Sinc numerical methods

In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques^[1] for finding approximate solutions of partial differential equations and integral equations based on the translates of sinc function and Cardinal function C(f,h) which is an expansion of f defined by

${\displaystyle C(f,h)(x)=\sum _{k=-\infty }^{\infty }f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)}$

where the step size h>0 and where the sinc function is defined by

${\displaystyle {\textrm {sinc}}(x)={\frac {\sin(\pi x)}{\pi x}}}$

Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers.

The truncated Sinc expansion of f is defined by the following series:

${\displaystyle C_{M,N}(f,h)(x)=\displaystyle \sum _{k=-M}^{N}f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)}$ .