Related papers: The Taylor series of the Gaussian kernel

We study the time-sliced thawed Gaussian propagation method, which was recently proposed for solving the time-dependent Schr\"odinger equation. We introduce a triplet of quadrature-based analysis, synthesis and re-initialization operators…

We prove a generalized Gauss-Kuzmin-L\'evy theorem for the $p$-numerated generalized Gauss transformation $$T_p(x)=\{\frac{p}{x}\}.$$ In addition, we give an estimate for the constant that appears in the theorem.

Gaussian Processes (GPs) are widely recognized as powerful non-parametric models for regression and classification. Traditional GP frameworks predominantly operate under the assumption that the inputs are either accurately known or subject…

Using a recent Mergelyan type theorem for products of planar compact sets we establish generic existence of Universal Taylor Series on products of planar simply connected domains Omegai, i=1, . . . , d. The universal approximation is…

We study query time bounds for the fundamental problem of estimating the kernel mean $\frac1{|X|}\sum_{x\in X}\mathbf{k}(x,y)$ of a query $y$ in a finite dataset $X\subset\mathbb{R}^d$ up to a prescribed additive error $\varepsilon$. The…

This paper derives a way to express differentiable complex-valued functions as the sum of powers of $(1-e^{\lambda x})$, where $\lambda\in\mathbb{R}$, with an explicit formula for the remainder. This formulation is then used to associate an…

It has long been agreed by academics that the inversion method is the method of choice for generating random variates, given the availability of the quantile function. However for several probability distributions arising in practice a…

We present some observations on the tau-function for the fourth Painlev\'e equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary…

In this paper we outline an algorithmic approach to compute Puiseux series expansions for algebraic surfaces. The series expansions originate at the intersection of the surface with as many coordinate planes as the dimension of the surface.…

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are…

In general, the distribution of residuals cannot be obtained explicitly. We give an asymptotic formula for the density of Pearson residuals in continuous generalized linear models corrected to order $n^{-1}$, where $n$ is the sample size.…

This tutorial aims to provide an intuitive introduction to Gaussian process regression (GPR). GPR models have been widely used in machine learning applications due to their representation flexibility and inherent capability to quantify…

We begin by explaining how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight thanks to a theorem of Borisov-Gunnells and explicit expansions of Eisenstein series at all cusps. Using this, we…

We establish a link between the phenomenon of Taylor dispersion and the theory of empirical distributions. Using this connection, we derive, upon applying the theory of large deviations, an alternative and much more precise description of…

In this work, series expansions in terms of Bessel functions of the first kind are given for the sine and cosine integrals. These representations differ from many of the known Neumann-type series expansions for the sine and cosine…

A fundamental drawback of kernel-based statistical models is their limited scalability to large data sets, which requires resorting to approximations. In this work, we focus on the popular Gaussian kernel and on techniques to linearize…

We consider both finite and infinite power chi expansions of $f$-divergences derived from Taylor's expansions of smooth generators, and elaborate on cases where these expansions yield closed-form formula, bounded approximations, or analytic…

Using the Kantorovitch method in combination with a Gaussian ansatz, we derive the equations of motion for spatial, temporal and spatiotemporal optical propagation in a dispersive Kerr medium with a general transverse and spectral gain…

We introduce the notion of an approximation system as a generalization of Taylor approximation, and we give some first examples. Next we develop the general theory, including error bounds and a sufficient criterion for convergence. More…

This paper is an extension program of the notion of circle of partition developed in our first paper \cite{CoP}. As an application we prove the Erd\H{o}s-Tur\'{a}n additive base conjecture.