Show fourier series approximations are bounded by norms.

Question

I am trying to find a proof on Fourier series approximation.

Answer 1

This is not true for $j>s$; if $j\le s$ it's immediate from the expression of the $H_2^s$ norm in terms of Fourier coefficients.

Of various equivalent forms we take $$||f||_s^2=\sum_n(1+|n|)^{2s}|\hat f(n)|^2.$$ If $j\le s$ and $|n|>N$ then $(1+|n|)^j=(1+|n|)^s(1+|n|)^{(j-s)} \le N^{j-s}(1+|n|)^s$. So $$||f-f_N||_j^2=\sum_{|n|>N}(1+|n|)^{2j}|\hat f(n)|^2\le N^{2(j-s)}\sum_{|n|>N}(1+|n|)^{2s}|\hat f(n)|^2 \le N^{2(j-s)}||f||_s^2.$$