On a functional equation associated with the trapezoidal rule
by Prasanna K. Sahoo  

Vol. 2 No. 1 (2007) P.67~P.82

ABSTRACT

The present work aims to determine the solution $f, g, h, k: \mathbb{R} \to \mathbb{R}$ of the equation $g(y)-h(x) = (y-x)[f(x) + 2k(sx+ty) + 2k(tx +sy) + f(y)]$
for all real numbers $x$ and $y$. Here $s$ and $t$ are any two a priori chosen real parameters. This functional equation arises in connection with the trapezoidal rule for the numerical evaluation of definite integrals. In the book [9], it was an open problem to find the general solution of the functional equation $g(y)- g(x) = (y-x) [f(x) + 2k(x+2y) + 2k(2x+y) + f(y)]$.
This paper also determines the differentiable solution of this functional equation.

KEYWORDS
Additive map, functional equation, trapezoidal rule

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 39B22

MILESTONES

Received: 2006-04-06
Revised : 2006-08-28
Accepted:

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