On the Unit Circle
نویسنده
چکیده
New characterizations are given for orthogonal polynomials on the unit circle and the associated measures in terms of the reflection coefficients in the recurrence equation satisfied by the polynomials.
منابع مشابه
Bertrand’s Paradox Revisited: More Lessons about that Ambiguous Word, Random
The Bertrand paradox question is: “Consider a unit-radius circle for which the length of a side of an inscribed equilateral triangle equals 3 . Determine the probability that the length of a ‘random’ chord of a unit-radius circle has length greater than 3 .” Bertrand derived three different ‘correct’ answers, the correctness depending on interpretation of the word, random. Here we employ geomet...
متن کاملOn The Mean Convergence of Biharmonic Functions
Let denote the unit circle in the complex plane. Given a function , one uses t usual (harmonic) Poisson kernel for the unit disk to define the Poisson integral of , namely . Here we consider the biharmonic Poisson kernel for the unit disk to define the notion of -integral of a given function ; this associated biharmonic function will be denoted by . We then consider the dilations ...
متن کاملOn Orthogonal Polynomials with Respect to Varying Measures on the Unit Circle
Let {4>„{dfi)} be a system of orthonormal polynomials on the unit circle with respect to d/i and {y/„,m(dß)} be a system of orthonormal polynomials on the unit circle with respect to the varying measures dß/\wn(z)\2, z = e'e , where {w„(z)} is a sequence of polynomials, degw« = n , whose zeros w„ i, ... , wn,n lie in \z\ < 1 The asymptotic behavior of the ratio of the two systems on and outside...
متن کاملNewman Polynomials, Reducibility, and Roots on the Unit Circle
A length k Newman polynomial is any polynomial of the form za1 + · · ·+zak (where a1 < · · · < ak). Some Newman polynomials are reducible over the rationals, and some are not. Some Newman polynomials have roots on the unit circle, and some do not. Defining, in a natural way, what we mean by the “proportion” of length k Newman polynomials with a given property, we prove that • 1/4 of length 3 Ne...
متن کاملVariations of the Circle of Willis in 100 Cadavers in Kerman Province
Background & Aims: The relationship between variations of Willis circle and some cerebrovascular diseases has been shown in previous studies. A few studies have been conducted in Iran on these variations. Methods: This study was performed on 100 recently deceased Iranians due to car accidents who had been referred to Kerman Forensic Medicine Center for autopsy. The length and diameter of cerebr...
متن کاملIntroduction to circular data
In many diverse scientific fields, the measurements are directions. For instance, a biologist may be measuring the direction of flight of a bird or the orientation of an animal. A series of such observations is called ”directional data”. Since a direction has no magnitude, these can be conveniently represented as points on the circumference of a unit circle centered at the origin or as unit ...
متن کامل