Fourier partial sum ($s_n(f)$)

A Fourier partial sum retains the modes whose integer frequency multipliers satisfy $|k|\le n$ in a complex Fourier series:For a real function of period $2\pi$, this is equivalently $a_0/2+{\sum }_{k=1}^n[a_k\cos (kx)+b_k\sin (kx)]$. It is a trigonometric polynomial, even when the original function is discontinuous. For a square wave, retaining $M$ nonzero odd sine modes gives $S_{2M-1}f$; all even-frequency coefficients are zero. The Gibbs phenomenon illustrates why more retained modes need not reduce the maximum error.

For each integer $m\ge 1$, the set of continuous functions satisfying $|S_N(f,0)|\le m$ for every $N$ is closed in the uniform norm. If it contained a ball of radius $r$, subtracting two members of that ball would bound every partial-sum functional norm by $4m/r$. The Dirichlet kernel harmonic lower bound excludes this, so these sets are nowhere dense. By the Baire category theorem, their complement is a dense $G_{\delta }$ set. Every function in that complement has unbounded Fourier partial sums at zero. Translation gives the same conclusion at any prescribed point. This is a direct Uniform boundedness principle argument on the real Banach space of continuous functions on the circle.

This average of Fourier partial sums reproduces every degree-at-most-$n$ trigonometric polynomial. With ${\sigma }_r=r^{-1}{\sum }_{j=0}^{r-1}{s}_j$,The bound follows because Fejér summation is a uniform-norm contraction. For $n=0$, the term involving ${\sigma }_0$ is omitted.

A continuous complex-valued function can have uniformly bounded Fourier partial sums which fail to converge at one point. Normalize harmonic sine polynomials by their harmonic coefficient sums, modulate them into disjoint high-frequency intervals and choose their degrees so that the complete norms are summable. At zero, complete blocks vanish while their midpoint prefixes have a common nonzero value. The frequency-separated Fourier block series gives a uniform bound on every partial sum as well as two distinct subsequential values.

If trigonometric polynomials have spectra in increasing disjoint positive-frequency intervals, the displayed summability gives a continuous uniform sum. At any cutoff its Fourier partial sum contains complete earlier blocks, at most one active prefix and no later blocks. Differences across an active prefix therefore isolate that block exactly. Fixed-size excursions occurring in arbitrarily late blocks contradict the Cauchy criterion, even when the complete blocks vanish at the point under consideration.

For a compact circle set with normalized Lebesgue measure at most $\exp (-8\pi M/\epsilon )$, where $M\ge 1$ and $0<\epsilon \le 1$, the displayed bounds can be realized by a trigonometric polynomial with spectrum in a positive-frequency interval arbitrarily far from zero. Take the logarithm of a positive-real-part Schwarz integral on the unit disk of a small-mean cutoff, truncate after radial dilation, and modulate its bounded imaginary part. A prefix isolates one analytic half and thus reveals the large real part. The separation between small function norm and large Fourier partial sums drives the Kahane-Katznelson divergence theorem.

Let finitely many arcs have centers ${\theta }_j$, half-lengths $0<{\delta }_j\le 1$ and $D={\sum }_j{\delta }_j$. Each summand has positive real part on the closed disk, and $\Phi (0)=1$. On the $j$th arc its contribution has real part at least $1/(4D)$. Thus the analytic logarithm $A=\log \Phi$ obeys $A(0)=0$, $|\mathrm{Im}A|<\pi /2$, and $\mathrm{Re}A\ge \log (1/(4D))$ throughout the arc union. Approximate $A$ by an analytic polynomial and modulate its bounded imaginary part; a partial Fourier sum isolates one half of the polynomial, producing a large value from a bounded function.

For $D_N(t)=\sin ((N+1/2)t)/\sin (t/2)$, define ${\Lambda }_N=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }|D_N|$. Evenness and $\sin (t/2)\le t/2$ on $(0,\pi ]$ give ${\Lambda }_N\ge (2/\pi ){\int }_0^{(N+1/2)\pi }|\sin v|/vdv$. The first $N$ complete sine periods of length $\pi$ have absolute sine integral two, and on the $j$th such interval $1/v\ge 1/(j\pi )$. This proves the displayed unbounded lower bound. The real functional $f\mapsto S_N(f,0)$ on continuous functions has norm exactly ${\Lambda }_N$: the upper bound is its kernel integral, and the continuous functions $D_N/\sqrt{D_N^2+{\delta }^2}$ approach that bound as $\delta \downarrow 0$ by dominated convergence.