Notes on uniform convergence of trigonometrical series, II

Uniform convergence of hypergeometric series

The considered problem is uniform convergence of sequences of hypergeometric series. We give necessary and sufficient conditions for uniformly dominated convergence of infinite sums of proper bivariate hypergeometric terms. These conditions can be checked algorithmically. Hence the results can be applied in Zeilberger type algorithms for nonterminating hypergeometric series.

Uniform Convergence of Double Trigonometric Series

It is a classical problem in Fourier analysis to give conditions for a single sine or cosine series to be uniformly convergent. Several authors gave conditions for this problem supposing that the coefficients are monotone, non-negative or more recently, general monotone. There are also results for the regular convergence of double sine series to be uniform in case the coefficients are monotone ...

On statistical type convergence in uniform spaces

The concept of ${mathscr{F}}_{st}$-fundamentality is introduced in uniform spaces, generated by some filter ${mathscr{F}}$. Its equivalence to the concept of ${mathscr{F}}$-convergence in uniform spaces is proved. This convergence generalizes many kinds of convergence, including the well-known statistical convergence.

A note on “Convergence rates and asymptotic normality for series estimators”: uniform convergence rates

This paper establishes improved uniform convergence rates for series estimators. Series estimators are least-squares fits of a regression function where the number of regressors depends on sample size. I will specialize my results to the cases of polynomials and regression splines. These results improve upon results obtained earlier by Newey, yet fail to attain the optimal rates of convergence....

Uniform convergence of single and double sine series and integrals