2007 / March Volume 2 No.1
On a functional equation associated with the trapezoidal rule
| Published Date |
2007 / March
|
|---|---|
| Title | On a functional equation associated with the trapezoidal rule |
| Author | |
| Keyword | |
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| Pagination | 67-82 |
| 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. |
| AMS Subject Classification |
39B22
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| Received |
2006-08-28
|