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The classical Fourier partial sums areHere $a_0={\pi }^{-1}\int f$, and functions are extended periodically. One version of the Dini test for Fourier series states that if one-sided limits $L_{+},L_{-}$ at $t$ exist andthen $s_n^f(t)\to (L_{+}+L_{-})/2$. A step function on a finite interval is constant on each side of any given point for sufficiently small distance, so its displayed numerator is zero there. Dini test for Fourier series gives exactly the asserted midpoint of the two one-sided values, including at jumps.

Let $P_n=s_n$ on $L^2$. It is the orthogonal projection onto the span of the displayed trigonometric functions through frequency $n$, hence $\Vert P_n\Vert \le 1$. For a step-function approximation $\psi$,The convergence on a dense set under a uniform operator bound, here applied using the allowed convergence for $\psi$ and density of step functions, proves $P_{n}f\to f$ in $L^2$. Consequently the union of these finite trigonometric spans is dense, which is precisely completeness of the orthonormal system. It also proves Parseval's equality by passing to the limit in the Pythagorean projection identity.

For $f(t)=t^2$, oddness gives $b_j=0$, direct integration gives $a_0=2{\pi }^2/3$, and two integrations by parts give $a_j=4(-1{)}^j/j^2$. ThusDini test for Fourier series justifies this pointwise on the interval, or absolute coefficient summability gives its continuous extension after the $L^2$ identification. At zero it yields ${\sum }_{j\ge 1}(-1{)}^{j-1}/j^2={\pi }^2/12$. Parseval givesTherefore ${\sum }_{j\ge 1}{j}^{-4}={\pi }^4/90$.