem and B,D are two of the Nevanlinna functions. It is also proved that sets like S are optimal in the sense that if one vector is removed, then the span is no longer dense.
We show that under very mild conditions on a measure $\mu$ on the interval $[0,\infty)$, the span of $\{x^k\}_{k=n}^{\infty}$ is dense in $L^2(\mu)$ for any $n=0,1,\ldots$. We present two different proofs of this result, one based on the density index of Berg and Thill and one based on the Hilbert space $L^2(\mu)\oplus \mathbb{C}^{n+1}$. Using the index of determinacy of Berg and Dur\'an we prove that if the measure $\mu$ on $\mathbb{R}$ has infinite index of determinacy then the polynomial ideal $R(x)\mathbb{C}[x]$ is dense in $L^2(\mu)$ for any polynomial $R$ with zeros having no mass under $\mu$.
We consider a normalized indeterminate Hamburger moment sequence s which is supposed to be Stieltjes. We revisit old results about determinacy/indeterminacy in the sense of Stieltjes for s and we prove some new results about the concepts involved.
Let $f_r(x)=\log(1+rx)/\log(1+x)$ for $x>0$. We prove that $f_r$ is a complete Bernstein function for $0\le r\le 1$ and a Stieltjes function for $1\le r$. This answers a conjecture of David Bradley that $f_r$ is a Bernstein function when $0\le r\le 1$.
We consider the Jacobi operator (T,D(T)) associated with an indeterminate Hamburger moment problem, i.e., the operator in $\ell^2$ defined as the closure of the Jacobi matrix acting on the subspace of complex sequences with only finitely many non-zero terms. It is well-known that it is symmetric with deficiency indices (1,1). For a complex number z let $\mathfrak{p}_z, \mathfrak{q}_z$ denote the square summable sequences (p_n(z)) and (q_n(z)) corresponding to the orthonormal polynomials p_n and polynomials q_n of the second kind. We determine whether linear combinations of $\mathfrak{p}_u,\mathfrak{p}_v,\mathfrak{q}_u,\mathfrak{q}_v$ for complex u,v belong to D(T) or to the domain of the self-adjoint extensions of T in $\ell^2$. The results depend on the four Nevanlinna functions of two variables associated with the moment problem. We also show that D(T) is the common range of an explicitly constructed family of bounded operators on $\ell^2$.
In a paper from 2016 D. R. Yafaev initiated a study of closable Hankel forms associated with the moments $(m_n)$ of a positive measure with infinite support on the real line. If $m_n=o(1)$ Yafaev characterized the closure of the form based on earlier work on quasi-Carleman operators. We give a new proof of the description of the closure based entirely on moment considerations. The main purpose of the present paper is a description of the self-adjoint Hankel operators associated with closed Hankel forms in the Hilbert space of square summable sequences. We do this not only in the case $m_n=o(1)$ studied by Yafaev but also in two other cases, where the Hankel form is closable, namely if the moment sequence is indeterminate or if the moment sequence is determinate with finite index of determinacy.
A family of recently investigated Bernstein functions is revisited and those functions for which the derivatives are logarithmically completely monotonic are identified. This leads to the definition of a class of Bernstein functions, which we propose to call Horn-Bernstein functions because of the results of Roger A. Horn.
We investigate conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions whose arguments are linear functions of the variable. We give necessary and sufficient conditions in terms of nonnegativity of a certain explicitly written measure in the q case and of a certain elementary function in the classical q=1 case. In the latter case we further provide simple new sufficient conditions leading to many new examples of logarithmically completely monotonic gamma ratios. Finally, we apply some of our results to study monotonicity of some gamma ratios and rational functions.
We give a characterization of positive definite integrable functions on a product of two Gelfand pairs as an integral of positive definite functions on one of the Gelfand pairs with respect to the Plancherel measure on the dual of the other Gelfand pair. In the very special case where the Gelfand pairs are Euclidean groups and the compact subgroups are reduced to the identity, the characterization is a much cited result in spatio-temporal statistics due to Cressie, Huang and Gneiting. When one of the Gelfand pairs is compact the characterization leads to results about expansions in spherical functions with positive definite expansion functions, thereby recovering recent results of the author in collaboration with Peron and Porcu. In the special case when the compact Gelfand pair consists of orthogonal groups, the characterization is important in geostatistics and covers a recent result of Porcu and White.
We show that a large collection of special functions, in particular Nielsen's beta function, are generalized Stieltjes functions of order 2, and therefore logarithmically completely monotonic. This includes the Laplace transform of functions of the form $xf(x)$, where $f$ is itself the Laplace transform of a sum of dilations and translations of periodic functions. Our methods are also applied to ratios of Gamma functions, and to the remainders in asymptotic expansions of the double Gamma function of Barnes.
