On the Moduli of Smoothness with Jacobi Weights
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We introduce the moduli of smoothness with Jacobi weights (1 − x)𝛼(1 + x)β for functions in the Jacobi weighted spaces Lp[−1, 1], 0 < p ≤ ∞. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted spaces Lp . If 1 ≤ p ≤ 1, then these moduli are equivalent to certain weighted K-functionals (and, hence, they are equivalent to certain weighted Ditzian–Totik moduli of smoothness for these p), while for 0 < p < 1 they are equivalent to certain “realization functionals.”
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