Sinc numerical methods

In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a> and <a href="/facts/Applied_mathematics/qyqklXXm">applied mathematics</a>, sinc numerical methods are numerical techniques for finding approximate solutions of <a href="/facts/Partial_differential_equations/DyPsBlv7">partial differential equations</a> and <a href="/facts/Integral_equations/LCu4Imfq">integral equations</a> based on the translates of <a href="/facts/Sinc/lSvDtHy0">sinc</a> function and Cardinal function C(f,h) which is an expansion of f defined by

C
        (
        f
        ,
        h
        )
        (
        x
        )
        =
        
          ∑
          
            k
            =
            −
            ∞
          
          
            ∞
          
        
        f
        (
        k
        h
        )
        
        
          
            sinc
          
        
        
          (
          
            
              
                
                  x
                  h
                
              
            
            −
            k
          
          )
        
      
    
    {\displaystyle C(f,h)(x)=\sum _{k=-\infty }^{\infty }f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)}

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

sinc
          
        
        (
        x
        )
        =
        
          
            
              sin
              ⁡
              (
              π
              x
              )
            
            
              π
              x
            
          
        
      
    
    {\displaystyle {\textrm {sinc}}(x)={\frac {\sin(\pi x)}{\pi x}}}

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
        )
        =
        
          
            ∑
            
              k
              =
              −
              M
            
            
              N
            
          
          f
          (
          k
          h
          )
          
          
            
              sinc
            
          
          
            (
            
              
                
                  
                    x
                    h
                  
                
              
              −
              k
            
            )
          
        
      
    
    {\displaystyle C_{M,N}(f,h)(x)=\displaystyle \sum _{k=-M}^{N}f(kh)\,{\textrm {sinc}}\left({\dfrac {x}{h}}-k\right)}
  
 .

Sinc numerical methods open-in-new

Sinc numerical methods