In a paper from 2016 D. R. Yafaev considers Hankel operators associated with Hamburger moment sequences q_n and claims that the corresponding Hankel form is closable if and only if the moment sequence tends to 0. The claim is not correct, since we prove closability for any indeterminate moment sequence but also for certain determinate moment sequences corresponding to measures with finite index of determinacy. It is also established that Yafaev's result holds if the moments satisfy \root{2n}\of{q_{2n}}=o(n).
Motivated by the problem of determining the values of $\alpha>0$ for which $f_\alpha(x)=e^\alpha - (1+1/x)^{\alpha x},\ x>0$ is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family $\varphi_\alpha$, $\alpha>0$, of entire functions such that $f_\alpha(x) =\int_0^\infty e^{-sx}\varphi_\alpha(s)\,ds, \ x>0.$ We show that each function $\varphi_\alpha$ has an expansion in power series, whose coefficients are determined in terms of Bell polynomials. This expansion leads to several properties of the functions $\varphi_\alpha$, which turn out to be related to the well known Bessel function $J_1$ and the Lambert $W$ function. On the other hand, by numerically evaluating the series expansion, we are able to show the behavior of $\varphi_\alpha$ as $\alpha$ increases from $0$ to $\infty$ and to obtain a very precise approximation of the largest $\alpha>0$ such that $\varphi_\alpha(s)\geq0,\, s>0$, or equivalently, such that $f_\alpha$ is completely monotonic.
We prove that s_n(a,b)=\Gamma(an+b)/\Gamma(b), n=0,1,\ldots is an infinitely divisible Stieltjes moment sequence for arbitrary a,b>0. Its powers s_n(a,b)^c, c>0 are Stieltjes determinate if and only if ac\le 2. The latter was conjectured in a paper by Lin (ArXiv: 1711.01536) in the case b=1. We describe a product convolution semigroup \tau_c(a,b), c>0 of probability measures on the positive half-line with densities e_c(a,b) and having the moments s_n(a,b)^c. We determine the asymptotic behaviour of e_c(a,b)(t) for t\to 0 and for t\to\infty, and the latter implies the Stieltjes indeterminacy when ac>2. The results extend previous work of the author and J. L. L\'opez and lead to a convolution semigroup of probability densities (g_c(a,b)(x))_{c>0} on the real line. The special case (g_c(a,1)(x))_{c>0} are the convolution roots of the Gumbel distribution with scale parameter a>0. All the densities g_c(a,b)(x) lead to determinate Hamburger moment problems.
Let $(s_n)_{n\ge 0}$ denote an indeterminate Hamburger moment sequence and let $\mathcal H=\{s_{m+n}\}$ be the corresponding positive definite Hankel matrix. We consider the question if there exists an infinite symmetric matrix $\mathcal A=\{a_{j,k}\}$, which is an inverse of $\mathcal H$ in the sense that the matrix product $\mathcal A\mathcal H$ is defined by absolutely convergent series and $\mathcal A\mathcal H$ equals the identity matrix $\mathcal I$, a property called (aci). A candidate for $\mathcal A$ is the coefficient matrix of the reproducing kernel of the moment problem, considered as an entire function of two complex variables. We say that the moment problem has property (aci), if (aci) holds for this matrix $\mathcal A$. We show that this is true for many classical indeterminate moment problems but not for the symmetrized version of a cubic birth-and-death process studied by Valent and co-authors. We consider mainly symmetric indeterminate moment problems and give a number of sufficient conditions for (aci) to hold in terms of the recurrence coefficients for the orthonormal polynomials. A sufficient condition is a rapid increase of the recurrence coefficients in the sense that the quotient between consecutive terms is uniformly bounded by a constant strictly smaller than one. We also give a simple example, where (aci) holds, but an inverse matrix of $\mathcal H$ is highly non-unique.
In 1998 G. Valent made conjectures about the order and type of certain indeterminate Stieltjes moment problems associated with birth and death processes having polynomial birth and death rates of degree p\ge 3. Romanov recently proved that the order is 1/p as conjectured, see \cite{Ro}. We prove that the type with respect to the order is related to certain multi-zeta values and that this type belongs to the interval [π/(p\sin(π/p)),π/(p\sin(π/p)\cos(π/p))], which also contains the conjectured value. This proves that the conjecture about type is asymptotically correct as p\to\infty. The main idea is to obtain estimates for order and type of symmetric indeterminate Hamburger moment problems when the orthonormal polynomials P_n and those of the second kind Q_n satisfy P_{2n}^2(0)\sim c_1n^{-1/\b} and Q_{2n-1}^2(0)\sim c_2 n^{-1/\a}, where 0<\a,\b<1 can be different, and c_1,c_2 are positive constants. In this case the order of the moment problem is majorized by the harmonic mean of \a,\b. Here α_n\sim β_n means that α_n/β_n\to 1. This also leads to a new proof of Romanov's Theorem that the order is 1/p.