Polynomial and rational inequalities on Jordan arcs and domains

In this paper we prove an asymptotically sharp Bernstein-type inequality for polynomials on analytic Jordan arcs. Also a general statement on mapping of a domain bounded by finitely many Jordan curves onto a complement to a system of the same number of arcs with rational function is presented here. This fact, as well as, Borwein-Erd\'elyi inequality for derivative of rational functions on the unit circle, Gonchar-Grigorjan estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials play key roles in the proof of the main result.Comment: Minor typos corrected, DOI adde

A Sharp Inequality of Markov Type for Polynomials Associated with Laguerre Weight

AbstractThe best possible constant An in an inequality of Markov type [ddx(e−xpn(x))][0, ∞)⩽An‖e−xpn(x)‖[0, ∞), where ‖·‖[0, ∞) denotes the sup-norm on the half real line [0, ∞) and pn is an arbitrary polynomial of degree at most n, is determined in terms of the weighted Chebyshev polynomials associated with the Laguerre weight e−x on [0, ∞)

The uniform closure of non-dense rational spaces on the unit interval

AbstractLet Pn denote the set of all algebraic polynomials of degree at most n with real coefficients. Associated with a set of poles {a1,a2,…,an}⊂R⧹[-1,1] we define the rational function spaces Pn(a1,a2,…,an):=f:f(x)=b0+∑j=1nbjx-aj,b0,b1,…,bn∈R.Associated with a set of poles {a1,a2,…}⊂R⧹[-1,1], we define the rational function spacesP(a1,a2,…):=⋃n=1∞Pn(a1,a2,…,an).It is an interesting problem to characterize sets {a1,a2,…}⊂R⧹[-1,1] for which P(a1,a2,…) is not dense in C[-1,1], where C[-1,1] denotes the space of all continuous functions equipped with the uniform norm on [-1,1]. Akhieser showed that the density of P(a1,a2,…) is characterized by the divergence of the series ∑n=1∞an2-1.In this paper, we show that the so-called Clarkson–Erdős–Schwartz phenomenon occurs in the non-dense case. Namely, if P(a1,a2,…) is not dense in C[-1,1], then it is “very much not so”. More precisely, we prove the following result.TheoremLet {a1,a2,…}⊂R⧹[-1,1]. Suppose P(a1,a2,…) is not dense in C[-1,1], that is,∑n=1∞an2-1<∞.Then every function in the uniform closure of P(a1,a2,…) in C[-1,1] can be extended analytically throughout the set C⧹{-1,1,a1,a2,…}

Polynomials with symmetric zeros

Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real self-reciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.Comment: Keywords: Self-inversive polynomials, self-reciprocal polynomials, Pisot and Salem polynomials, M\"obius transformations, knot theory, Bethe equation

Exact Reconstruction using Beurling Minimal Extrapolation

We show that measures with finite support on the real line are the unique solution to an algorithm, named generalized minimal extrapolation, involving only a finite number of generalized moments (which encompass the standard moments, the Laplace transform, the Stieltjes transformation, etc). Generalized minimal extrapolation shares related geometric properties with basis pursuit of Chen, Donoho and Saunders [CDS98]. Indeed we also extend some standard results of compressed sensing (the dual polynomial, the nullspace property) to the signed measure framework. We express exact reconstruction in terms of a simple interpolation problem. We prove that every nonnegative measure, supported by a set containing s points,can be exactly recovered from only 2s + 1 generalized moments. This result leads to a new construction of deterministic sensing matrices for compressed sensing.Comment: 27 pages, 3 figures version 2 : minor changes and new titl

Exact Markov inequalities for the Hermite and Laguerre weights

AbstractDenote by πn the set of all real algebraic polynomials of degree at most n and let Un≔{e-x2p(x):p∈πn}, Vn≔{e-xp(x):p∈πn}. We prove the following exact Markov inequalities:‖u(k)‖R⩽‖u*,n(k)‖R‖u‖R,∀u∈Un,∀k∈N,and‖v(k)‖R+⩽‖v*,n(k)‖R+‖v‖R+,∀u∈Vn,∀k∈N,where ‖·‖R (‖·‖R+) is the supremum norm on R (R+≔[0,∞)) and u*,n (v*,n) is the Chebyshev polynomial from Un (Vn)

Asymptotic sharpness of a Bernstein-type inequality for rational functions in H^{2}

A Bernstein-type inequality in the standard Hardy space H^{2} of the unit disc \mathbb{D}=\{z\in\mathbb{C}:\,|z|<1\}, for rational functions in \mathbb{D} having at most n poles all outside of \frac{1}{r}\mathbb{D},