A Study of King-Type Operators Connecting Wavelet and Quantum Calculus: Numerical and Graphical Analysis

This paper introduces a new class of King-type wavelet-assisted Kantorovich q-Baskakov operators, built to address the observation that classical operators of this type typically fail to reproduce the quadratic test function exactly. Moment formulas are derived for these operators, and a weighted Korovkin-type theorem is used to establish convergence. The speed of pointwise convergence is then quantified with the help of the modulus of smoothness and Lipschitz-type maximal function spaces; the resulting convergence factor δnqn turns out to be smaller than that reported for earlier q-Baskakov constructions, indicating an improvement in approximation power. Beyond the theoretical development, a numerical and graphical investigation plots the pointwise error |Υn,q*f(z)−f(z)| for several parameter pairs (n,q), with the error found to decrease steadily as n increases and q approaches 1, consistent with the theoretical estimates. Implementation details accompany this study to make the computations reproducible and to connect the abstract approximation-theoretic results with their practical execution.