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N. H. Bingham

Publications and source records attributed to N. H. Bingham.

At least 19 recordsLinked to original sources
arXiv2024

Homomorphisms from Functional Equations: The Goldie Equation, II

In this sequel to arXiv1407.4089 by the second author, we extend to multi-dimensional (or infinite-dimensional) settings the Goldie equation arising in the general regular variation of `General regular variation, Popa groups and quantifier weakening', J. Math. Anal. Appl. 483 (2020) 123610, 31 pp. (arXiv1901.05996). We extend the work there in two directions. First, we algebraicize the theory, by systematic use of certain groups -- the Popa groups arising in earlier work by Popa, and their relatives the Javor groups. Secondly, we extend from the original context on the real line to multi-dimensional (or infinite-dimensional) settings.

math.CA↗
arXiv2024

The Goldie Equation: III. Homomorphisms from Functional Equations

This is the second of three sequels to arXiv1407.4089 -- the third of the resulting quartet -- concerning the real-valued continuous solutions of the multivariate Goldie functional equation (GFE) below of Levi-Civita type. Following on from the preceding paper arXiv1910.05816, in which these solutions are described explicitly, here we characterize (GFE) as expressing homomorphy (in all but some exceptional "improper" cases) between multivariate Popa groups, defined and characterized earlier in the sequence. The group operation involves a form of affine addition (with local scalar acceleration) similar to the circle operation of ring theory. We show the affine action in (GFE) may be replaced by general continuous acceleration yielding a functional equation (GGE) which it emerges has the same solution structure as (GFE). The final member of the sequence arXiv2105.07794 considers the richer framework of a Banach algebra which allows vectorial acceleration, giving the closest possible similarity to the circle operation.

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arXiv2022

Tauberian Korevaar

We focus on the Tauberian work for which Jaap Korevaar is best known, together with its connections with probability theory. We begin (Section 1) with a brief sketch of the field up to Beurling's work. We follow with three sections on Beurling aspects: Beurling slow variation (Section 2); the Beurling Tauberian theorem for which it was developed (Section 3); Riesz means and Beurling moving averages (Section 4). We then give three applications from probability theory: extremes (Section 5), laws of large numbers (Section 6), and large deviations (Section 7). We turn briefly to other areas of Korevaar's work in Section 8. We close with a personal postscript (whence our title).

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arXiv2021

Gaussian random fields: with and without covariances

We begin with isotropic Gaussian random fields, and show how the Bochner-Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Matérn processes on Euclidean space, spheres, manifolds and graphs, using Bessel potentials and stochastic partial differential equations (SPDEs). We then turn from this continuous setting to approximating discrete settings, Gaussian Markov random fields (GMRFs), and the computational advantages they bring in handling large data sets, by exploiting the sparseness properties of the relevant precision (concentration matrices).

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arXiv2021

The life, work and legacy of P. L. Chebyshev

We survey briefly the life and work of P. L. Chebyshev, and his ongoing influence. We discuss his contributions to probability, number theory and mechanics, his pupils and mathematical descendants, and his role as the founding father of Russian mathematics in general and of the Russian school of probability in particular.

math.HO↗
arXiv2021

Prediction theory in continuous time

We consider prediction theory for stationary stochastic processes in continuous time. We discuss prediction using the whole (infinite) past, and using only a finite section of the past. The solutions to both these classical problems have long been known. Our focus is to provide short simple proofs which reveal the probabilistic meaning of the results.

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arXiv2021

The Gołąb-Schinzel and Goldie functional equations in Banach algebras

We are concerned below with the characterization in a unital commutative real Banach algebra $\mathbb{A}$ of continuous solutions of the Gołąb-Schinzel functional equation (below), the general Popa groups they generate and the associated Goldie functional equation. This yields general structure theorems involving both linear and exponential homogeneity in $\mathbb{A}$ for both these functional equations and also explict forms, in terms of the recently developed theory of multi-Popa groups [BinO3,4], both for the ring $C[0,1]$ and for the case of $\mathbb{R}^{d}$ with componentwise product, clarifying the context of recent developments in [RooSW]. The case $\mathbb{A}=\mathbb{C}$ provides a new viewpoint on continuous complex-valued solutions of the primary equation by distinguishing analytic from real-analytic ones.

math.FA↗
arXiv2021

Integral Representations of Ultraspherical Polynomials II

In the first part, by the first author's work of 1972, an integral representation for an ultraspherical polynomial of higher index in terms of one of lower index and an infinite series was obtained. While this representation works well from a theoretical point of view, it is not numerically satisfactory as it involves polynomials of high degree, which are numerically unstable. Here we sum this series to obtain an integral, which is numerically tractable.

