On some equations stemming from quadrature rules

Barbara Koclęga-Kulpa, Tomasz Szostok, Szymon Wąsowicz

Abstract


We deal with functional equations of the type

F(y)-F(x) =(y-x)∑k=1nfk((1-λk)x+λky),

connected to quadrature rules and, in particular, we find the solutions of the following functional equation

f(x)-f(y)=(x-y)[g(x)+h(x+2y)+h(2x+y)+g(y)].

We also present a solution of the Stamate type equation

yf(x)-xf(y)=(x-y)[g(x)+h(x+2y)+h(2x+y)+g(y)].

All results are valid for functions acting on integral domains.


Keywords


Functional equations, polynomial functions, quadrature rules

Mathematics Subject Classification


39B52

References


Koclega-Kulpa, B. and T. Szostok. "On some equations connected to Hadamard inequalities." Aequationes Math. 75 (2008): 119-129.

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Pawlikowska, I. "Solutions of two functional equations using a result of M. Sablik." Aequationes Math. 72 (2006): 177-190.

Riedel, T. and P.K. Sahoo. "Mean value theorems and functional equations." Singapore-New Jersey-Lodon-Hong Kong: World Scientific, 1998.

Sablik, M. and A. Lisak "On a problem of P.K. Sahoo." Report of meeting. The Seventh Katowice-Debrecen Winter Seminar on Functional Equations and Inequalities, January 31 - February 2, 2007. Ann. Math. Sil. 21 (2007): 70-71.

Sablik, M. "Taylor’s theorem and functional equations." Aequationes Math. 60 (2000): 258-267.

Sahoo, P.K. "On a functional equation associated with the trapezoidal rule." Report of meeting. The Forty-fourth International Symposium on Functional Equations, May 14-20, 2006. Aequationes Math. 73 (2007): 185.


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