[Paper Review] A Dunkl generalization of q-parametric Szasz-Mirakjan operators

M. Mursaleen, Nasiruzzaman|arXiv (Cornell University)|Nov 19, 2015

Approximation Theory and Sequence Spaces10 references3 citations

TL;DR

This paper introduces a new q-parametric generalization of Szász-Mirakjan operators using the q-Dunkl exponential function, establishing Korovkin-type approximation theorems and deriving convergence rates via modulus of continuity and Lipschitz classes. The key contribution is the establishment of quantitative convergence estimates for functions in the Lipschitz class with explicit dependence on q-parameters and Dunkl parameters.

ABSTRACT

In this paper, we construct a linear positive operators q-parametric Szasz-Mirakjan operators generated by the q-Dunkl generalization of the exponential function. We obtain Korovkin's type approximation theorem for these operators and compute convergence of these operators by using the modulus of continuity. Furthermore, the rate of convergence of the operators for functions belonging to the Lipschitz class is presented.

Motivation & Objective

• To extend q-parametric Szász-Mirakjan operators by incorporating the q-Dunkl exponential function for improved approximation properties.
• To establish a Korovkin-type approximation theorem for the new operators in the context of q-calculus and Dunkl theory.
• To analyze the convergence rate using the modulus of continuity and provide quantitative estimates for functions in the Lipschitz class.
• To generalize existing q-Szász-Mirakjan operators by integrating Dunkl-type symmetry and q-integer structures.

Proposed method

• The authors define a linear positive operator using the q-Dunkl exponential function $ E_{\mu,q}(x) = \sum_{n=0}^{\infty} \frac{q^{n(n-1)/2} x^n}{\gamma_{\mu,q}(n)} $, where $ \gamma_{\mu,q}(n) $ generalizes the gamma function via q-integers.
• The operator is constructed as $ D_{n_1,n_2}^*(f;q_{n_1},q_{n_2};x,y) = \frac{1}{E_{\mu_1,q_{n_1}}([n_1]_{q_{n_1}}x)} \sum_{k_1=0}^{\infty} \frac{([n_1]_{q_{n_1}}x)^{k_1}}{\gamma_{\mu_1,q_{n_1}}(k_1)} q_{n_1}^{k_1(k_1-1)/2} f\left(\frac{1 - q_{n_1}^{2\mu_1\theta_{k_1} + k_1}}{q_{n_1}^{k_1 - 2}(1 - q_{n_1}^{n_1})}\right) $ for the x-variable, with a similar form for y.
• The convergence analysis relies on the modulus of continuity $ \omega(f;\delta) $, with the bound $ \left| D_{n_1,n_2}^*(f;q_{n_1},q_{n_2};x,y) - f(x,y) \right| \leq \frac{1}{2} \omega(f;\delta_1,\delta_2) \left(1 + \sqrt{\lambda_{n_1}(x)} \right)\left(1 + \sqrt{\lambda_{n_2}(y)} \right) $.
• For functions in the Lipschitz class $ Lip_M(\nu_1,\nu_2) $, the rate of convergence is bounded by $ M \left( \lambda_{n_1}(x) \right)^{\nu_1/2} \left( \lambda_{n_2}(y) \right)^{\nu_2/2} $, where $ \lambda_{n_i}(z) = D_{n_1,n_2}^*((e_{i,0} - z)^2; q_{n_1}, q_{n_2}; x,y) $.
• The proof uses Hölder's inequality and properties of the q-Dunkl exponential function to control the error in terms of second moments of the operator.
• The framework integrates q-calculus, Dunkl theory, and approximation theory by combining q-integers, Dunkl weight functions, and generating functions for $ \gamma_{\mu,q}(n) $.

Experimental results

Research questions

• RQ1 How can q-Szász-Mirakjan operators be generalized using the q-Dunkl exponential function to incorporate symmetry from Dunkl theory?
• RQ2 What is the rate of convergence of the new operators in terms of the modulus of continuity for continuous functions on $[0,\infty) \times [0,\infty)$?
• RQ3 How does the Lipschitz class condition affect the convergence rate of the generalized operators?
• RQ4 What are the quantitative bounds for the approximation error when $ q \to 1 $ and $ \mu $ varies?

Key findings

• The proposed q-Dunkl generalization of q-parametric Szász-Mirakjan operators satisfies a Korovkin-type approximation theorem, ensuring convergence for continuous functions on $[0,\infty) \times [0,\infty)$.
• The convergence rate is bounded by $ \frac{1}{2} \omega(f;\delta_1,\delta_2) \left(1 + \sqrt{\lambda_{n_1}(x)} \right)\left(1 + \sqrt{\lambda_{n_2}(y)} \right) $, where $ \lambda_{n_i}(z) $ represents the second moment of the operator around the point z.
• For functions in the Lipschitz class $ Lip_M(\nu_1,\nu_2) $, the convergence rate is $ M \left( \lambda_{n_1}(x) \right)^{\nu_1/2} \left( \lambda_{n_2}(y) \right)^{\nu_2/2} $, showing explicit dependence on the smoothness parameters $ \nu_1, \nu_2 $.
• The second moment $ \lambda_{n_i}(z) $ is explicitly computed as $ D_{n_1,n_2}^*((e_{i,0} - z)^2; q_{n_1}, q_{n_2}; x,y) $, which quantifies the variance-like behavior of the operator.
• The bound improves as $ q_{n_i} \to 1 $, indicating convergence to classical operators in the limit, with the rate controlled by $ [n_i]_{q_{n_i}} $.
• The results demonstrate that the q-Dunkl generalization preserves the approximation quality of classical q-Szász-Mirakjan operators while incorporating additional symmetry and flexibility via the Dunkl parameter $ \mu $.