Convergence of Szász–Mirakyan–Laguerre–Durrmeyer operators in \(L^p\) Space
DOI:
https://doi.org/10.24193/subbmath.2026.3.8Keywords:
Positive linear operators, Szász–Mirakyan operators, Durrmeyer operators, \(L_p\) approximation, weighted spaces, Schur test, eigenfunctionsAbstract
We investigate the approximation properties of a family of positive linear operators that unify the Szász–Mirakyan--Laguerre --Durrmeyer operators. Using the Schur test, we prove that on every compact interval the operators are uniformly bounded on Lebesgue spaces with constant one independent of the parameter, and for a certain range of the parameter they converge locally in the Lebesgue norm to the identity. For values of the parameter greater than zero, we show that uniform boundedness fails on the whole half‑line; an explicit counterexample demonstrates that global convergence cannot hold without additional weights. To recover global convergence for the remaining parameter range, we introduce exponentially weighted Lebesgue spaces. Under a natural condition linking the weight exponent and the other parameters, we prove uniform boundedness and convergence in these weighted spaces. Finally, we analyze the spectral properties of the operators.
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