Fourier Transform 6 | Fourier Series in L² [dark version]

Fourier series in the L² setting are built as orthogonal projections onto a finite-dimensional space spanned by sines and cosines. Once the inner product for 2π-periodic, square-integrable functions is fixed, the cosine and sine system becomes an orthonormal basis (with a specific scaling), turning the “best trigonometric approximation” problem into standard linear algebra: for each n, the projection onto the span of the first 2n+1 basis functions minimizes the L² distance to the target function.

Concretely, the construction starts with the space of 2π-periodic functions and the L² inner product (using complex conjugation for the first factor). With the chosen normalization, the constant function and the cosine/sine functions up to frequency n form an orthonormal system. For a fixed n, the relevant subspace U_n has dimension 2n+1, and the orthogonal projection of a function f onto U_n is a trigonometric polynomial F_n(f). The Fourier coefficients appear directly as inner products with the basis elements: the constant coefficient is computed by integrating f against the constant basis function, cosine coefficients a_k come from integrating f against cos(kx), and sine coefficients b_k come from integrating f against sin(kx). The Fourier series of f is then the sequence of these projections as n grows, and the key interpretation in L² is that each partial sum is the closest trigonometric polynomial of that degree in the L² norm.

The transcript also notes a boundary of the theory: the projection interpretation relies on L². For functions only in L¹, the coefficient formulas still make sense because sine and cosine are bounded, so the integrals defining the coefficients exist. But without the L² inner-product structure, the “orthogonal projection” meaning becomes harder to justify, leaving interpretation for later.

To make the machinery tangible, the discussion works through a piecewise constant 2π-periodic step function defined as f(x)=1 on [-π,0] and f(x)=0 on [0,π], extended periodically. Computing coefficients shows that all cosine terms vanish due to symmetry on the integration interval. The sine coefficients depend on parity: b_k is zero when k is even, and equals -2/(πk) when k is odd. The resulting Fourier series therefore consists of a constant term 1/2 plus only odd sine terms with coefficients -2/(πk). Visualizations then compare partial sums: stopping after the first term yields a constant plus a scaled sine component, and adding more odd frequencies introduces increasingly rapid oscillations that better approximate the step function in the L² (integral-distance) sense.

Overall, the central takeaway is that Fourier series in L² are not just formal expansions: each partial sum is the optimal trigonometric approximation (in L²) obtained by projecting onto the span of sines and cosines up to a fixed frequency.