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The rotation $x\mapsto x+a$ of a circle of circumference $2\pi$ is ergodic exactly when $a/\pi$ is irrational. In the Fourier basis, an invariant function can have a nonzero $k$th coefficient only when $e^{ika}=1$.

The integer multiplication map on the circle is an ergodic transformation for Lebesgue measure. In the Fourier basis, an invariant function has coefficients $c_j=0$ when $K$ does not divide $j$, and $c_j=c_{j/K}$ otherwise. Every nonzero index eventually reduces to the first case, so only the constant coefficient remains.

Fourier analysis describes a function by the oscillations that compose it. A graph records values at successive positions or times; a frequency spectrum records how much of each frequency is present, together with its wave phase. The same complicated shape can become a short list of simple components. For periodic functions, the components form a Fourier series; on the whole real line, the Fourier transform uses a continuous range of frequencies, and the Fourier inversion theorem puts them back together.

For a concrete example, consider the trigonometric polynomialWith $t$ measured in seconds, its three Fourier harmonics have frequencies $1$, $3$, and $5$ cycles per second and amplitudes $1$, $1/2$, and $1/4$. Each cosine has zero wave phase. Adding them creates the sharper peaks in the figure. The frequency spectrum makes the recipe immediately visible; changing an amplitude changes the shape without changing the component's frequency. Fourier orthogonality lets us recover each Fourier coefficient by averaging against the corresponding sine or cosine, because the other Fourier harmonics cancel over a full period of a function.

The deeper reason these components are so useful is that they simplify operations. By Euler's formula, the exponential function $e^{i\omega x}$ packages a cosine and a sine into one Fourier mode. A translation or a derivative changes only its multiplier:Thus Fourier modes are simultaneous eigenfunctions of these operations. In Fourier series, this gives a literal diagonalization in the Fourier basis; on the real line, the Fourier transform gives the continuous counterpart, with each frequency handled separately. The Translation property of the Fourier transform and the Fourier transform of a derivative turn shifting and differentiating a whole function into multiplication of its Fourier transform. A convolution likewise becomes multiplication by the convolution theorem. This is beautiful because one description simplifies several apparently different operations at once.

For example, the heat equation $u_t=\kappa u_{xx}$, with thermal diffusivity $\kappa >0$, evolves a spatial Fourier mode asFine ripples have larger angular frequencies and decay faster: the formula explains why heating smooths a jagged temperature profile. For a general initial profile with sufficient regularity, the Fourier transform gives $\hat{u}(\omega ,t)=e^{-\kappa {\omega }^{2}t}\hat{u}(\omega ,0)$, and Fourier inversion recovers the temperature. Similarly, multiplying selected parts of a frequency spectrum suppresses unwanted components. Claude E. Shannon's Communication in the Presence of Noise (1949), Section III, describes ideal filtering as a projection that retains some frequency coordinates and removes others.^[ref]

The physical motivation is already visible in Joseph Fourier's Théorie analytique de la chaleur (1822). Chapter III constructs heat profiles using sums of sines and cosines; Chapter IX, articles 355–359, passes from Fourier series to integrals as the spatial interval becomes unbounded. These heat problems explain the motivation for the discrete and continuous descriptions. Their modern formulations specify which functions admit reconstruction and in what sense the expansions converge.^[ref][ref]

For irrational $\alpha$, the irrational skew shift is the skew product $T_{\alpha }(x,y)=(x+\alpha ,y+x)$ on the torus $(\mathbb{R}/\mathbb{Z}{)}^2$. It preserves normalized Lebesgue measure and has iterates $T_{\alpha }^n(x,y)=(x+n\alpha ,y+nx+\alpha n(n-1)/2)$ modulo one. Its Koopman operator sends the Fourier basis character $e_{r,s}=e^{2\pi i(rx+sy)}$ to $e^{2\pi ir\alpha }{e}_{r+s,s}$. The resulting infinite chains of Fourier coefficients show that it is an ergodic transformation. Moreover, uniform equidistribution of an irrational skew shift proves unique ergodicity.

