Sinc numerical methods

Sinc Numerical Methods are a main topic in Numerical Analysis and Applied mathematics. This methods are based on Sinc function defined by sinc(x)=sin(Pi*x)/(Pi*x) which have many interesting properties.

For detailed discussion see "Sinc Methods for quadrature and differential equations SIAM 1992 J. Lund" During the last three decades there have been developed a variety of Sinc numerical methods based on the Sinc approximation.

The 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 PDEs,
• approximate solution of integral equations,
• construction of conformal maps.

In the standard setup of the Sinc numerical methods, the errors are known to be O(exp(−�√n)) with some �k>0 , where n is the number of nodes or bases used in the methods. However, Sugihara has recently found that the errors in the Sinc numerical methods are O(exp(−k' n/log n)) with some ��k'>0, in another setup that is also meaningful both theoretically and practically. It has also been found that the error bounds of O(exp(−k�n/log n)) are best possible in a certain mathematical sense.

References

Recent developments of the Sinc numerical methods Masaaki Sugihara, Takayasu Matsuo Journal of Computational and Applied Mathematics 164–165 (2004)