scholarly journals Polynomials of Least Deviation from Zero in Sobolev p-Norm

2022 ◽  
Author(s):  
Abel Díaz-González ◽  
Héctor Pijeira-Cabrera ◽  
Javier Quintero-Roba
Keyword(s):  
Convex Hull ◽  
Asymptotic Distribution ◽  
Zeros Of Polynomials ◽  
Order Restriction ◽  
Distribution Of Zeros ◽  
Location Of Zeros ◽  
Discrete Part ◽  
Asymptotic Distribution Of Zeros ◽  
General Conditions

AbstractThe first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p-norm ($$1<p<\infty $$ 1 < p < ∞ ) for the case $$p=1$$ p = 1 . Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that the n-th polynomial of least deviation has at least $$n-\mathbf {d}^*$$ n - d ∗ zeros on the convex hull of the support of the measure, where $$\mathbf {d}^*$$ d ∗ denotes the number of terms in the discrete part.

scholarly journals CONVERGENCE OF FUBINI–STUDY CURRENTS FOR ORBIFOLD LINE BUNDLES

2013 ◽  
Vol 24 (07) ◽  
pp. 1350051 ◽  
Author(s):  
DAN COMAN ◽  
GEORGE MARINESCU
Keyword(s):  
Line Bundle ◽  
Asymptotic Distribution ◽  
Line Bundles ◽  
Singular Line ◽  
Distribution Of Zeros ◽  
Bergman Kernels ◽  
Holomorphic Sections ◽  
Asymptotic Distribution Of Zeros

We discuss positive closed currents and Fubini–Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini–Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curvature is strictly positive, generalizing a well-known theorem of Tian. We include applications to the asymptotic distribution of zeros of random holomorphic sections.

scholarly journals On the Distribution of Zeros and Poles of Rational Approximants on Intervals

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
V. V. Andrievskii ◽  
H.-P. Blatt ◽  
R. K. Kovacheva
Keyword(s):  
Asymptotic Distribution ◽  
Polynomial Approximation ◽  
Additional Assumption ◽  
Equilibrium Measure ◽  
Distribution Of Zeros ◽  
Rational Approximants ◽  
Bounded Degree ◽  
Asymptotic Distribution Of Zeros ◽  
Best Rational Approximants ◽  
Zeros And Poles

The distribution of zeros and poles of best rational approximants is well understood for the functions , . If is not holomorphic on , the distribution of the zeros of best rational approximants is governed by the equilibrium measure of under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, -values, and poles of best real rational approximants of degree at most to a function that is real-valued, but not holomorphic on . Generalizations to the lower half of the Walsh table are indicated.

scholarly journals Exact Formulae for Variances of Functionals of Convex Hulls

2013 ◽  
Vol 45 (04) ◽  
pp. 917-924
Author(s):  
Christian Buchta
Keyword(s):  
Convex Hull ◽  
Asymptotic Distribution ◽  
Convex Polygon ◽  
Convex Hulls ◽  
Affine Invariance ◽  
Uniform Sample

The vertices of the convex hull of a uniform sample from the interior of a convex polygon are known to be concentrated close to the vertices of the polygon. Furthermore, the remaining area of the polygon outside of the convex hull is concentrated close to the vertices of the polygon. In order to see what happens in a corner of the polygon given by two adjacent edges, we consider—in view of affine invariance—n points P 1,…, P n distributed independently and uniformly in the interior of the triangle with vertices (0, 1), (0, 0), and (1, 0). The number of vertices of the convex hull, which are close to the origin (0, 0), is then given by the number Ñ n of points among P 1,…, P n , which are vertices of the convex hull of (0, 1), P 1,…, P n , and (1, 0). Correspondingly, D̃ n is defined as the remaining area of the triangle outside of this convex hull. We derive exact (nonasymptotic) formulae for var Ñ n and var . These formulae are in line with asymptotic distribution results in Groeneboom (1988), Nagaev and Khamdamov (1991), and Groeneboom (2012), as well as with recent results in Pardon (2011), (2012).

A jump-driven Markovian electric load model

1990 ◽  
Vol 22 (3) ◽  
pp. 564-586 ◽  
Author(s):  
Roland Malhamé
Keyword(s):  
Power Systems ◽  
Load Management ◽  
Water Heating ◽  
State Behavior ◽  
Physically Based ◽  
The Individual ◽  
Discrete Part ◽  
Heating Loads ◽  
Transient And Steady State

Electric water heating loads, in power systems, can be adequately modeled by Markov processes comprising a mix of continuous and discrete states. A physically-based characterization of the dynamic behavior of large aggregates of electric water heating loads is obtained by deriving the forward Kolmogorov equations associated with the individual hybrid-state processes. In addition, by focusing on the discrete part of the state, a Markov renewal viewpoint of the processes is developed. Both viewpoints are used to analyze and predict the transient and steady-state behavior of these loads, of great importance in load management applications.

Sign in / Sign up

Export Citation Format

Share Document