Sinc numerical methods

<h2 id="sinc-numerical-methods-cover">Sinc numerical methods cover</h2>
<ul><li>function approximation,</li>
<li>approximation of <a href="/facts/Derivative/7Ito70Dh">derivatives</a>,</li>
<li>approximate definite and indefinite <a href="/facts/Integral/V7lyV997">integration</a>,</li>
<li>approximate solution of initial and boundary value ordinary <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> (ODE) problems,</li>
<li>approximation and inversion of <a href="/facts/Fourier_transform/XXhKJtc8">Fourier</a> and <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace</a> transforms,</li>
<li>approximation of <a href="/facts/Hilbert_transforms/RoQaC8MS">Hilbert transforms</a>,</li>
<li>approximation of definite and indefinite <a href="/facts/Convolution/PVPrdz9J">convolution</a>,</li>
<li>approximate solution of partial differential equations,</li>
<li>approximate solution of <a href="/facts/Integral_equations/LCu4Imfq">integral equations</a>,</li>
<li>construction of conformal maps.</li></ul>
<p>Indeed, Sinc are ubiquitous for approximating every operation of calculus
</p><p>In the standard setup of the sinc numerical methods, the errors (in <a href="/facts/Big_O_notation/weFFjSWg">big O notation</a>) are known to be 
  
    
      
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    {\displaystyle 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<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> has recently found that the errors in the Sinc numerical methods based on double exponential transformation are 
  
    
      
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    {\displaystyle O\left(e^{-{\frac {kn}{\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.
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<h2 id="reading">Reading</h2>
<ul><li>Stenger, Frank (2011). <i>Handbook of Sinc Numerical Methods</i>. Boca Raton, Florida: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781439821596.</li>
<li>Lund, John; Bowers, Kenneth (1992). <i>Sinc Methods for Quadrature and Differential Equations</i>. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780898712988.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Stenger, F. (2000). "Summary of sinc numerical methods". Journal of Computational and Applied Mathematics. 121 (1–2): 379–420. doi:10.1016/S0377-0427(00)00348-4. <a href="https://doi.org/10.1016%2FS0377-0427%2800%2900348-4" target="_blank">https://doi.org/10.1016%2FS0377-0427%2800%2900348-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Sugihara, M.; Matsuo, T. (2004). "Recent developments of the Sinc numerical methods". Journal of Computational and Applied Mathematics. 164–165: 673–689. doi:10.1016/j.cam.2003.09.016. <a href="https://doi.org/10.1016%2Fj.cam.2003.09.016" target="_blank">https://doi.org/10.1016%2Fj.cam.2003.09.016</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
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Sinc numerical methods open-in-new

Sinc numerical methods