Sinc numerical methods explained

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 ^{inf ty} _{k =-inf ty} f(kh) rm{sinc} \left(\dfrac{x}{h} - k \right) where the step size h>0 and where the sinc function is defined byrm{sinc}(x)= (\sin(\pix))/(\pix) 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 ^{N} _{k =- M} f(kh) rm{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 ^{- (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

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

Notes and References

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