I am having a little trouble understanding what I have to actually do here. What does differentiate with respect to bn? I thinks after differentiation I must use some calculus theorem about extreme but how do I get there?
2026-04-12 05:52:41.1775973161
Find the coefficients of the Fourier series that minimise the error.
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If $v_1,…,v_n$ are orthogonal vectors wrt. some scalar product and its norm, then $$ \|y-(c_1v_1+…+c_nv_n)\| $$ gets minimized by the orthogonal projection of $y$ onto the subspace spanned by $v_1,…v_n$. Which means that the coefficients are obtained as $$ c_k=\frac{\langle y,v_k\rangle}{\|v_k\|^2} $$