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:

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

Sinc numerical methods cover

• function approximation,
• approximation of derivatives,
• approximate definite and indefinite integration,
• approximate solution of initial and boundary value ordinary differential equation (ODE) problems,
• approximation and inversion of Fourier and Laplace transforms,
• approximation of Hilbert transforms,
• approximation of definite and indefinite convolution,
• approximate solution of partial differential equations,
• approximate solution of integral equations,
• construction of conformal maps.

Indeed, Sinc are ubiquitous for approximating every operation of calculus

In the standard setup of the sinc numerical methods, the errors (in big O notation) are known to be ${\displaystyle O\left(e^{-c{\sqrt {n}}}\right)}$ with some c>0, where n is the number of nodes or bases used in the methods. However, Sugihara^[2] has recently found that the errors in the Sinc numerical methods based on double exponential transformation are ${\displaystyle O\left(e^{-{\frac {kn}{\ln n}}}\right)}$ with some k>0, in a setup that is also meaningful both theoretically and practically and are found to be best possible in a certain mathematical sense.

Reading

• Stenger, Frank (2011). Handbook of Sinc Numerical Methods. Boca Raton, Florida: CRC Press. ISBN 9781439821596.
• Lund, John; Bowers, Kenneth (1992). Sinc Methods for Quadrature and Differential Equations. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). ISBN 9780898712988.