Original function

Transformed function

$u\left(x,t\right)=u_1\left(x,t\right)\pm u_2\left(x,t\right)$

$U\left(i,j\right)=U_1\left(i,j\right)\pm U_2\left(i,j\right)$

$u\left(x,t\right)=cv\left(x,t\right)$

$U\left(i,j\right)=cV\left(i,j\right)$

$u\left(x,t\right)=u_1\left(x,t\right)u_2\left(x,t\right)$

$U\left(i,j\right)={\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{U}_1\left(i,n-j\right)U_2\left(m-i,j\right)}}$

$u\left(x,t\right)=\frac{\partial v\left(x,t\right)}{\partial t}$

$U\left(i,j\right)=\left(j+1\right)V\left(i,j+1\right)$

$u\left(x,t\right)=\frac{{\partial }^{k}v\left(x,t\right)}{\partial x^k}$

$U\left(i,j\right)=\frac{\left(i+k\right)!}{k!}V\left(i+k,j\right)$

$u\left(x,t\right)=\frac{{\partial }^{k}v\left(x,t\right)}{\partial t^k}$

$U\left(i,j\right)=\frac{\left(j+k\right)!}{k!}V\left(i,j+k\right)$