This paper estimates upper and lower bounds for the approximation rates of iterated Boolean sums of multivariate Bernstein polynomials. Both direct and inverse inequalities for the approximation rate are established in terms of a certain K -functional. From these estimates, one can also determine the class of functions yielding optimal approximations to the iterated Boolean sums. c © 2008 Elsev...

We show that a function F : F2m → C whose multiplicative Fourier coefficients are given by certain character sums, which we call Jacobi-like sums, is in fact the ±1-valued characteristic function of a difference set in F2m with Singer parameters. We show further that the difference sets arising from such functions are precisely the so-called DD difference sets constructed by the first author an...

We introduce the new price probability measure, which entirely depends on measures of value and volume market trades. define nth statistical moment as ratio to all trades performed during an averaging time interval Δ. The set moments determines characteristic function its Fourier transform defines measure. volatility 1st 2nd prediction requires a description sums squares Δ we call it second-ord...

We prove a simple formula for the main value of r-even functions and give applications of it. Considering the generalized Ramanujan sums cA(n, r) involving regular systems A of divisors we show that it is not possible to develop a Fourier theory with respect to cA(n, r), like in the the usual case of classical Ramanujan sums c(n, r).

By a standard approximation argument it follows that S[f ] may be meaningfully defined as a continuous function in ξ for almost every x whenever f ∈ L and the a priori bound of the theorem continues to hold for such functions. Theorem 1.1 is intimately related to almost everywhere convergence of partial Fourier sums for functions in L[0, 1]. Via a transference principle [12], it is indeed equiv...