Id like to know methods for numerical computation of $L^2$ norm of a two dimensional trigonometric polynomial.

I have the coefficients. If I want to compute the L^1 norm, I can do so by sampling in time at a rate greater than twice of the order of the trig polynomial (Shannon sampling theorem) and then doing summation of samples which can approximate the L^1 norm integral quite well. But for L^2 norm I need to square in time domain and cannot use this procedure as squaring is non linear?

PS : I dont want to use Parseval's theorem, as it wont help if I want to compute L^3 norm of a 3 dimensional trigonometric polynomial. I want a general procedure for $L^p$ norm of a $p$ dimensional trig polynomial.

  • 3
    $\begingroup$ The trapezoidal rule should be accurate for even integer $p$, because then it's exponentially convergent. For other $p$, the chief problem would be that the integrand is not analytic at the roots. For general $p$, there is a way to do it, as implemented in chebfun and similar packages, which can also handle the otherwise difficult case $p=\infty$. $\endgroup$
    – Kirill
    Apr 28 '17 at 2:33
  • $\begingroup$ For the $L^1$ norm, you need to take the absolute value, which is also a nonlinear operation. $\endgroup$
    – cfh
    Apr 29 '17 at 7:34
  • $\begingroup$ @cfh Its non linear but does not any problem for area computation, just need to have sufficient sample rate. For p norm in p dimensions i guees we need to sample at a rate 2p times the degree of trig polynomial in each dimension. So it is just a matter of going for higher samples to get accuracy. $\endgroup$
    – Rajesh D
    Apr 29 '17 at 8:07
  • $\begingroup$ Can you give an example of the kind of function for which you try to compute the norm? $\endgroup$ Apr 29 '17 at 20:32

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