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arXiv2020

Extremes and regular variation

We survey the connections between extreme-value theory and regular variation, in one and higher dimensions, from the algebraic point of view of our recent work on Popa groups.

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arXiv2019

On subadditive functions upper bounded on a 'large' set

The notion of a shift-compact set in an abelian topological group $X$ plays a significant role in functional equations and inequalities, especially so since each Borel set that is not Haar-meagre, alternatively not Haar-null, is necessarily shift-compact for $X$ completely metrizable (see \cite{BJ} and \cite{BinO8}). Although in general boundedness of a subadditive function does not imply its continuity, here we prove that each subadditive function $f:X\rightarrow \mathbb{R}$ (i.e. with the function satisfying $f(x+y)\leq f(x)+f(y)$ for $x,y\in X$) bounded above on a~shift-compact (non-Haar-null, non-Haar-meagre) set is locally bounded at each point of the domain. Our results refer to \cite[Chapter~XVI]{Kuczma} and papers by N.H.~Bingham and A.J.~Ostaszewski \cite{BO,BinO1,BinO2,BinO6,BinO7}.

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arXiv2019

Gaussian random fields on the sphere and sphere cross line

We review the Dudley integral for the Belyaev dichotomy applied to Gaussian processes on spheres, and discuss the approximate (or restricted) continuity of paths in the discontinuous case. We discuss also the spatio-temporal case, of sphere cross line. In the continuous case, we investigate the link between the smoothness of paths and the decay rate of the angular power spectrum, following Tauberian work of the first author, Malyarenko, and Lang and Schwab.

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arXiv2019

Beyond Haar and Cameron-Martin: the Steinhaus support

Motivated by a Steinhaus-like interior-point property involving the Cameron-Martin space of Gaussian measure theory, we study a group-theoretic analogue, the Steinhaus triple $(H,G,μ)$, and construct a Steinhaus support, a Cameron-Martin-like subset, $H(μ)$ in any Polish group $G$ corresponding to `sufficiently subcontinuous' measures $μ$, in particular for `Solecki-type' reference measures.

math.FA↗
arXiv2019

Sequential regular variation: extensions of Kendall's theorem

Regular variation is a continuous-parameter theory; we work in a general setting, containing the existing Karamata, Bojanic-Karamata/de Haan and Beurling theories as special cases. We give sequential versions of the main theorems, that is, with sequential rather than continuous limits. This extends the main result, a theorem of Kendall's (which builds on earlier work of Kingman and Croft), to the general setting.

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arXiv2019

General Regular Variation, Popa Groups and Quantifier Weakening

We introduce general regular variation, a theory of regular variation containing the existing Karamata, Bojanic-Karamata/de Haan and Beurling theories as special cases. The unifying theme is the Popa groups of our title viewed as locally compact abelian ordered topological groups, together with their Haar measure and Fourier theory. The power of this unified approach is shown by the simplification it brings to the whole area of quantifier weakening, so important in this field.

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arXiv2018

The Steinhaus-Weil property: its converse, subcontinuity and Solecki amenability

The Steinhaus-Weil theorem that concerns us here is the simple, or classical, `interior-points' property -- that in a Polish topological group a non-negligible set B has the identity as an interior point of $BB^{-1}$. There are various converses; the one that mainly concerns us is due to Simmons and Mospan. Here the group is locally compact, so we have a Haar reference measure $η$. The Simmons-Mospan theorem states that a (regular Borel) measure has such a Steinhaus-Weil property if and only if it is absolutely continuous with respect to the Haar measure. In Part I (Propositions 1-7, Theorems 1-4) we exploit the connection between the interior-points property and a selective form of infinitesimal invariance afforded by a certain family of selective reference measures $σ$, drawing on Solecki's amenability at 1 (and using Fuller's notion of subcontinuity). In Part II (Propositions 8, 9, Theorems 5, 6) we develop a number of relatives of the Simmons-Mospan theorem. In Part III (Theorems 7, 8) we link this with topologies of Weil type. We close in Part IV with Propositions 12-13 and Theorem B -- concerning the `composite interior-point' property (of $AB^{-1}$) and the Borell `relative interior-point' property (relative to the Cameron-Martin space) -- and complements.

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arXiv2018

Set Theory and the Analyst

This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure -- category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley [Kel]: "what every young analyst should know".

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arXiv2017

Variants on the Berz sublinearity theorem

We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice which Berz used. We consider thinned versions, in which conditions are imposed on only part of the domain of the function -- results of quantifier-weakening type. There are connections with classical results on subadditivity. We close with a discussion of the extensive related literature.

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