Existence of best approximations by exponential sums in several independent variables

Quasi-sums in Several Variables

In this note we introduce the notions of quasi-sums and of the local quasi-sums in several variables, respectively. We prove that the local quasi-sums are also quasi-sums. We show how this result can be applied to find the continuous solutions of the functional equation g(u11 + · · ·+ u1N , . . . , uM1 + · · ·+ uMN ) = f(g1(u11, . . . , uM1), . . . , gN (u1N , . . . , uMN )) that are strictly m...

Exponential Approximations of Geometri Sums for Reliability

Exponential Approximations Using Fourier Series Partial Sums

The problem of accurately reconstructing a piece-wise smooth, 2π-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to approximate the locations and magnitudes of the discontinuities in f an...

Best Approximation of Monomials in Several Variables

For a monomial x in d variables, the problem of best approximation to x by polynomials of lower degrees is studied on the unit sphere, the unit ball and the standard simplex. For the uniform norm we discuss what is known in d = 2, for which complete solutions are known, and in d ≥ 3 for which only a few cases have been successfully solved. For the L norm, we present the complete solution, inclu...

D ec 2 00 4 , INCOMPLETE QUADRATIC EXPONENTIAL SUMS IN SEVERAL VARIABLES

We consider incomplete exponential sums in several variables of the form S(f, n, m) = 1 2n ∑