Remark on extremal measure extensions

1980 ◽  
pp. 79-80 ◽  
Author(s):  
Heinrich v. Weizsäcker
Keyword(s):  
Extremal Measure

scholarly journals Zero distribution of sequences of classical orthogonal polynomials

2003 ◽  
Vol 2003 (17) ◽  
pp. 985-993
Author(s):  
Plamen Simeonov
Keyword(s):  
Orthogonal Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Limit Measure ◽  
Zero Distribution ◽  
Extremal Measure

We obtain the zero distribution of sequences of classical orthogonal polynomials associated with Jacobi, Laguerre, and Hermite weights. We show that the limit measure is the extremal measure associated with the corresponding weight.

scholarly journals The Gelfond–Schnirelman Method in Prime Number Theory

2005 ◽  
Vol 57 (5) ◽  
pp. 1080-1101 ◽  
Author(s):  
Igor E. Pritsker
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Prime Number ◽  
Potential Theory ◽  
Integer Coefficients ◽  
Polynomial Type ◽  
Weighted Capacity ◽  
Theoretic Problem ◽  
The Subject ◽  
Extremal Measure

AbstractThe original Gelfond–Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on [0, 1] to give a Chebyshev-type lower bound in prime number theory. We study a generalization of thismethod for polynomials inmany variables. Ourmain result is a lower bound for the integral of Chebyshev's ψ-function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support.

scholarly journals Finite-infinite range inequalities in the complex plane

1991 ◽  
Vol 14 (4) ◽  
pp. 625-638
Author(s):  
H. N. Mhaskar
Keyword(s):  
Weight Function ◽  
Compact Subset ◽  
Complex Plane ◽  
Borel Measure ◽  
Positive Borel Measure ◽  
Christoffel Functions ◽  
Extremal Measure ◽  
Infinite Range

LetE⫅Cbe closed,ωbe a suitable weight function onE,σbe a positive Borel measure onE. We discuss the conditions onωandσwhich ensure the existence of a fixed compact subsetKofEwith the following property. For anyp,0<P≤∞, there exist positive constantsc1, c2depending only onE,ω,σandpsuch that for every integern≥1and every polynomialPof degree at mostn,∫E\K|ωnP|pdσ≤c1exp(−c2n)∫K|ωnP|pdσ. In particular, we shall show that the support of a certain extremal measure is, in some sense, the smallest setKwhich works. The conditions onσare formulated in terms of certain localized Christoffel functions related toσ.

Sign in / Sign up

Export Citation Format

Share Document