Positivity of Turán determinants for orthogonal polynomials
نویسنده
چکیده
The orthogonal polynomials pn satisfy Turán’s inequality if p 2 n(x)− pn−1(x)pn+1(x) ≥ 0 for n ≥ 1 and for all x in the interval of orthogonality. We give general criteria for orthogonal polynomials to satisfy Turán’s inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the q–ultraspherical polynomials.
منابع مشابه
Turán Inequalities and Zeros of Orthogonal Polynomial
We use Turán type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term recurrence pk+1 = xpk − ckpk−1, with a nondecreasing sequence {ck}. As a special case they include a non-asymptotic version of Máté, Nevai and Tot...
متن کاملTurán Inequalities and Zeros of Orthogonal Polynomials∗
We use Turán type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term recurrence pk+1 = xpk − ckpk−1, with a nondecreasing sequence {ck}. As a special case they include a non-asymptotic version of Máté, Nevai and Tot...
متن کاملTur/.n Inequalities for Symmetric Orthogonal Polynomials
A method is outlined to express a Tur,n determinant of solutions of a three term recurrence relation as a weighted sum of squares. This method is shown to imply the positivity of Tur determinants of symmetric Pollaczek polynomials, Lommel polynomials and q-Bessel functions.
متن کاملHigher Order Turán Inequalities
The celebrated Turán inequalities P 2 n(x) − Pn−1(x)Pn+1(x) ≥ 0, x ∈ [−1, 1], n ≥ 1, where Pn(x) denotes the Legendre polynomial of degree n, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities γ2 n − γn−1γn+1 ≥ 0, n ≥ 1, which hold for the Maclaurin coefficients of the real entire function ψ in the L...
متن کاملSolving singular integral equations by using orthogonal polynomials
In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...
متن کامل