Simultaneous approximation by Gauss–Weierstrass–Wachnicki operators


In this note, we spotlight a generalization of Weierstrass integral operators introduced by Eugeniusz Wachnicki. The construction involves modified Bessel functions. The operators are correlated with diffusion equation. Our main result consists in obtaining the asymptotic expansion of derivatives of any order of Wachnicki’s operators. All coefficients are explicitly calculated, and distinct expressions are provided for analytical functions


Ulrich Abel
Fachbereich MND, Technische Hochschule Mittelhessen, Wilhelm-Leuschner-Strasse 13, 61169, Friedberg, Germany

Octavian Agratini
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania


Gauss–Weierstrass operator; Bessel function; Gamma function; hypergeometric function; asymptotic expansion

Paper coordinates

U. Abel, O. Agratini, Simultaneous approximation by Gauss-Weierstrass-Wachnicki operators, Mediterr. J. Math., 19 (2022), art. no. 267,


About this paper


Mediterranean Journal of Mathematics

Publisher Name


Print ISSN


Online ISSN


google scholar link

[1] Abel, U., Agratini, O.: On Wachnicki’s generalization of the Gauss-Weierstrass integral, In: recent advances. In: Analysis, Mathematical (ed.) Anna Maria Candela. Mirella Cappelletti Montano, Elisabeta Mangino), Springer (2022) Google Scholar

[2] Abramowitz, M., Stegun, I.A. (Eds.): Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards Applied Mathematics Series 55, Issued June 1964, Tenth Printing, with corrections (December 1972)

[3] Bragg, L.R.: The radial heat polynomials and related functions. Trans. Amer. Math. Soc. 119, 270–290 (1965) Article MathSciNet Google Scholar

[4] Krech, G., Krech, I.: On some bivariate Gauss-Weierstrass operators. Constr. Math. Anal. 2(2), 57–63 (2019) MathSciNet MATH Google Scholar

[5] Wachnicki, E.: On a Gauss-Weierstrass generalized integral. Rocznik Naukowo-Dydaktyczny Akademii Pedagogicznej W Krakowie, Prace Matematyczne 17, 251–263 (2000) MathSciNet MATH Google Scholar

[6] Zayed, A.I.: Handbook of function and generalized function transformations, 1st 1st Edition. CRC Press, London (1996) Google Scholar

[7] Digital Library of Mathematical Functions, Download references

Related Posts