Fourier series expand periodic functions in an orthogonal trigonometric basis. The cleanest theory lives first in \(L^2\), where orthogonality gives projection.
Trigonometric system
Complex exponentials form the natural orthonormal system on the circle.
Definition. On \([-\pi,\pi]\), set \(e_n(x)=(2\pi)^{-1/2}e^{inx}\), \(n\in\mathbb Z\). The Fourier coefficient of \(f\in L^1\) is \(\widehat f(n)=(2\pi)^{-1/2}\int_{-\pi}^{\pi}f(x)e^{-inx}\,\mathrm{d}x\).
Proposition. The functions \(e_n\) are orthonormal in \(L^2([-\pi,\pi])\).
Proof. \(\langle e_n,e_m\rangle=(2\pi)^{-1}\int_{-\pi}^{\pi}e^{i(n-m)x}\,\mathrm{d}x\). The integral is \(1\) if \(n=m\) and \(0\) otherwise.
Bessel inequality
Orthogonal projection controls the size of coefficients.
Theorem. For \(f\in L^2([-\pi,\pi])\), \[ \sum_{n\in\mathbb Z}|\widehat f(n)|^2\le\|f\|_2^2. \]
Proof. For a finite set \(F\), orthogonality gives \[ \left\|f-\sum_{n\in F}\widehat f(n)e_n\right\|_2^2 =\|f\|_2^2-\sum_{n\in F}|\widehat f(n)|^2\ge0. \] Take an increasing sequence of finite sets exhausting \(\mathbb Z\).
Dirichlet kernels
Partial sums are convolutions with oscillatory kernels.
Definition. The Dirichlet kernel and symmetric partial sum are \[ D_N(x)=\sum_{n=-N}^Ne^{inx} =\frac{\sin((N+\tfrac12)x)}{\sin(x/2)}, \qquad S_Nf(x)=\frac1{2\pi}\int_{-\pi}^{\pi}f(x-y)D_N(y)\,\mathrm{d}y. \]
Remark. The \(L^1\)-norm of \(D_N\) grows like \(\log N\), so pointwise convergence of Fourier partial sums is subtle and cannot be treated like a positive approximate identity.
Fej?r means
Averaged partial sums have a positive kernel and converge better.
Theorem. If \(f\in C(\mathbb T)\), the Fej?r means \(\sigma_N f\) converge uniformly to \(f\). If \(f\in L^p\), \(1\le p\lt \infty\), then \(\sigma_Nf\to f\) in \(L^p\).
Proof. Averaging the partial sums gives \[ \sigma_Nf=\frac1{N+1}\sum_{k=0}^NS_kf =\frac1{2\pi}f*K_N, \] where \[ K_N(x)=\frac1{N+1}\left(\frac{\sin((N+1)x/2)}{\sin(x/2)}\right)^2. \] Thus \(K_N\ge0\), \((2\pi)^{-1}\int K_N=1\), and for every \(\delta>0\), \[ \sup_{\delta\le|x|\le\pi}K_N(x)\le\frac{1}{(N+1)\sin^2(\delta/2)}\longrightarrow0. \] For continuous \(f\), \[ |\sigma_Nf(x)-f(x)| \le\frac1{2\pi}\int K_N(y)|f(x-y)-f(x)|\,\mathrm{d}y. \] Uniform continuity controls \(|y|<\delta\), and concentration controls the complement, uniformly in \(x\). For \(L^p\), Minkowski gives the same estimate with \(\|f(\cdot-y)-f\|_p\), and translation continuity completes the proof.
Completeness and Parseval identity
Theorem. The trigonometric system \(\{e_n:n\in\mathbb Z\}\) is a complete orthonormal basis of \(L^2([-\pi,\pi])\). Consequently, \[ \|f\|_2^2=\sum_{n\in\mathbb Z}|\widehat f(n)|^2, \qquad S_Nf\longrightarrow f\text{ in }L^2. \]
Proof. If \(f\in L^2\) is orthogonal to every \(e_n\), then every Fourier coefficient vanishes. Since \(\mu(\mathbb T)<\infty\), \(f\in L^1\), and every Fej?r mean is identically zero. But Fej?r's theorem gives \(\sigma_Nf\to f\) in \(L^2\), so \(f=0\). Hence the orthogonal complement of the closed span of the trigonometric system is trivial, proving completeness.
The orthogonal projection \(S_Nf\) onto \(\operatorname{span}\{e_n:|n|\le N\}\) therefore converges in \(L^2\) to \(f\). Pythagoras gives \[ \|S_Nf\|_2^2=\sum_{|n|\le N}|\widehat f(n)|^2, \] and passage to the limit yields Parseval.
Theorem (Dirichlet-Jordan pointwise criterion). If a \(2\pi\)-periodic function has bounded variation near \(x\), then \[ S_Nf(x)\longrightarrow\frac{f(x+)+f(x-)}2. \] In particular, convergence is to \(f(x)\) at every continuity point.
Proof outline. Symmetrize the convolution formula to write the error as an oscillatory integral involving \[ \frac{f(x+t)+f(x-t)-f(x+)-f(x-)}{2\sin(t/2)} \sin((N+\tfrac12)t). \] Bounded variation makes the amplitude integrable in the Riemann-Stieltjes sense near zero. The Dirichlet test then makes the oscillatory integral vanish. The midpoint value appears because the symmetric kernel samples the two one-sided limits equally.
Least-squares projection onto frequencies
Fourier series are least-squares approximations in \(L^2\).
Remark. The \(N\)-term Fourier partial sum is the orthogonal projection onto the span of frequencies \(|n|\le N\). This is the same Hilbert-space geometry behind regression projections and conditional expectation.