The descendant of a face with cyclic vertices $p_0,\dots ,p_{n-1}$ has vertices $p_i^{\prime }=(p_i+p_{i+1})/2$. Their arithmetic mean is unchanged. On the cyclic Fourier basis, a mode of frequency $j$ is multiplied by $(1+e^{2\pi ij/n})/2$. Every nonconstant mode has modulus less than one, so the descendant face shrinks to the original vertex centroid. This puts that centroid on the subdivision limit, without requiring the original face to be planar.

On the circle group $\mathbb{T}$, normalized Haar measure is the pushforward measure of uniform angle on $[0,2\pi )$. For a continuous function $f:\mathbb{T}\to \mathbb{C}$, defineThe factor $1/(2\pi )$ makes the integral of the constant function $1$ equal to $1$. For $g=e^{i\varphi }$, a change of angle givesThe last equality uses the $2\pi$-periodic function $u\mapsto f(e^{iu})$. Since the circle group is an abelian group, this also verifies right invariance. An arc of angular length $L$ has normalized Haar measure $L/(2\pi )$. In particular, the arc $A=\{e^{i\theta }:\pi /6\le \theta \le \pi /2\}$ and its quarter-turn $iA$ both have measure $(\pi /2-\pi /6)/(2\pi )=1/6$.

The continuous unitary characters ${\chi }_n(e^{i\theta })=e^{in\theta }$, indexed by integers, give a fully computed family of integrals:Here $n$ must be an integer, so $z^n$ is well defined and $e^{2\pi in}=1$. Every nonconstant mode cancels exactly over a full turn. For example, the integral of $f(e^{i\theta })=3+2\cos \theta -\sin (2\theta )$ is $3$, and the integral of ${\cos }^2\theta =(1+\cos (2\theta ))/2$ is $1/2$. More generally,This is the orthogonality of complex exponentials underlying the Fourier basis and Fourier series: invariant averaging separates distinct frequencies and retains the constant component.

This is mid-edge subdivision, rather than a vertex-retaining refinement. Use the usual closed two-manifold mesh setting: each edge has two incident facets and each vertex has a cyclic fan. For boundary meshes, a boundary refinement rule has to be specified separately. The relevant generic-position condition excludes collapsed local tangent directions and singular regular patches.

Topology and the manifold property. Regard the abstract mesh as a cell decomposition of a surface. Join the edge midpoints in each facet. The inner polygon is the new face associated with that facet; the corner regions around an old vertex assemble into the face associated with that vertex. At an old edge midpoint, the four adjacent corner connections form a cyclic link. Thus every new interior vertex has valence four, every new edge has two incident faces, and the refined cell complex is homeomorphic to the original surface. If the original counts are $V,E,F$, the new counts are $E,2E,V+F$, preserving $V-E+F$. The cyclic-link argument, rather than Euler characteristic alone, verifies the local manifold condition. Convergence and local regularity, justified below, transfer this abstract surface structure to the limit.

Convex hull. Each newly created point is an average of two previous points. By induction it is a convex combination of original vertices. The same is true of points of any triangulation of the refined facets, and taking limits keeps them in the closed convex hull. Hence the entire limit stays in that hull, not necessarily inside the original nonconvex enclosed solid.

Facet centroids. Follow the descendant inner face of an original $n$-sided facet, with cyclic vertices $p_i$. Its vertices satisfy $p_i^{(r+1)}=(p_i^{(r)}+p_{i+1}^{(r)})/2$. Summing proves mid-edge facet centroid invariance:For a cyclic Fourier mode $e^{2\pi iji/n}$ the multiplier is $(1+e^{2\pi ij/n})/2$, whose modulus is $|\cos (\pi j/n)|<1$ for $j\ne 0$. Every mean-zero mode therefore decays, so all descendant face vertices converge to $\overline{p}$. Consequently the original vertex centroid belongs to the limit surface. No planarity assumption is needed.

