We derive approximate numerical solutions for an ordinary differential equation common in engineering using two different types of basis functions, polynomial and Gaussian, and a maximum discrepancy error measure. We compare speed and accuracy of the two solutions. The basic finding for our example is that while Gaussian basis functions can be used, the computational effort is greater than that...

Abstract It is known that if $$f:{{\mathbb R}}^2\rightarrow {\mathbb R}$$ f : R 2 → a polynomial in each variable, then f polynomial. We present generalizations of this fact, when $${{\mathbb R}}^...

A read-once formula is one in which each variable appears on at most a single input. Angluin, Hellerstein, and Karpinski give a polynomial time algorithm that uses membership and equivalence queries to identify exactly read-once boolean formulas over the basis fAND;OR;NOTg AHK93]. The goal of this work is to consider natural generalizations of these gates, in order to develop exact identiicatio...

As it is not clear how to choose an appropriate kernel function for a specific learning problem, this paper examines the performance of three kernel functions with varied parameters for online handwritten character recognition. The three kernel functions evaluated are the polynomial, complete polynomial, and radial basis functions. Results show that the three functions perform comparably with t...

and infinitely many integers n, multiplication in a normal basis of Fqn over Fq can be computed with O(n logn loglogn), division with O(n log n loglogn) operations in Fq , and exponentiation of an arbitrary element in Fqn with O(n2 loglogn) operations in Fq . We also prove that using a polynomial basis exponentiation in F2n can be done with the same number of operations in F2 for all n. The pre...

In this article, by using Chebyshev’s polynomials and Chebyshev’s expansion, we obtain the best uniform polynomial approximation out of P2n to a class of rational functions of the form (ax2+c)-1 on any non symmetric interval [d,e]. Using the obtained approximation, we provide the best uniform polynomial approximation to a class of rational functions of the form (ax2+bx+c)-1 for both cases b2-4a...

In this paper, we consider a new class of analytic functions in the unit disk using polynomials of order alpha. We give some sufficient conditions for functions belonging to this class.

In many cases, multivariate interpolation by smooth radial basis functions converges towards polynomial interpolants, when the basis functions are scaled to become “wide”. In particular, examples show that interpolation by scaled Gaussians seems to converge towards the de Boor/Ron “least” polynomial interpolant. The paper starts by providing sufficient criteria for the convergence of radial int...