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Use the Fourier basisof $L^2([-\pi ,\pi {]}_{\mathrm{per}})$. For the rotation $F(x)=x+a$,If $a/\pi$ is irrational, $e^{ika}=1$ implies $k=0$. Hence every fixed $L^2$ function has only its constant Fourier coefficient, and part (i) proves ergodicity.

If $a/\pi =p/q$ is rational, thenis a nonconstant fixed function because $e^{i2qa}=e^{i2\pi p}=1$. Part (i) now shows that the system is not ergodic. Therefore the ergodicity criterion for a circle rotation is

For a real $2L$-periodic square-integrable function with the convention $f\sim a_0/2+{\sum }_{n\ge 1}(a_n\cos (n\pi x/L)+b_n\sin (n\pi x/L))$, Parseval's identity isIt states that the squared L2 norm equals the sum of the squared components in the orthogonal Fourier basis, with the normalization factors shown. In the present even-function case every $b_n$ vanishes.

A standard complete orthogonal family is the real Fourier basisCompleteness gives expansion in the L2 space for a square-integrable function. Mere integrability alone does not guarantee pointwise convergence of a Fourier series; the explicit function in part (b) is continuous and piecewise smooth, so this issue does not obstruct its expansion.

Let $S_N$ be the Fourier partial sum through frequency $N$. By orthogonality and the coefficient formulas,Expanding the nonnegative squared L2 norm of the error gives the exact finite identityFor $f$ in the L2 space, completeness of the Fourier basis makes the left-hand side tend to zero. Dividing the resulting equality by $2L$ proves Parseval's identity:This argument uses convergence in L2 rather than an unqualified exchange of an infinite sum with the integral of its square. The function in part (c) is continuous and piecewise polynomial, so these hypotheses hold.

Expand every transverse closed string coordinate in a real orthonormal Fourier basis of the spatial circle. The canonical momenta are $p_{Ia}=T{\dot{q}}_{Ia}$, and spatial orthogonality reduces the field Hamiltonian to the displayed independent harmonic oscillators. Canonical quantization gives ${\omega }_a(a_{Ia}^{†}{a}_{Ia}+1/2)$ for each nonzero-frequency mode; a zero-frequency mode retains the free particle term $p_{Ia}^2/(2T)$. The formal zero-point energy requires a regulator; it is not automatically the massless flat-background intercept. Opposite travelling-wave oscillator combinations diagonalize the translation generator and satisfy closed-string level matching.

For an irrational skew shift, the averages of every continuous function $g$ converge uniformly in the starting point to $\int gd{m}_2$, with $m_2$ normalized Lebesgue measure. Nonconstant Fourier basis characters have either linear or quadratic phases. Linear phases are bounded geometric series; for quadratic phases the Van der Corput inequality for finite scalar sequences reduces to linear correlations of irrational frequency, uniformly in the starting point. Approximation by trigonometric polynomials proves the assertion and identifies every invariant Borel probability measure as $m_2$.