Quadratic triangular pieces. On a regular quadrilateral mesh, apply two midpoint steps. Each new point combines three old neighboring controls with weights $1/2,1/4,1/4$. For example, around a face with cyclic corners $p_{00},p_{10},p_{11},p_{01}$, the four new points areThis is the binary refinement mask obtained by expandingwith the translation chosen to place the refined grid at quarter offsets. The limit is therefore represented by shifts of the quadratic four-direction box spline with directions $e_1,e_2,e_1+e_2,e_1-e_2$. Projection of a four-cube onto the plane has piecewise total degree $4-2=2$, and its knot grid is triangular. Coordinatewise linear combinations preserve that degree, giving parametric quadratic triangle patches. Around a nonregular center there are infinitely many shrinking rings of such patches; the isolated center is their limit, not generally part of a finite polynomial patch representation.

Only finitely many initially possible smoothness exceptions. Every vertex has valence four after the first step. Only the descendants of original facets and the faces created around original vertices can remain nonquadrilateral. All subsequently created faces around four-valent vertices are quadrilaterals. Thus every point except the limiting centers of those persistent nonquadrilateral faces eventually lies in a regular patch neighborhood. The number of possible exceptional centers is at most $V+F$. For the regular box spline, any three distinct directions still span the plane, so three directions must be removed to make the remainder nonspanning. The box spline continuity criterion gives $C^{3-2}=C^1$ across its triangle boundaries. This proves $C^1$ continuity away from at most those finitely many centers.

Removing the apparent exceptions. Center a finite neighborhood of an $n$-sided exceptional face, and analyze a full two-step subdivision matrix so the mesh orientation is restored. Cyclic Fourier basis modes diagonalize its sector symmetry. The face mode has eigenvalueThe local ring blocks also contribute eigenvalues $1/4$ and zero. For $n>3$, the constant mode is the single eigenvalue one, and the two first-frequency tangent modes dominate all nonconstant remaining modes. Construct their characteristic map of a subdivision surface; its quadratic derivative pieces must have a nonzero consistently oriented Jacobian, and its fundamental sector must be injective. These checks, not the eigenvalue ordering alone, give a regular chart and a continuously limiting tangent plane for generic data.

The case $n=3$ needs a separate Jordan analysis: $\lambda =1/4$ is eightfold, so the usual semisimple double-eigenvalue test does not apply. Its first-frequency Jordan chains supply the leading tangent behavior, while the other contributions become relatively negligible. Regularity and injectivity of that generalized characteristic map complete the argument. These concrete spectral and characteristic-map checks are established for this scheme in Peters and Reif, Sections 3 and Appendix. Thus the generic limit is geometrically $C^1$ also at the exceptional centers; compatible local charts need not preserve the original uniform refinement parameterization.

The manifold statement concerns the abstract limit surface and its regular local parametrizations. It does not, from combinatorics and generic tangent data alone, prove global injectivity of the image in three-dimensional space. A particular model may require an additional global self-intersection test before claiming that its image is an embedded manifold.

The string sigma model in a spacetime metric is the Polyakov actionTake conformal gauge $h_{ab}=\mathrm{diag}(-1,1)$, and choose spatial period $2\pi$ for the closed string. Substitution of the plane-wave metric givesThe stipulated light-cone gauge in string theory $X^{+}=\tau$ sets ${\dot{X}}^{+}=1$ and $X^{\prime +}=0$. Its longitudinal term $-T\int {\dot{X}}^{-}$ is a total time derivative. The transverse string sigma model in an isotropic plane wave is thereforeThe resulting Euler-Lagrange equations are the massive wave equationsThese conventions fix the mass term directly from $X^{+}=\tau$. More generally $X^{+}=\kappa \tau$ would replace $m$ by $|m\kappa |$, with $p^{+}=2\pi T\kappa$ for this spatial period.

