Asymptotic formulae of multivariate Bernstein approximation

Asymptotic Formulae

Let ts,n be the n-th positive integer number which can be written as a power p, t ≥ s, of a prime p (s ≥ 1 is fixed). Let πs(x) denote the number of prime powers p, t ≥ s, not exceeding x. We study the asymptotic behaviour of the sequence ts,n and of the function πs(x). We prove that the sequence ts,n has an asymptotic expansion comparable to that of pn (the Cipolla’s expansion).

Multivariate Bernstein polynomials and convexity

It is well known that in two or more variables Bernstein polynomi-als do not preserve convexity. Here we introduce two variations, one stronger than the classical notion, the other one weaker, which are preserved. Moreover, a weaker suucient condition for the monotony of subsequent Bernstein polynomials is given. linearly independent, in the course of which d has to be greater than or equal to ...

Asymptotic Formulae for Partition Ranks

Using an extension of Wright’s version of the circle method, we obtain asymptotic formulae for partition ranks similar to formulae for partition cranks which where conjectured by F. Dyson and recently proved by the first author and K. Bringmann.

Asymptotic Behavior of Multivariate Reward Processes with Nonlinear Reward Functions

Approximation with Bernstein-Szegö polynomials

We present approximation kernels for orthogonal expansions with respect to Bernstein-Szegö polynomials. The construction is derived from known results for Chebyshev polynomials of the first kind and does not pose any restrictions on the Bernstein-Szegö polynomials.