3
$\begingroup$

I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as $$ \sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x) $$ where $d\ge 2$ is an integer, $a,b \in \mathbb{R}$ and $f:\mathbb{R}^d \longrightarrow \mathbb{R}$ is a smooth function in $[a,b]^d$. I am particularly interested in such expansion with a control of the error term.

I appreciate any reference or suggestions.

$\endgroup$
4
$\begingroup$

See e.g. Y. Karshon, S. Sternberg, and J.Weitsman. The Euler-Maclaurin formula for simple integral polytopes and Y. Karshon, S. Sternberg, and J. Weitsman. Euler-Maclaurin with remainder for a simple integral polytope.

For an alternative summation formula -- in terms of integrals only, without using derivatives -- see Approximating sums by integrals only: multiple sums and sums over lattice polytopes.

In these papers, you will also find explicit expressions for and/or simple bounds on the remainder.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.