Moment Structure and Approximation Properties of Univariate and Bivariate Durrmeyer–Beta-Type Operators

We study a Durrmeyer–Beta-type positive linear operator on the half-line, constructed from rising-factorial weights and a normalized beta-prime kernel. Exact raw moments through degree four and the corresponding central moments are derived under the required moment-existence conditions. These identities yield local approximation estimates, weighted Korovkin convergence, and a weighted O(n−1) rate for functions with bounded first and second derivatives. A tensor-product extension provides bivariate convergence and error bounds involving partial moduli of continuity and coordinatewise regularity. Comparisons with related Durrmeyer constructions clarify the operator’s structure, while a matched-index comparison with a Dunkl–Beta operator examines its numerical accuracy. The numerical examples include smooth, oscillatory, and nonsmooth bounded functions, with an additional experiment on [0,20]. We also investigate reconstruction from noisy observations using 30 signal–noise realizations at each of five Gaussian noise levels. The comparison includes a Gaussian kernel smoother, an ARIMA model, and an LSTM reconstruction network. The Gaussian smoother gives the best mean accuracy at the lowest noise level, whereas D80 gives the lowest mean normalized errors and the highest mean R2 at the four higher levels. These experiments illustrate the approximation and smoothing behavior of the operator under the stated computational settings.