[Paper Review] Approximating classes of functions defined by operators of differentiation or operators of generalised translation by means of algebraic polynomials

Nimete Sh. Berisha, Faton M. Berısha|arXiv (Cornell University)|Jul 30, 2012

Approximation Theory and Sequence Spaces1 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for the best approximation of functions in weighted $L_p$ spaces on $[-1,1]$ by algebraic polynomials, using a generalized modulus of smoothness defined via an asymmetric operator of generalized translation. The key contribution is a characterization of approximation classes via smoothness conditions on derivatives under a generalized differential operator $D_{x, u, u}$, linking approximation error decay to the smoothness of higher-order derivatives measured by the generalized modulus of continuity.

ABSTRACT

In this paper, approximation by means of algebraic polynomials of classes of functions defined by a generalised modulus of smoothness of operators of differentiation of these functions is considered. We give structural characteristics of classes of functions defined by the order of best approximation by algebraic polynomials.

Motivation & Objective

• To characterize classes of functions in weighted $L_p$ spaces on $[-1,1]$ via generalized modulus of smoothness defined through an asymmetric operator of generalized translation.
• To establish necessary and sufficient conditions for the best approximation of such functions by algebraic polynomials of degree $\leq n-1$.
• To link the decay rate of the best approximation error $E_n(f)_{p,\alpha,\beta}$ to the smoothness of the $2r$-th derivative of $f$ under the generalized differential operator $D_{x,\nu,\mu}$.

Proposed method

• Introduces an asymmetric operator of generalized translation $\tilde{T}_t(f,x)$ based on a kernel involving $R = x\cos t - z\sqrt{1-x^2}\sin t$, tailored to Jacobi-weighted spaces.
• Defines a generalized modulus of smoothness $\tilde{\omega}(f,\delta)_{p,\alpha,\beta} = \sup_{|t|\leq\delta} \|\tilde{T}_t f - f\|_{p,\alpha,\beta}$ to measure smoothness in weighted $L_p$ norms.
• Uses the generalized differential operator $D_{x,\nu,\mu} = (1-x^2)\frac{d^2}{dx^2} + (\mu - \nu - (\nu + \mu + 2)x)\frac{d}{dx}$ to define higher-order smoothness of functions.
• Applies weighted polynomial approximation theory, leveraging bounds on derivatives of algebraic polynomials in Jacobi-weighted $L_p$ norms.
• Establishes equivalence between the decay rate of $E_n(f)_{p,\alpha,\beta}$ and the smoothness of $D_{x,\nu,\mu}^r f$ measured via $\tilde{\omega}(D_{x,\nu,\mu}^r f, \delta)_{p,\alpha,\beta}$.
• Employs function classes of modulus of continuity type $\varphi$ satisfying subadditivity and doubling-type conditions to quantify smoothness.

Experimental results

Research questions

• RQ1 What conditions on the generalized modulus of smoothness $\tilde{\omega}(f,\delta)_{p,\alpha,\beta}$ are necessary and sufficient for the best approximation error $E_n(f)_{p,\alpha,\beta}$ to decay as $\varphi(1/n)$?
• RQ2 How does the smoothness of the $2r$-th derivative $D_{x,\nu,\mu}^r f$ in the generalized sense relate to the approximation error $E_n(f)_{p,\alpha,\beta}$?
• RQ3 What are the precise ranges of parameters $p$, $\alpha$, $\beta$, $\nu$, $\mu$ for which the equivalence between approximation error and generalized smoothness holds?

Key findings

• The best approximation error $E_n(f)_{p,\alpha,\beta}$ satisfies $E_n(f)_{p,\alpha,\beta} \leq C_1 \varphi(1/n)$ if and only if the generalized modulus of smoothness satisfies $\tilde{\omega}(f,\delta)_{p,\alpha,\beta} \leq C_2 \varphi(\delta)$, for a function $\varphi$ of modulus of continuity type.
• For $r \geq 0$, the decay rate $E_n(f)_{p,\alpha,\beta} \leq \frac{C_1}{n^{2r}} \varphi(1/n)$ is equivalent to $\tilde{\omega}(D_{x,\nu,\mu}^r f, \delta)_{p,\alpha,\beta} \leq C_2 \varphi(\delta)$, with $D_{x,\nu,\mu}^0 f = f$
• The conditions on $\alpha$, $\beta$, $\nu$, $\mu$, and $p$ are explicitly given, ensuring the equivalence holds under constraints such as $\alpha > 1 - \frac{1}{2p}$, $\beta > 1 - \frac{1}{2p}$ for $1 < p < \infty$, and $\alpha, \beta \geq 1$ for $p = \infty$
• The results generalize classical Jackson-type theorems to weighted $L_p$ spaces with Jacobi weights, using a non-symmetric generalized translation operator.
• The equivalence holds under mild conditions on $\varphi$, including doubling and comparability properties, ensuring robustness of the characterization.
• For $\varphi(\delta) = \delta^\lambda$ with $2\lambda_0 < \lambda < 2$, the results recover and extend known sharp estimates from prior work (e.g., [13]) for $r=0$.