In numerical analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

 and applied mathematics

Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

, sinc numerical methods are numerical techniques 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=-∞)^∞ sinc(x/h-k) where the step size h>0 and where the sinc function is defined by Sinc(x)=sin⁡(\pi x)/(\pi x)
Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers.

Sinc numerical methods cover

• function approximation,
• approximation of derivative
  Derivative
  In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
  
  s,
• approximate definite and indefinite integration
  Integral
  Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
  
  ,
• approximate solution of initial and boundary value ordinary differential equation
  Differential equation
  A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
  
   (ODE) problems,
• approximation and inversion of Fourier
  Fourier transform
  In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
  
   and Laplace transforms,
• approximation of Hilbert transforms,
• approximation of definite and indefinite convolution
  Convolution
  In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
  
  ,
• 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 are known to be O(exp(−c√n)) with some c>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 based on double exponential transformation are O(exp(−k n/log n)) with some k>0, in a 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.

Reading

J. Lund and K.L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, 1992.
F.Stenger Handbook of Sinc Numerical Methods, Taylor & Francis,2010.
M. Sugihara and Takayasu Matsuo , Recent developments of the Sinc numerical methods, Journal of Computational and Applied Mathematics 164–165, 2004.
F. Stenger, Summary of Sinc numerical methods, Journal of Computational and Applied Mathematics (121) 379-420 (2000)