Use the real orthonormal Fourier basisFor the plane-wave string mode frequencies, let ${\omega }_0=|m|$ and ${\omega }_{n,c}={\omega }_{n,s}=\sqrt{n^2+m^2}$. The complete real mode expansion for nonzero frequencies isEvery term obeys the wave equation and is periodic. Orthonormality and completeness let its coefficients match arbitrary appropriate initial position and velocity data. For $m=0$, replace the zero-mode bracket by $q_0^I(0)+\tau p_0^I/T$; the zero mode becomes a free particle rather than an oscillator.

The canonical momentum density is ${\Pi }_I=T{\dot{X}}^I$. The transverse Hamiltonian isWrite $X^I={\sum }_a{q}_a^I{f}_a$ and $p_a^I=T{\dot{q}}_a^I$. Since $\int f_a{f}_b={\delta }_{ab}$ and $\int f_a^{\prime }{f}_b^{\prime }=n_a^2{\delta }_{ab}$, substitution gives the plane-wave string mode HamiltonianThis is a collection of independent harmonic oscillators. The last equality follows by substituting the sine-and-cosine time dependence; each oscillator energy is constant.

For canonical quantization of each nonzero-frequency mode, putThen $H_{\perp }={\sum }_{I,a}{\omega }_a(a_a^{I†}{a}_a^I+1/2)$. Opposite travelling-wave oscillator combinations give the equivalent formFor $m\ne 0$, the formal zero-point energy is $E_0=\frac{24}{2}[|m|+2{\sum }_{n\ge 1}\sqrt{n^2+m^2}]$. At $m=0$, replace the zero-mode occupation term by ${\sum }_I(p_0^I{)}^2/(2T)$. The zero-point expression is divergent and needs a regulator or a specified normal ordering convention; no mass-independent intercept has been assumed.

The remaining Virasoro constraints determine the longitudinal coordinate:Periodicity of $X^{-}$ imposes ${\int }_0^{2\pi }{\Pi }_I{X}^{\prime I}d\sigma =0$, the closed-string level matching condition. Labelling the travelling oscillators by their opposite worldsheet momenta gives ${\sum }_{n\ge 1,I}n(N_n^I-{\tilde{N}}_n^I)=0$. The light-cone energy $p^{-}=-p_{+}$ equals $T\int ({\dot{X}}^{-}+m^2{X}^I{X}^I)d\sigma =H_{\perp }$, consistent with the positive mass term in the Hamiltonian.

This is the classical reduction in the specified background. As a full quantum background with constant dilaton and no Kalb–Ramond field, this metric has $R_{++}=24m^2$ and fails the lowest-order sigma-model beta function equation when $m\ne 0$; additional background fields would be required for quantum Weyl invariance.

Inside the infinite square well, the time-independent Schrödinger equation is $-{\hslash }^2{u}^{\prime \prime }/(2m)=Eu$ with $u(-a)=u(a)=0$. For $E<0$ the exponential solutions cannot satisfy both conditions except trivially, and for $E=0$ neither can a linear function. For $E>0$ put $k=\sqrt{2mE}/\hslash$. The first boundary condition selects $u=C\sin (k(x+a))$, and the second gives $\sin (2ak)=0$. Thuson $[-a,a]$, with $u_n=0$ outside. Their squared integrals are one, and the sine orthogonality identity gives an orthonormal set. The sine Fourier basis on the interval is complete: odd extension to an interval of length $4a$ reduces this to completeness of the Fourier series basis. Hence these are a complete set of normalized energy eigenstates.

Choosing the harmless sign of the second energy eigenstate so that $v_1=a^{-1/2}\cos (\pi x/(2a))$ and $v_2=a^{-1/2}\sin (\pi x/a)$, the specified wave function is $v_1/\sqrt{5}+2v_2/\sqrt{5}$. The Born rule therefore giveswith no other possible measured energies. The two energy eigenvalues are $E_1={\hslash }^2{\pi }^2/(8m{a}^2)$ and $E_2=4E_1$.

