Amos J. Carpenter

On the bernstein conjecture in approximation theory

WithE2n(|x|) denoting the error of best uniform approximation to |x| by polynomials of degree at most 2n on the interval [−1, +1], the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constantβ for which lim 2nE2n(|x|)=β.n→∞ Moreover, by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds forβ: 0.278<β<0.286. Now, the average of these bounds is 0.282, which, as Bernstein noted as a “curious coincidence,” is very close to 1/(2√π)=0.2820947917... This observation has over the years become known as the Bernstein Conjecture: Isβ=1/(2√π)? We show here that the Bernstein conjecture isfalse. In addition, we determine rigorous upper and lower bounds forβ, and by means of the Richardson extrapolation procedure, estimateβ to approximately 50 decimal places.

Asymptotics for the zeros of the generalized Bessel polynomials

SummaryWe investigate the location of the zeros of the normalized generalized Bessel polynomials and the normalized reversed generalized Bessel polynomials. Also, the rate at which these zeros approach certain well-defined curves is investigated. On the basis of numerical computations and graphs, four new conjectures are proposed.

Zeros of the partial sums of cos(z) and sin(z) II

Summary. We study here in detail the location of the real and complex zeros of the partial sums of

Angular Distribution of Zeros of the Partial Sums of ez via the Solution of Inverse Logarithmic Potential Problem

\cos (z)

Scientific computation on some mathematical problems

and

Extended numerical computations on the “1/9” conjecture in rational approximation theory

\sin (z)

Zeros of the partial sums of cos(z) and sin(z). III

, which extends results of Szegö (1924) and Kappert (1996).

Zeros of the partial sums of cos (z) and sin (z). I.

We continue the work of Szegő [18] on describing the angular distribution of the zeros of the normalized partial sum sn(nz) of ez, where

Some numerical results on best uniform polynomial approximation of X α on [0, 1]

s _{n}(z):={\sum _{k=0} ^{n}}z ^k/k!

Asymptotics for the zeros and poles of normalized Pad\'{e} approximants to\({\rm e}^{z}\)

. We imbed this problem in some inverse problem of potential theory and prove a so-called Erdős-Turán-type theorem, which is of interest in itself.