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

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

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

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

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:

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

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 $O\left(e^{-c\sqrt{n}}\right)$ 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 $O\left(e^{-\frac{k n}{\ln n}}\right)$ 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, FL: 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.

References

1. Stenger, F. (2000). "Summary of sinc numerical methods". Journal of Computational and Applied Mathematics 121: 379–420. doi:10.1016/S0377-0427(00)00348-4.
2. Sugihara, M.; Matsuo, T. (2004). "Recent developments of the Sinc numerical methods". Journal of Computational and Applied Mathematics. 164-165: 673. doi:10.1016/j.cam.2003.09.016.