On a functional equation connected to Gauss quadrature rule
Abstract
We consider the functional equation
F(y)−F(x) = (y−x)[f(αx+βy)+f(βx+αy)]
stemming from Gauss quadrature rule. In previous results equations of this type with rational only coefficients α and β were considered. In this paper we allow these numbers to be irrational. We find all solutions of this equation for functions acting on ℝ. However, some results are valid also on integral domains.
Keywords
functional equations on integral domains; quadrature rules
References
2. Haruki Sh., A property of quadratic polynomials, Amer. Math. Monthly 86 (1979), no. 7, 577–579.
3. Koclęga-Kulpa B., Szostok T., On some equations connected to Hadamard inequalities, Aequationes Math. 75 (2008), 119–129.
4. Koclęga-Kulpa B., Szostok T., Wąsowicz Sz., Some functional equations characterizing polynomials, Tatra Mt. Math. Publ. (to appear).
5. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, Państwowe Wydawnictwo Naukowe (Polish Scientific Publishers) and Uniwersytet Śląski, Warszawa–Kraków–Katowice, 1985.
6. Pawlikowska I., Solutions of two functional equations using a result of M. Sablik, Aequationes Math. 72 (2006), 177–190.
7. Riedel T., Sahoo P.K., Mean value theorems and functional equations, World Scientific, Singapore–New Jersey–London–Hong Kong, 1998.
8. Sablik M., Taylor’s theorem and functional equations, Aequationes Math. 60 (2000), 258–267.
9. Sablik M., On a problem of P.K. Sahoo – joint work with Arkadiusz Lisak, talk at the 7th KDWS, Będlewo, Poland, January 31 – February 3, 2007.
10. Report of Meeting. The Fifth Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities, Ann. Math. Sil. 19 (2005), 65–78.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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