A Chebyshev Polynomial Rate-of-Convergence Theorem for Stieltjes Functions

A Chebyshev Polynomial Rate-of-Convergence Theorem for Stieltjes Functions

The theorem proved here extends the author's previous work on Chebyshev series [4] by showing that if f(x) is a member of the class of so-called "Stieltjes functions" whose asymptotic power series 2 anx" about x = 0 is such that ttlogjaj _ hm —;-= r, «log n then the coefficients of the series of shifted Chebyshev polynomials on x e [0, a],2b„Tf(x/a), satisfy the inequality 2 . m log | (log |M) ...

Separation Theorem of Chebyshev - Markov - Stieltjes type forLaurent Polynomials Orthogonal on ( 0 ; 1 )

Multipeakons and a theorem of Stieltjes

A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts. Camassa and Holm [2],[3] introduced the strongly nonlinear equation (1 − 1 4 D 2)u t = 3 2 (u 2) x − 1 8 (u 2 x) x − 1 4 (uu xx) x , (1) as a possible model for dispersive waves in shallow water and show...

Titchmarsh theorem for Jacobi Dini-Lipshitz functions

Our aim in this paper is to prove an analog of Younis's Theorem on the image under the Jacobi transform of a class functions satisfying a generalized Dini-Lipschitz condition in the space $mathrm{L}_{(alpha,beta)}^{p}(mathbb{R}^{+})$, $(1< pleq 2)$. It is a version of Titchmarsh's theorem on the description of the image under the Fourier transform of a class of functions satisfying the Dini-Lip...

A Generalization of the Heine–stieltjes Theorem

The Heine–Stieltjes Theorem describes the polynomial solutions, (v, f) such that T (f) = vf , to specific second order differential operators, T , with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro. The new methods developed are used to describe intri...