After the first measurement gives the ground state, the old state is $v_1$. Doubling the width puts the walls at $\pm 2a$, and the new normalized ground state is $w_1=(2a{)}^{-1/2}\cos (\pi x/(4a))$. The stipulated unchanged state is zero outside the original interval, so the probability amplitude isSquaring givesThe ground-state overlap after sudden expansion of a square well stays unchanged during subsequent evolution in the fixed new well, because the energy-basis coefficients acquire only phases. The answer is conditional on the already observed old ground state; its earlier probability $1/5$ is not another factor in this conditional question.

Identify the circle group with $[0,1)$ modulo its endpoints, and let $\lambda$ be normalized Lebesgue measure. We first check invariance. On the $r$th inverse branch, $x=(y+r)/m$ for $0\le r<m$. For an integrable function $g$, substitution givesIn particular, testing indicator functions shows that the integer multiplication map on the circle preserves $\lambda$.

To prove ergodicity, let $B$ be a measurable set with $\lambda (E_m^{-1}B△B)=0$ and put $h=1_B$. Then $h(E_{m}x)=h(x)$ almost everywhere. Its Fourier coefficients areUsing invariance of $h$ and substituting separately on the same $m$ branches givesIn the last line, the finite geometric series of $m$th roots of unity is zero unless $m$ divides $k$, when it equals $m$. For any nonzero integer $k$, divide repeatedly by $m$ until the resulting integer is no longer divisible by $m$. The recurrence then implies $c_k=0$.

Since the Fourier basis is a complete orthonormal basis of $L^2([0,1),\lambda )$, the absence of every nonconstant Fourier coefficient implies $h=c_0=\lambda (B)$ almost everywhere. An indicator function takes only the values zero and one, so $\lambda (B)$ must be zero or one. This proves

The discrete Bochner–Herglotz theorem says that a sequence $r:\mathbb{Z}\to \mathbb{C}$ is a positive-definite function, meaningif and only if there is a unique finite positive Borel measure $\sigma$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ such thatThe mass is $\sigma (\mathbb{T})=r(0)$. The necessity of positive definiteness follows by integrating $|{\sum }_j{c}_j{e}^{2\pi ijt}{|}^2$.

For sufficiency, positive definiteness makes the relevant Gram matrices Hermitian positive semidefinite, so $r(-n)=\overline{r(n)}$ and $|r(n)|\le r(0)$. DefineTaking $c_j=e^{-2\pi ijt}$ in the positive-definiteness condition gives $p_N(t)\ge 0$, while ${\int }_{\mathbb{T}}{p}_N(t)dt=r(0)$. Moreover,Thus the integrals against every trigonometric polynomial converge. These polynomials are uniformly dense in continuous functions on the circle: smooth periodic approximation followed by Fourier inversion for smooth functions gives this density. The uniform mass bound $r(0)$ then makes the integrals against every continuous $\phi$ converge as well. Their limit is a positive linear functional $L$ with $|L(\phi )|\le r(0)\Vert \phi {\Vert }_{\infty }$. The standard Riesz-Markov-Kakutani representation theorem gives a finite positive Borel measure $\sigma$ with $L(\phi )=\int \phi d\sigma$. The preceding coefficient computation proves its required Fourier coefficients. Finally, two such measures agree on all trigonometric polynomials, hence on all continuous functions by uniform density, and therefore are identical. This proves the Bochner–Herglotz theorem, including existence and uniqueness; if $r(0)=0$, the estimate $|r(n)|\le r(0)$ gives the zero measure directly.

