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

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

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:

.

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 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 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