This document contains detailed proofs of theorems stated in the main article entitled Random Feature Maps for Dot Product Kernels. 1 Proof of Theorem 1 We first recollect Schoenberg’s result in its original form Theorem 1 (Schoenberg (1942), Theorem 2). A function f : [−1, 1]→ R constitutes a positive definite kernel K : S∞ × S∞ → R, K : (x,y) 7→ f(〈x,y〉) iff f is an analytic function admittin...

A Taylor matrix method is developed to find an approximate solution of the most general linear Fredholm integrodifferential-difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. This method transforms the given general linear Fredholm integrodifferential-difference equations and the mixed conditions to matrix equations with unknown Taylor coe...

There are many ways to approximate cosine curve. In this study we have examined the way how curve can be written as any order Bezier As a result using Maclaurin series 4th and 6th based on control points with matrix form in E2. We give of coefficients. Also coefficients too.

Abstract The objective of this paper is to introduce new classes m-fold symmetric bi-univalent functions. We discuss estimates on the Taylor–Maclaurin coefficients $|a_{m+1}|$ | a m + 1 and $|a_{2...

Velleda Baldoni, Nicole Berline, Michèle Vergne. Local Euler-Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of a rational polytope. Matthias Beck, Christian Haase, Bruce Reznick, Michèle Vergne, Volkmar Welker and Ruriko Yoshida. Integer Points in Polyhedra-Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, American Mathematical Society, pp.1...

In this paper, we first modify one of the most famous theorems on principle differential subordination to hold true for normalized analytic functions with a fixed initial Taylor-Maclaurin coefficient. By using modified form, generalize and improve several results, which appeared recently in literature geometric function theory complex analysis. We also prove some simple conditions starlikeness,...

The object of this article is to explore a ?-pseudo-?-convex ?-fold symmetric bi-univalent function family satisfying subordinations condition generalizing certain previously examined families. We originate the initial Taylor–Maclaurin coefficient estimates functions in defined family. classical Fekete–Szegö inequalities for also estimated. Furthermore, we present some special cases main result...

Let F(z) be a vector-valued function, F : e _ eN, which is analytic at z= 0 and meromorphic .. in a neighbourhood ofz = 0, and let its Maclaurin series be given. In this work we develop vector­ valued rational approximation procedures for F(z) by applying vector extrapolation methods to the sequence of partial sums of its Maclaurin series. We analyze some of the algebraic and analytic propertie...