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Trapezoidal Rule Revisited

Vol. 6 No. 3 (2011) P.347~P.360

ABSTRACT

P. K. Sahoo in [7] has arrived at the functional equation stemming from trapezoidal rule $g(y)-g(x)=\frac{y-x}{6}\left[f(x)+2f\left(\frac{2x+y}{3}\right)+2f\left(\frac{x+2y}{3}\right)+f(y)\right],$ for $x,y\in\mathbb{R},$ where $f$ and $g$ are unknown functions. In fact, Sahoo considered more general equations $$\label{Sahoo1} g(y)-h(x)=(y-x)[f(x)+2k(sx+ty)+2k(tx+sy)+f(y)]$$ with four unknown functions (cf. [8]) and $$\label{Sahoo2} f_{1}(y)-g_{1}(x)=(y-x)[f_{2}(x)+f_{3}(sx+ty)+f_{4}(tx+sy)+f_{5}(y)]$$ with six unknown functions (cf. [8]), where $s$ and $t$ are two fixed real parameters. The equations have been solved in [7] and [8] for $s^{2}=t^{2}$ or $s=0$ or $t=0$ without any regularity assumptions, and in the case $s^{2}\neq t^{2}$ (with $st\neq0$) the solutions have been determined under high regularity assumptions on unknown functions (differentiability of second or fourth order).
In this paper we solve equations (1) and (2) in the case of $s^{2}\neq t^{2}$ (with $st\neq0$) with no regularity assumptions on unknown functions for rational parameters $s$ and $t$, and under very weak assumptions in other cases.

KEYWORDS
Functional equations, trapezoidal rule, generalized polynomials, no regularity assumptions

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 39B05, 39B22

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