Numerical Integration Using Sums of Exponential Functions

A Numerical Integration Method by Using Generalized Series of Functions

Bilinear Sums with Exponential Functions

Let g = 0,±1 be a fixed integer. Given two sequences of complex numbers (φm) ∞ m=1 and (ψn) ∞ n=1 and two sufficiently large integers M and N , we estimate the exponential sums ∑ p≤M gcd(ag,p)=1 ∑ 1≤n≤N φpψnep (ag ) , a ∈ Z, where the outer summation is taken over all primes p ≤ M with gcd(ag, p) = 1.

a numerical integration method by using generalized series of functions

On Approximation of Functions by Exponential Sums

We introduce a new approach, and associated algorithms, for the efficient approximation of functions and sequences by short linear combinations of exponential functions with complex-valued exponents and coefficients. These approximations are obtained for a finite but arbitrary accuracy and typically have significantly fewer terms than Fourier representations. We present several examples of thes...

Using Stepanov’s Method for Exponential Sums Involving Rational Functions

For a non-trivial additive character ψ and multiplicative character χ on a finite field Fq , and rational functions f, g in Fq(x), we show that the elementary Stepanov-Schmidt method can be used to obtain the corresponding Weil bound for the sum ∑ x∈Fq\S χ(g(x))ψ(f(x)) where S is the set of the poles of f and g. We also determine precisely the number of characteristic values ωi of modulus q1/2 ...