2024/07/23 by Abel, Ulrich, Acu, Ana Maria, Heilmann, Margareta +1 · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2407.16474
In this paper we introduce a general family of Szász--Mirakjan--Durrmeyer type operators depending on an integer parameter j ∈ ℤ. They can be viewed as a generalization of the Szász--Mirakjan--Durrmeyer operators [9], Phillips operators [11] and corresponding Kantorovich modifications of higher order. For j∈ ℕ, these operators possess the exceptional property to preserve constants and the monomial xj. It turns out, that an extension of this family covers certain well-known operators studied before, so that the outcoming results could be unified. We present the complete asymptotic expansion for the sequence of these operators. All its coefficients are given in a concise form. In order to prove the expansions for the class of locally integrable functions of exponential growth on the positive half-axis, we derive a localization result which is interesting in itself.