Related papers: The Taylor series of the Gaussian kernel

We present a sufficient condition of existence of asymptotic expansion in negative power series for a function defined by Taylor series and unitary formulas for coefficients of this expansion. An example of computing scheme for arctangent…

Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…

An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results…

The Askey-Wilson function transform is a q-analogue of the Jacobi function transform with kernel given by an explicit non-polynomial eigenfunction of the Askey-Wilson second order q-difference operator. The kernel is called the Askey-Wilson…

We study the Borel summability of the small time expansion of the heat kernel associated to a first order perturbation of a Laplacian. An explicit formula for this kernel plays a central role. As a consequence, we get a Poisson formula on…

In this paper, we derive by using elementary methods some continued fractions, certain identities involving derivatives of tanx, several expressions for log coshx and an identity for {\pi}2, from a series expansion of tan x, which gives the…

It is well known that the Taylor series expansion of $(1+ z)^{A}$ does not converge for $|z|>1$ where A is a real number which is not equal to zero or a positive integer. A limited series expansion of this expression is obtained in this…

Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…

Herein, the theory of Bergman kernel is developed to the weighted case. A general form of weighted Bergman reproducing kernel is obtained, by which we can calculate concrete Bergman kernel functions for specific weights and domains.

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painleve equation. For the general solution, explicit recursive formulae for the Taylor expansion of the…

We consider linear approximation based on function evaluations in reproducing kernel Hilbert spaces of certain analytic weighted power series kernels and stationary kernels on the interval $[-1,1]$. Both classes contain the popular Gaussian…

Choosing the most adequate kernel is crucial in many Machine Learning applications. Gaussian Process is a state-of-the-art technique for regression and classification that heavily relies on a kernel function. However, in the Gaussian…

Let $G$ be a finitely generated group, and let $\Bbbk{G}$ be its group algebra over a field of characteristic $0$. A Taylor expansion is a certain type of map from $G$ to the degree completion of the associated graded algebra of $\Bbbk{G}$…

We use Taylor's formula with Lagrange remainder to make a modern adaptation of Poisson's proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left (or…

We analyse the perturbative series expansion of the vacuum expectation value of a Wilson loop in Chern-Simons gauge theory in the temporal gauge. From the analysis emerges the notion of the kernel of a Vassiliev invariant. The kernel of a…

By using the theory of maximal $L^{q}$-regularity and methods of singular analysis, we show a Taylor's type expansion--with respect to the geodesic distance around an arbitrary point--for solutions of quasilinear parabolic equations on…

We derive general relativistic Gaussian equations for osculating elements for orbits under the influence of a perturbing force without any restrictions in an underlying Schwarzschild space-time. Such a formulation provides a way to describe…

Ulmer and Kaissl formulas for the deconvolution of one-dimensional Gaussian kernels are generalized to the three-dimensional case. The generalization is based on the use of the scalar version of the Grad's multivariate Hermite polynomials…

We establish Taylor series expansions in rational (and elliptic) function bases using E. Rains' elliptic extension of the Askey-Wilson divided difference operator. The expansion theorem we consider extends M.E.H. Ismail's expansion for the…

The theory of normality for base $g$ expansions of real numbers in $[0,1)$ is rich and well developed. Similar theories have been developed for many other numeration systems, such as the regular continued fraction expansion,…