Asymptotic zero distribution for a class of extremal polynomials

We consider extremal polynomials with respect to a Sobolev-type [Formula: see text]-norm, with [Formula: see text] and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures (i.e. supported on disjoint subsets of the real line), it is proved that their critical points are simple and contained in the interior of the convex hull of the support of the measures involved and the asymptotic critical point distribution is studied. We also find the [Formula: see text]th root asymptotic behavior of the corresponding sequence of Sobolev extremal polynomials and their derivatives.

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A one dimensional random space-filling problem

Journal of Applied Probability ◽

10.1017/s0021900200110101 ◽

1968 ◽

Vol 5 (02) ◽

pp. 427-435 ◽

Cited By ~ 5

Author(s):

John P. Mullooly

Keyword(s):

Asymptotic Behavior ◽

Finite Number ◽

Real Line ◽

Random Variables ◽

Random Interval ◽

Space Filling ◽

One Dimensional ◽

The Real ◽

Fixed Constant ◽

Filling Problem

Consider an interval of the real line (0, x), x > 0; and place in it a random subinterval S(x) defined by the random variables Xx and Yx , the position of the center of S(x) and the length of S(x). The set (0, x)– S(x) consists of two intervals of length δ and η. Let a > 0 be a fixed constant. If δ ≦ a, then a random interval S(δ) defined by Xδ, Yδ is placed in the interval of length δ. If δ < a, the placement of the second interval is not made. The same is done for the interval of length η. Continue to place non-intersecting random subintervals in (0, x), and require that the lengths of all the random subintervals be ≦ a. The process terminates after a finite number of steps when all the segments of (0, x) uncovered by random subintervals are of length < a. At this stage, we say that (0, x) is saturated. Define N(a, x) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N(a, x), for large x.

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Logarithmic asymptotics of contracted Sobolev extremal polynomials on the real line

Journal of Approximation Theory ◽

10.1016/j.jat.2006.03.014 ◽

2006 ◽

Vol 143 (1) ◽

pp. 62-73 ◽

Cited By ~ 4

Author(s):

G. López Lagomasino ◽

F. Marcellán Español ◽

H. Pijeira Cabrera

Keyword(s):

Real Line ◽

The Real ◽

Extremal Polynomials ◽

Logarithmic Asymptotics

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A one dimensional random space-filling problem

Journal of Applied Probability ◽

10.2307/3212263 ◽

1968 ◽

Vol 5 (2) ◽

pp. 427-435 ◽

Cited By ~ 12

Author(s):

John P. Mullooly

Keyword(s):

Asymptotic Behavior ◽

Finite Number ◽

Real Line ◽

Random Variables ◽

Random Interval ◽

Space Filling ◽

One Dimensional ◽

The Real ◽

Fixed Constant ◽

Filling Problem

Consider an interval of the real line (0, x), x > 0; and place in it a random subinterval S(x) defined by the random variables Xx and Yx, the position of the center of S(x) and the length of S(x). The set (0, x)– S(x) consists of two intervals of length δ and η. Let a > 0 be a fixed constant. If δ ≦ a, then a random interval S(δ) defined by Xδ, Yδ is placed in the interval of length δ. If δ < a, the placement of the second interval is not made. The same is done for the interval of length η. Continue to place non-intersecting random subintervals in (0, x), and require that the lengths of all the random subintervals be ≦ a. The process terminates after a finite number of steps when all the segments of (0, x) uncovered by random subintervals are of length < a. At this stage, we say that (0, x) is saturated. Define N(a, x) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N(a, x), for large x.

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Searching for a particle on the real line

Advances in Applied Probability ◽

10.1017/s0001867800039720 ◽

1974 ◽

Vol 6 (01) ◽

pp. 79-102 ◽

Cited By ~ 3

Author(s):

Bert Fristedt ◽

David Heath

Keyword(s):

Wide Class ◽

Real Line ◽

Optimization Problems ◽

Optimal Strategies ◽

Cost Functions ◽

Expected Cost ◽

The Real

In this paper we consider two optimization problems and two game problems. In each problem, a particle is hidden on the real line (sometimes randomly, and sometimes by an antagonistic hider), and a seeker, starting at the origin, wishes to find the particle with minimal expected cost. We consider a fairly wide class of cost functions depending upon the position of the particle and the time used to discover it. For the games we obtain the values and (∊-) optimal strategies. For the optimization problems we obtain qualitative features of (∊-) optimal searches.

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Searching for a particle on the real line

Advances in Applied Probability ◽

10.2307/1426208 ◽

1974 ◽

Vol 6 (1) ◽

pp. 79-102 ◽

Cited By ~ 17

Author(s):

Bert Fristedt ◽

David Heath

Keyword(s):

Wide Class ◽

Real Line ◽

Optimization Problems ◽

Optimal Strategies ◽

Cost Functions ◽

Expected Cost ◽

The Real

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Higher Order Sobolev-Type Spaces on the Real Line

Journal of Function Spaces ◽

10.1155/2014/261565 ◽

2014 ◽

Vol 2014 ◽

pp. 1-13

Author(s):

Bogdan Bojarski ◽

Juha Kinnunen ◽

Thomas Zürcher

Keyword(s):

Sobolev Spaces ◽

Real Line ◽

Finite Differences ◽

Higher Order ◽

Sobolev Type ◽

The Real ◽

Sobolev Functions ◽

Sobolev Type Spaces ◽

Type Spaces

This paper gives a characterization of Sobolev functions on the real line by means of pointwise inequalities involving finite differences. This is also shown to apply to more general Orlicz-Sobolev, Lorentz-Sobolev, and Lorentz-Karamata-Sobolev spaces.

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The properties of Bessel-type fractional derivatives and integrals

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i8.203 ◽

2016 ◽

pp. 3973-3982

Author(s):

V. R. Lakshmi Gorty

Keyword(s):

Fractional Order ◽

Real Line ◽

Computer Algebra System ◽

Fractional Derivatives ◽

Graphical Representation ◽

Fractional Integrals ◽

The Real ◽

Fractional Derivatives And Integrals ◽

Half Axis ◽

Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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