Michael I. Ganzburg

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

In this paper, we establish new asymptotic relations for the errors of approximation in Lp?1,1], 00, by the Lagrange interpolation polynomials at the Chebyshev nodes of the first and second kind. As a corollary, we show that the Bernstein constantB?,p?limn?∞n?+1/p infck??x????k=0nmckxk?Lp?1,1] is finite for ?>0 and p?13,∞).

Strong asymptotics in Lagrange interpolation with equidistant nodes

In this paper we prove three conjectures of Revers on Lagrange interpolation for fλ(t)= |t|λ, λ > 0, at equidistant nodes. In particular, we describe the rate of divergence of the Lagrange interpolants LN(fλ, t) for 0<|t| < 1, and discuss their convergence at t=0. We also establish an asymptotic relation for max|t|≤1||t|λ -LN(fλ, t)|. The proofs are based on strong asymptotics for |t|λ -- LN(fλ, t), 0 ≤ |t| < 1.

An extremal problem for trigonometric polynomials

Let T, (x) = Zk=0 (ak cos kx + bk sin kx) be a trigonometric polynomial of degree n. The problem of finding Cnp, the largest value for C in the inequality maxflaol, lall,..., lan? lbil,..., 1bn1} < (1/C) 11Tn11P is studied. We find Cnp exactly provided p is the conjugate of an even integer 2s and n > 2s 1, s = 1, 2,.... For general p, 1 < p < oo,we get an interval estimate for Cnp, where the interval length tends to 0 as n tends to oo.

Lagrange Interpolation and New Asymptotic Formulae for the Riemann ZetaFunction

An asymptotic representation for the Riemann zeta function ζ(s) in terms of the Lagrange interpolation error of some function f s,2N at the Chebyshev nodes is found. The representation is based on new error formulae for the Lagrange polynomial interpolation to a function of the form \(f(y) ={ \int \nolimits \nolimits }_{\mathbb{R}} \frac{\varphi (t)} {t-iy}\mathrm{d}t.\) As the major application of this result, new criteria for ζ(s)=0 and ζ(s)≠0 in the critical strip 0

Sharp constants in V. A. MarkovBernstein type inequalities of different metrics

We study relations between sharp constants in the V. A. MarkovBernstein inequalities of different Lr-metrics for algebraic polynomials on an interval and for entire functions on the real line or half-line. In a number of cases, we prove that the sharp constant in the inequality for entire functions of exponential type or semitype is the limit of sharp constants in the corresponding inequalities for algebraic polynomials of degree n as n.

Exact errors of best approximation for complex-valued nonperiodic functions

Abstract We extend Markov’s and Nagy’s theorems on best approximation by entire functions of exponential type in the L 1 ( R ) metric to some complex-valued integrable and locally integrable functions. We use these results for finding sharp constants of best approximation in L 1 ( R ) and L ∞ ( R ) on some complex convolution classes. For classes of real-valued convolutions those constants were found by Akhiezer. As an example, we apply these results to the Schwarz-type kernel and to the corresponding convolution classes.

A Markov-Type Inequality for Multivariate Polynomials on a Convex Body

A Markov-type inequality for the k-homogeneous part of a multivariate polynomial on a convex centrally symmetric body is given and an extremal polynomial is found. This generalizes and extends some estimates for univariate and multivariate polynomials obtained by Markov, Bernstein, Visser, Reimer, and Rack.

Best Fourier Approximation and Application in Efficient Blurred Signal Reconstruction

We expand upon the known results on sharp linear Fourier methods of approximation where the approximation is the best in terms of both rate and constant among all polynomial procedures of approximation. So far these results have been studied due to their mathematical beauty rather than their practical importance. In this paper we show that they are the core mathematics underlying best statistical methods of solving noisy ill-posed problems. In particular, we suggest a procedure for recovery of noisy blurred signals based on samples of small sizes where a traditional statistics concludes that the complexity of such a setting makes the problem not worthy of a further study. Thus, we present a problem where a combination of the classical approximation theory and statistics leads to interesting practical results.