Nikol’skii inequality between the uniform norm and L q -norm with jacobi weight of algebraic polynomials on an interval

2016 ◽  
Vol 42 (2) ◽  
pp. 91-120 ◽  
Author(s):  
V. Arestov ◽  
M. Deikalova
Keyword(s):  
Uniform Norm ◽  
Algebraic Polynomials ◽  
Jacobi Weight ◽  
Nikol’Skii Inequality

scholarly journals INEQUALITIES FOR ALGEBRAIC POLYNOMIALS ON AN ELLIPSE

2020 ◽  
Vol 6 (2) ◽  
pp. 87
Author(s):  
Tatiana M. Nikiforova
Keyword(s):  
Approximation Theory ◽  
Upper Bound ◽  
Algebraic Polynomial ◽  
Uniform Norm ◽  
Algebraic Polynomials

The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci \(\pm 1\) of the derivative of an algebraic polynomial with real coefficients normalized on the segment \([- 1,1]\).

Convex Polynomial Approximation in the Uniform Norm: Conclusion

2005 ◽  
Vol 57 (6) ◽  
pp. 1224-1248 ◽  
Author(s):  
K. A. Kopotun ◽  
D. Leviatan ◽  
I. A. Shevchuk
Keyword(s):  
Convex Function ◽  
Polynomial Approximation ◽  
Uniform Norm ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Jackson Type ◽  
Algebraic Polynomials ◽  
Type Estimate ◽  
Open Problems ◽  
Open Questions

AbstractEstimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each r-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian–Totik kth modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs (k, r) we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

Exceptional solutions to the Painlevé VI equation associated with the generalized Jacobi weight

2017 ◽  
Vol 06 (01) ◽  
pp. 1750003
Author(s):  
Shulin Lyu ◽  
Yang Chen
Keyword(s):  
Asymptotic Expansions ◽  
Hankel Determinant ◽  
Generalized Jacobi Weight ◽  
Jacobi Weight ◽  
Special Cases ◽  
Painlevé Vi Equation

We consider the generalized Jacobi weight [Formula: see text], [Formula: see text]. As is shown in [D. Dai and L. Zhang, Painlevé VI and Henkel determinants for the generalized Jocobi weight, J. Phys. A: Math. Theor. 43 (2010), Article ID:055207, 14pp.], the corresponding Hankel determinant is the [Formula: see text]-function of a particular Painlevé VI. We present all the possible asymptotic expansions of the solution of the Painlevé VI equation near [Formula: see text] and [Formula: see text] for generic [Formula: see text]. For four special cases of [Formula: see text] which are related to the dimension of the Hankel determinant, we can find the exceptional solutions of the Painlevé VI equation according to the results of [A. Eremenko, A. Gabrielov and A. Hinkkanen, Exceptional solutions to the Painlevé VI equation, preprint (2016), arXiv:1602.04694 ], and thus give another characterization of the Hankel determinant.

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