Now let $U$ be the Koopman operator $Uf=f\circ T$ on the complex L2 space of the probability measure-preserving system. We use $⟨f,g⟩=\int f\overline{g}d\mu$, linear in the first entry. Since $U$ is an isometry, the sequence $r_f(n)=⟨U^{n}f,f⟩$ for $n\ge 0$, extended by $r_f(-n)=\overline{r_f(n)}$, is positive-definite: its matrix is the Gram matrix of $f,Uf,\dots$. The Bochner–Herglotz theorem gives its spectral measure of a Koopman observable ${\sigma }_f$.

We first prove exactly how an atom of a measure detects an eigenfunction. Fix $\zeta =e^{2\pi i{t}_0}$ and setThe mean ergodic theorem, applied to the isometry $V={\zeta }^{-1}U$, says these averages converge in L2 space to the orthogonal projection $P_{\zeta }f$ onto $\ker (U-\zeta I)$. Its usual Hilbert-space proof works without invertibility: the orthogonal complement of $\overline{\mathrm{ran}(I-V)}$ is $\ker (I-V^{\ast })=\ker (I-V)$, and the averages of $(I-V)h$ telescope to $(h-V^{N}h)/N\to 0$. The equality of the two kernels follows from $V$ being an isometry, by expanding $\Vert Vh-h{\Vert }^2$ when $V^{\ast }h=h$.

Expanding the averaged norm through the spectral measure of a Koopman observable givesThe integrand is at most one and converges to $1_{\{t_0\}}$. By the dominated convergence theorem,Thus a nonzero atom corresponds to a nonzero Koopman operator eigenfunction, with eigenvalue $\zeta$, in the cyclic spectral data of $f$.

We also need the Wiener Fourier coefficient lemma, which here has a short proof. For any finite positive circle measure $\sigma$ with coefficients $r(n)$,The bracket is bounded in absolute value by one and tends to the indicator of the diagonal $t=s$. Another application of the dominated convergence theorem givesIn particular the limit is zero for an atomless measure.

Suppose there are no nonconstant Koopman operator eigenfunctions. For every $f$ in the mean-zero L2 space, all its projections $P_{\zeta }f$ vanish: a nonzero projection would be a nonconstant eigenfunction, since it has mean zero. Hence ${\sigma }_f$ is atomless, and the Wiener Fourier coefficient lemma givesThe Cauchy-Schwarz inequality also gives vanishing averaged absolute autocorrelations. For two mean-zero observables $f,g$, the sesquilinear polarization identitythen gives vanishing averaged absolute cross-correlations. Taking $f=1_A-\mu (A)$ and $g=1_B-\mu (B)$ proveswhich is weak mixing.

Conversely, this set formulation of weak mixing extends to all pairs of mean-zero L2 space observables by finite-simple-function approximation and the uniform estimate $|⟨U^{n}f,g⟩|\le \Vert f{\Vert }_2\Vert g{\Vert }_2$. If $Uh=\zeta h$ and $h\ne 0$, the isometry property gives $|\zeta |=1$. If $\zeta \ne 1$, invariance of the integral gives $\int h=0$. If $\zeta =1$ and $h$ is nonconstant, replace it by $h-\int h$, a nonzero mean-zero eigenfunction. But in either casefor every $n$, contradicting weak mixing. We have proved weak mixing is equivalent to absence of nonconstant eigenfunctions, including for noninvertible transformations.

Finally, the doubling map $Tx=2x(\mathrm{mod}1)$ preserves normalized Lebesgue measure, as substitution on its two inverse branches shows. Let $f\in L^2(\mathbb{T})$ be an eigenfunction, $f(2x)=\zeta f(x)$, with Fourier coefficients $c_k$. The Fourier basis calculation givesSince $\zeta \ne 0$, all odd coefficients vanish. Every nonzero integer is $2^{r}m$ with $m$ odd, and repeated use of $\zeta c_{2^{r}m}=c_{2^{r-1}m}$ makes all nonzero coefficients vanish. Completeness of the Fourier basis then says $f$ is constant. Therefore the doubling map has no nonconstant eigenfunctions, and

The musical isomorphisms are the pointwise maps induced by the Riemannian metric:In local coordinates they are $(X^{♭}{)}_i=g_{ij}{X}^j$ and $({\alpha }^{♯}{)}^i=g^{ij}{\alpha }_j$. The inverse metric makes these inverse bundle maps, hence inverse maps between smooth vector fields and smooth differential one-forms.

