On Wachnicki’s Generalization of the Gauss–Weierstrass Integral


The paper aims at a generalization of the Gauss–Weierstrass integral introduced by Eugeniusz Wachnicki two decades ago. It is intimately connected to a generalization of the heat equation. The main result is an asymptotic expansion for the operators when applied to a function belonging to a rather large class. An essential auxiliary result is a localization theorem which is interesting in itself.


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; Kummer function; Asymptotic expansion; Degree of approximation

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U. Abel and O. Agratini, On Wachnicki’s Generalization of the Gauss-Weierstrass Integral, In: Candela, A.M., Cappelletti Montano, M., Mangino, E. (eds) Recent  advances in Mathematical Analysis. Trends in Mathematics,  Birkhäuser, Cham., pp 1–13,


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Recent Advances in Mathematical Analysis 

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[1] Abel, U.: A Voronovskaya type result for simultaneous approximation by Bernstein–Chlodovsky polynomials. Results Math. 74, Article number 117 (2019) Google Scholar
[2] Abel, U.: Voronovskaja type theorems for positive linear operators related to squared fundamental functions. In: Draganov, B., Ivanov, K., Nikolov, G., Uluchev, R. (eds.) Constructive Theory of Functions, Sozopol 2019, pp. 1–21. Prof. Marin Drinov Publishing House of Bas, Sofia (2020) Google Scholar
[3] Abel, U., Ivan, M.: Simultaneous approximation by Altomare operators. Suppl. Rend. Circ. Mat. Palermo 82(2), 177–193 (2010) MathSciNet MATH Google Scholar
[4] Abel, U., Ivan, M.: Complete asymptotic expansions for Altomare operators. Mediterr. J. Math. 10, 17–29 (2013) CrossRef MathSciNet MATH Google Scholar
[5] Abel, U., Karsli, H.: A complete asymptotic expansion for Bernstein-Chodovsky polynomials for functions on R. Mediterr. J. Math., 17, Article number 201 (2020) Google Scholar
[6] Abel, U., Agratini, O., Păltănea, R.: A complete asymptotic expansion for the quasi-interpolants of Gauss–Weierstrass operators. Mediterr. J. Math. 15, Article number 156 (2018) Google Scholar
[7] Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. Tenth Printing (Issued, June 1964) (With corrections, December 1972) Google Scholar
[8] Altomare, F., Campiti, M.: Korovkin-type Approximation Theory and its Applications. de Gruyter Series Studies in Mathematics, vol. 17. Walter de Gruyter, Berlin/New York (1994) Google Scholar
[9] Altomare, F., Milella, S.: Integral-type operators on continuous function spaces on the real line. J. Approx. Theory 152(2), 107–124 (2008) CrossRef MathSciNet MATH Google Scholar
[10] Aral, A., Gal, S.G.: q-Generalizations of the Picard and Gauss–Weierstrass singular integrals. Taiwanese J. Math. 12(9), 2051–2515 (2008) Google Scholar
[11] Bardaro, C., Mantellini, I., Uysal, G., Yilmaz, B.: A class of integral operators that fix exponential functions. Mediterr. J. Math. 18, Article number 179 (2021) Google Scholar
[12] Bragg, L.R.: The radial heat polynomials and related functions. Trans. Am. Math. Soc. 119, 270–290 (1965) CrossRef MathSciNet MATH Google Scholar
[13] Ifantis, E.K., Siafarikas, P.D.: Bounds for modified Bessel functions. Rend. Circ. Mat. Palermo II 40(3), 347–356 (1991) CrossRef MathSciNet MATH Google Scholar
[14] Krech, G., Krech, I.: On some bivariate Gauss–Weierstrass operators. Constr. Math. Anal. 2(2), 57–63 (2019) MathSciNet MATH Google Scholar
[15] Luke, Y.L.: Inequalities for generalized hypergeometric functions. J. Approx. Theory, 5, 41–65 (1972) CrossRef MathSciNet MATH Google Scholar
[16] NIST Digital Library of Mathematical Functions. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds. https://dlmf.nist.gov
[17] Rosenbloom, P., Widder, D. V.: Expansions in heat polynomials and associated functions. Trans. Am. Math. Soc. 92, 220–266 (1959) CrossRef MathSciNet MATH Google Scholar
[18]  Sikkema, P.C.: On some linear positive operators. Ind. Math. 32, 327–337 (1970) CrossRef MathSciNet MATH Google Scholar
[19]  Wachnicki, E.: On a Gauss–Weierstrass generalized integral. Rocznik Nauk.-Dydakt. Akad. Pedagogícznej W Krakow. Prace Mat. 17(2000), 251–263 MathSciNet MATH Google Scholar
[20]  Yilmaz, B.: Approximation properties of modified Gauss–Weierstrass integral operators in exponential weighted Lp spaces. Facta Univ. (Nis̆) Ser. Math. Inform. 36(1), 89–100 (2021)


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