Weighted integrability of double cosine series with nonnegative coefficients

On Nonnegative Cosine Polynomials with Nonnegative Integral Coefficients

We show that there exist Po > 0 ar>d P\, ■■■ , Pn nonnegative integers, such that 0 < p(x) = Po + Pi cosx + • ■ • + Pn cos Nx and po -C s1/3 for .s = J2'J=oPj < improving on a result of Odlyzko who showed the existence of such a polynomial p that satisfies Po < (.slogs)1/3 . Our result implies an improvement of the best known estimate for a problem of Erdos and Szekeres. As our method is noncon...

Exact Estimates for Integrals Involving Dirichlet Series with Nonnegative Coefficients

We consider the Dirichlet series ∞ ∑ k=2 akk −1−x =: f(x), x > 0, with coefficients ak ≥ 0 for all k. Among others, we prove exact estimates of certain weighted Lp-norms of f on the unit interval (0, 1) for any 0 < p <∞, in terms of the coefficients ak . Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for int...

Wiener Type Theorems for Jacobi Series with Nonnegative Coefficients

This paper gives three theorems regarding functions integrable on [−1, 1] with respect to Jacobi weights, and having nonnegative coefficients in their (Fourier–)Jacobi expansions. We show that the L-integrability (with respect to the Jacobi weight) on an interval near 1 implies the L-integrability on the whole interval if p is an even integer. The Jacobi expansion of a function locally in L∞ ne...

Polynomials with Nonnegative Coefficients

The authors verify the conjecture that a conjugate pair of zeros can be factored from a polynomial with nonnegative coefficients so that the resulting polynomial still has nonnegative coefficients. The conjecture was originally posed by A. Rigler, S. Trimble, and R. Varga arising out of their work on the Beauzamy-Enflo generalization of Jensen's inequality. The conjecture was also made independ...

Certain Estimates for Double Sine Series with Multiple-monotone Coefficients