For a covariant two-tensor $\beta$, its metric trace is ${\mathrm{tr}}_g\beta =g^{ij}{\beta }_{ij}={\sum }_i\beta (e_i,e_i)$ in a Riemannian orthonormal frame. The contraction is independent of the chosen frame. There is a terminology distinction here: if $\beta$ is literally an alternating differential two-form, this trace is zero. The object needed below is the covariant two-tensor $D(X^{♭})$, generally not alternating. Interpreting that tensor as an alternating form would make the stated divergence construction impossible.

Let $D$ be the Levi-Civita connection. The Riemannian gradient, divergence and positive Laplace-Beltrami operator areThus $g({\mathrm{grad}}_{g}f,Y)=Yf$, and in coordinatesThis choice makes the Laplacian eigenvalues nonnegative on a closed Riemannian manifold. The alternative convention $+\mathrm{div}\mathrm{grad}$ reverses all eigenvalue signs, but none of the comparisons below.

For the Riemannian submersion, let $F=f\circ \pi$ and choose orthonormal horizontal lifts $E_i$ of a base frame and an orthonormal vertical frame $V_{\alpha }$. The horizontal part of $D_{E_i}{E}_i$ projects to the base Levi-Civita connection; this follows directly from its Koszul formula applied to basic horizontal fields, because their inner products and projected brackets are pulled back from the base. Therefore the horizontal trace of the Riemannian Hessian of $F$ is the pulled-back base Hessian trace. Since $V_{\alpha }F=0$, its vertical terms areWriting $H={\sum }_{\alpha }(D_{V_{\alpha }}{V}_{\alpha }{)}^{\mathrm{hor}}$ for the unnormalised fibre mean curvature vector givesThe totally geodesic submanifold condition on the fibres makes $H=0$. Consequently eigenfunction pullback by a Riemannian submersion givesHere the Riemannian submersion is onto, so the map is injective. Its image is exactly the members of the upstairs eigenspace constant on each whole fibre: such a function descends smoothly by local sections of the submersion, and the displayed identity forces the descended function to have the same eigenvalue. In particular, whenever these eigenspaces are finite dimensional, the downstairs multiplicity is no larger than the upstairs multiplicity. The upstairs eigenspace may also contain functions varying along fibres.

A flat torus is ${\mathbb{R}}^d/\Lambda$ with the metric induced by the Euclidean metric, where $\Lambda$ is a full-rank Euclidean lattice. Its dual lattice is ${\Lambda }^{\ast }=\{w:⟨w,v⟩\in \mathbb{Z}\text{ for all }v\in \Lambda \}$. The characters $e_w(x)=e^{2\pi i⟨w,x⟩}$ descend to the quotient and form an orthogonal Fourier basis of $L^2$ on it. Direct differentiation gives the spectrum of a flat torus:with multiplicity equal to the number of dual lattice vectors of each length. In particular zero has multiplicity one.

Let the given four-dimensional isospectral manifolds be ${\mathbb{R}}^4/{\Lambda }_1$ and ${\mathbb{R}}^4/{\Lambda }_2$. For $d=4+k$, take their products with the same $k$ circles of length $\epsilon$, equivalently the lattices ${\Lambda }_i\oplus \epsilon {\mathbb{Z}}^k$. The dual lattices are ${\Lambda }_i^{\ast }\oplus {\epsilon }^{-1}{\mathbb{Z}}^k$, so their Laplacian eigenvalues areThe given equality of base spectra with multiplicities makes these product spectra identical.

