Myron Goldstein

A sequence of extremal problems for trigonometric polynomials

The sequence of extremal problems In = sup{(2π)−1 ∝02π¦p(θ)¦2 dθ¦pϵ Pn}, where Pn denotes the set of nonnegative trigonometric polynomials of degree ⩽n having constant term 1, is studied. It is shown that (n + 1) C1 ⩽ In < 1 + (n + 1) C1, where C1 = 0.686981293….

CHARACTERIZATIONS OF THIN AND SEMI-THIN SETS

Under mild assumptions about a function φ : (0,1] -+ (0,+°°), sets E which are thin at the origin 0 of R m (m > 3) are characterized by the existence of a Newtonian potential u for which lim inf <(>(|x|)u(x) > 0 and x-K),xeE t if>(t)M(u,t)dt < where M(u,t) is the mean value of u on the sphere 0 of centre 0 and radius t. Various generalizations and extensions are indicated. AMS-Classification: 31B15

Characterization of Open Strips by Harmonic Quadrature

Let D be an open subset of ℝ n (n ≥ 2) of finite n-dimensional Lebesgue-measure λ n (D). Assume furthermore that the point 0 of ℝ n belongs to D. Then a theorem of Kuran states, if

Best Harmonic L1 Approximation to Subharmonic Functions

{1 \over {{\lambda _n}\left( D \right)}}\int_D {hd{\lambda _n} = h\left( 0 \right)}

Quadrature and Harmonic Approximation of Subharmonic Functions in Strips

for all harmonic and integrable functions on D, then D is an open ball centred at 0. The main aim of this paper is to show that a similar characterization holds for the open strip, too.

On the inverse mean value property of harmonic functions on strips

We derive an estimate for Δn, 1 = sup{(2π)−1 ∝02π¦p(eit)¦dt: p(z) = 1 + a1z + · · · + anzn, Re(p(z)) > 0 for ¦z¦ < 1}. In particular it is shown that Δn, 1 ⩽ 1 + log(C1(n + 1) + 1), where C1 = 0.686981293…, It is also shown that 2π ⩽ lim infn → ∞ Δn, 1log n. Finally, upper bounds are found for the Lp-norms of polynomials with positive real part on the unit disk.