Fourier Transform 6 | Fourier Series in L²

Fourier series in the square-integrable setting are built by projecting a function onto a finite-dimensional space spanned by orthonormal trigonometric functions. The key move is to work in L²: for 2π-periodic functions, the inner product lets cosine and sine (and the constant function) behave like an orthonormal basis, so the “best” trigonometric polynomial approximation is not guesswork—it is the orthogonal projection, which automatically minimizes the L² distance.

The construction starts by recalling the L² framework for 2π-periodic functions: functions are treated as equivalence classes, and the inner product is defined using an integral over one period. With a specific normalization factor (chosen so the constant, cos(kx), and sin(kx) become orthonormal), the family forms an orthonormal system in L². For a fixed n, the span of these functions up to frequency n produces a (2n+1)-dimensional subspace. Linear algebra then supplies the projection: for any f in L², the orthogonal projection onto this subspace is a trigonometric polynomial, denoted F_N f, whose coefficients are computed directly from inner products with the basis functions.

Those coefficients become the Fourier coefficients. The constant term comes from the inner product with the constant function, while each cosine coefficient a_k comes from the inner product with cos(kx), and each sine coefficient b_k comes from the inner product with sin(kx). Once these coefficients are defined for every n, the Fourier series is the sequence of these trigonometric polynomials as n grows. Importantly, the orthogonal-projection interpretation relies on L²: the approximation is “best” specifically with respect to the L² norm (the distance measured by the square-integral). The transcript notes that the coefficient formulas can still be written for functions in L¹ because cosine and sine are bounded, but the geometric meaning as an orthogonal projection becomes unclear there.

To make the machinery concrete, the example uses a 2π-periodic step function: f(x)=1 on (−π,0] and f(x)=0 on (0,π], extended periodically. Because the function is piecewise constant, the Fourier coefficients can be computed by straightforward integrals. Symmetry forces the cosine coefficients to vanish, leaving only a constant term and sine terms. The sine coefficients depend on whether the harmonic index is even or odd: even indices contribute zero, while odd indices contribute values proportional to −1/k. The resulting Fourier series is therefore a constant plus a sum of scaled sine functions at odd frequencies.

Finally, the approximation is visualized by truncating the series after a small number of terms. Each truncation produces the orthogonal projection onto a larger subspace, so the approximation improves in the L² sense—meaning the integral of squared error decreases—even if pointwise behavior is not yet the focus. The step function’s jumps lead to oscillatory partial sums, and increasing the number of terms adds more oscillations that better mimic the original shape in the L² framework.