Abstract
We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as
where \(\varphi (x) = \sqrt {1-x^2}\), \(\Delta _h^k(f,x)\) is the kth symmetric difference of f on [−1, 1],
and α, β > −1∕p if 0 < p < ∞, and α, β ≥ 0 if p = ∞.
We show, among other things, that for all \(m, n\in \mathbb N\), 0 < p ≤∞, polynomials P n of degree < n and sufficiently small t,
where w α,β(x) = (1 − x) α(1 + x)β is the usual Jacobi weight.
In the spirit of Yingkang Hu’s work, we apply this to characterize the behavior of the polynomials of best approximation of a function in a Jacobi weighted L p space, 0 < p ≤∞. Finally we discuss sharp Marchaud and Jackson type inequalities in the case 1 < p < ∞.
Dedicated to the memory of our friend, colleague, and collaborator Yingkang Hu (July 6, 1949–March 11, 2016)
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Acknowledgement
The first author was supported by NSERC of Canada Discovery Grant RGPIN 04215-15.
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Kopotun, K.A., Leviatan, D., Shevchuk, I.A. (2019). On Some Properties of Moduli of Smoothness with Jacobi Weights. In: Abell, M., Iacob, E., Stokolos, A., Taylor, S., Tikhonov, S., Zhu, J. (eds) Topics in Classical and Modern Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12277-5_1
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DOI: https://doi.org/10.1007/978-3-030-12277-5_1
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