Related papers: Schoenberg's Theorem via the law of large numbers

We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr's doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a…

Let F(R^n) be the algebra of Fourier transforms of functions from L_1(R^n), K(R^n) be the algebra of Fourier transforms of bounded complex Borel measures in R^n and W be Wiener algebra of continuous 2pi-periodic functions with absolutely…

It is a theorem of Denker and Urba\'nski ('91) that if $T:\mathbb C\to\mathbb C$ is a rational map of degree at least two and if $\phi:\mathbb C\to\mathbb R$ is H\"older continuous and satisfies the "thermodynamic expanding" condition…

A classical theorem due to Borel asserts that any formal serie with real coefficients is the Taylor expansion of a germ of $\mathcal{C}^{\infty}- {\rm function}$. We study such a problem in the context of Lie algebras of vector fields or of…

A well-known theorem of Schoenberg states that if f(z) generates a PF_r sequence then 1/f(-z) generates a PF_r sequence. We give two counterexamples which show that this is not true, and give a correct version of the theorem. In the…

We prove two versions of Bochner's theorem for locally compact quantum groups. First, every completely positive definite "function" on a locally compact quantum group $\G$ arises as a transform of a positive functional on the universal…

Let \phi be a self-map of B^n, the unit ball in C^n, fixing 0, and having full-rank at 0. If \phi (0)= 0, Koenigs proved in 1884 that in the well- known case n = 1, Schroeder's equation, f \circ \phi = \phi '(0) f has a solution f, which is…

Multi-class systems having possibly both finite and infinite classes are investigated under a natural partial exchangeability assumption. It is proved that the conditional law of such a system, given the vector of the empirical measures of…

The quantum measurement problem as was formulated by von Neumann in 1933 can be solved by going beyond the operational quantum formalism. In our "prequantum model" quantum systems are symbolic representations of classical random fields. The…

An exact uncertainty principle, formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position, leads from the…

For a phenomenon $\boldsymbol{f}$ that is a function of $n$ factors, defined on a finite abelian group $G$, we derive its population statistics solely from its Fourier transform $\hat{\boldsymbol{f}}$. Our main result is an…

Assume that $g(|\xi|^2)$, $\xi\in\mathbb{R}^k$, is for every dimension $k\in\mathbb{N}$ the characteristic function of an infinitely divisible random variable $X^k$. By a classical result of Schoenberg $f:=-\log g$ is a Bernstein function.…

A theorem of Hoischen states that given a positive continuous function $\varepsilon:\mathbb{R}\to\mathbb{R}$, an integer $n\geq 0$, and a closed discrete set $E\subseteq\mathbb{R}$, any $C^n$ function $f:\mathbb{R}\to\mathbb{R}$ can be…

Based on the Goldbach conjecture and arithmetic fundamental theorem, the Goldbach conjecture was extended to more general situations, i.e., any positive integer can be written as summation of some specific prime numbers, which depends on…

The sum of the absolute values of the Fourier coefficients of a function $f:\mathbb{F}_2^n \to \mathbb{R}$ is called the spectral norm of $f$. Green and Sanders' quantitative version of Cohen's idempotent theorem states that if the spectral…

Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $\mu$ on $[0,1)$, every $f\in L^2(\mu)$ possesses a Fourier series of the form $f(x)=\sum_{n=0}^{\infty}c_ne^{2\pi inx}$. We show that the coefficients…

For $k$ a positive integer let $S_k(n) = 1^k + 2^k + \cdots + n^k$, i.e., $S_k(n)$ is the sum of the first $k$-th powers. Faulhaber conjectured (later proved by Jacobi) that for $k$ odd, $S_k(n)$ could be written as a polynomial of…

We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The…

We consider a notion of uniform thinning for a finite sequence of random variables $(X_1,...,X_n)$ obtained by removing one random variable, uniformly at random. If a triangular array of random variables $(X_{n,k} : n \in \mathbb{N}_+, 1…

Sturm's strong law of large numbers in $\mathrm{CAT}(0)$ spaces has been an influential tool to study the geometric mean or also called Karcher barycenter of positive definite matrices. It provides an easily computable stochastic…