Fejer completeness: a continuous one-periodic function with all integrals against the characters vanishing is zero

Given: The continuous one-periodic $h$ and the vanishing of all the displayed integrals.

The Fejer kernel is $F_N(t)=\frac{1}{N+1}\mid {\sum }_{j=0}^N{e}_j(t){\mid }^2$ and equals ${\sum }_{\mathrm{\mid }k\mathrm{\mid }\le N}(1-\frac{\mathrm{\mid }k\mathrm{\mid }}{N+1})e_k(t)$; it is nonnegative with ${\int }_0^1{F}_N=1$ (The Fejer kernel is a positive approximate identity, Cesaro and Abel means of a Fourier series).

The Fejer means ${\sigma }_{N}h=h\ast F_N$ converge to $h$ uniformly: ${\sup }_x\mathrm{\mid }{\sigma }_{N}h(x)-h(x)\mathrm{\mid }\to 0$ (Cesaro and Abel means of a Fourier series, Fejer means converge uniformly for continuous periodic functions).

For continuous one-periodic complex $\phi$, affine substitution applied to each real component gives ${\int }_0^1\phi (x-t)dt={\int }_{x-1}^x\phi (u)du={\int }_0^1\phi (u)du$. For the second equality, write $x=m+r$ with $m\in \mathbb{Z}$, $0\le r<1$; translate by $m$, split $[r-1,r]$ at $0$, and translate $[r-1,0]$ by $1$ to $[r,1]$, using periodicity and additivity (Substitution: if $\phi$ is differentiable on $[c,d]$ with ${\phi }^{\mathrm{\prime }}$ integrable and $f$ is continuous on an interval containing $\phi ([c,d])$, then ${\int }_{\phi (c)}^{\phi (d)}f={\int }_c^d(f\circ \phi ){\phi }^{\mathrm{\prime }}$, The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$, For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$).

Character periodicity: $\exp (i\pi )=-1$ by $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$, hence $\exp (2\pi i)=1$ by exponential addition. Iteration gives $\exp (2\pi ik)=1$ for $k\ge 0$; for $k<0$, $\exp (2\pi ik)\exp (-2\pi ik)=\exp (0)=1$ gives the same conclusion. Thus $e_k(t+1)=e_k(t)$. Exponential addition: $e_k(x-t)=e_k(x)e_k(-t)=e_k(x)e_{-k}(t)$ ($\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential, The complex exponential by its power series).

Continuous real components are Riemann integrable on compact intervals. The componentwise convention makes the integral complex-linear: expand complex multiplication into its two real components and apply real linearity and additivity (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$, For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$).