The sharp Jackson inequality for \(L^2\)-approximation on the periodic cylinder. (English) Zbl 1340.41013

Gu, Yi; Liu, Yongping

Acta Math. Sci., Ser. B, Engl. Ed. 35, No. 2, 375-382 (2015).

Summary: We consider the Jackson inequality in \(L^2(B^d \times T, W^B_{\kappa,\mu}(x))\), where the weight function \(W^B_{\kappa,\mu}(x)\) is defined on the ball \(B^d\) and related to reflection group, and obtain the sharp Jackson inequality \[ E_{n-1, m-1}(f)_2\leq \mathcal{K}_{n,m}(\tau, r)\omega_r(f,t)_2, \qquad \tau\geq 2\tau_{n,\lambda}, \] where \(\tau_{n,\lambda}\) is the first positive zero of the Gegenbauer cosine polynomial \(C^\lambda_n(\cos\theta)\) (\(n\in \mathbb N\)).

MSC:

41A17	Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A50	Best approximation, Chebyshev systems

Keywords:

Jackson inequality; best approximation; modulus of continuity; weight function