Choose $\epsilon$ smaller than the shortest nonzero vector length in either original Euclidean lattice. By short-vector cancellation for flat tori, the shortest nonzero vectors in the product lattice are exactly $\pm \epsilon e_j$ in its new factor. To spell out the nonisometry argument, any isometry of the product flat tori lifts to an isometry of their universal Euclidean covers. This is affine, and its orthogonal linear part maps one product lattice onto the other. It must map the span of their shortest vectors, namely the new $k$-dimensional factor, onto itself; orthogonality then preserves the original four-dimensional factor and maps ${\Lambda }_1$ onto ${\Lambda }_2$. That would give an isometry of the original flat tori, contrary to the assumption. Isospectral nonisometric flat tori therefore exist in every dimension $d\ge 4$.

An orthonormal multiresolution analysis consists of closed subspaces $V_j\subset L_2(\mathbb{R})$, $j\in \mathbb{Z}$, with $V_j\subset V_{j+1}$, $\overline{{\bigcup }_j{V}_j}=L_2(\mathbb{R})$, ${\bigcap }_j{V}_j=\{0\}$, and the dilation equivalence $h\in V_j⟺h(2\cdot )\in V_{j+1}$. A scaling function $\varphi$ has integer translates forming an orthonormal basis of $V_0$. Henceis an orthonormal basis of $V_j$. Set $W_j=V_{j+1}\ominus V_j$, the orthogonal complement of $V_j$ inside $V_{j+1}$. A function $\psi$ whose integer translates form an orthonormal basis of $W_0$ produces a complete orthonormal wavelet system $\{2^{j/2}\psi (2^j\cdot -k):j,k\in \mathbb{Z}\}$: different $W_j$ are orthogonal, their sum contains every difference $V_b\ominus V_a$, and density together with the zero intersection exhausts $L_2(\mathbb{R})$.

Here is the filter construction that guarantees such a wavelet. Nesting puts $\varphi$ in $V_1$, so expansion in its orthonormal basis gives the scaling refinement equation $\varphi ={\sum }_n{a}_n\varphi (2\cdot -n)$, with $a\in {\ell }_2$. Define the MRA low-pass filter and its complementary quadrature mirror filter bySplitting the orthonormal translates and Fourier periodization identity at $2t$, as in part (b), gives $|m(t){|}^2+|m(t+\pi ){|}^2=1$ almost everywhere. Consequently the rows ofhave length one and are orthogonal; the matrix is unitary. Put $b_n=(-1{)}^n\overline{a_{1-n}}$ and $\psi ={\sum }_n{b}_n\varphi (2\cdot -n)$, so $q(t)=\frac{1}{2}{\sum }_n{b}_n{e}^{-int}$.

To see completeness as well as orthogonality, identify $V_1$ with its fine-scale coefficient sequences and take their Fourier series. Group the frequencies $t$ and $t+\pi$ over a half-period. The two rows of the displayed matrix are the normalized coarse and complementary channels; their unitarity is exactly an onto isometry from these two channels to the fine-scale coefficient space. Translates of $\varphi$ and $\psi$ correspond to the Fourier basis in the respective channels. They therefore jointly form an orthonormal basis of $V_1$. The orthogonal complement channel gives the wavelet completion of a multiresolution filter, namely the integer translates of $\psi$ span $W_0$. This proves the asserted relation with an orthonormal wavelet.

A continuous map $F$ of a compact metric space is uniquely ergodic if it has exactly one invariant measure that is a Borel probability measure. The uniqueness requirement ranges over all invariant Borel probability measures.

For an irrational rotation of the circle, let $R_{\alpha }x=x+\alpha$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$, and let $\nu$ be any invariant Borel probability measure. For each integer $r$, set ${\chi }_r(x)=e^{2\pi irx}$. Invariance givesFor $r\ne 0$, irrationality forces $e^{2\pi ir\alpha }\ne 1$, so the corresponding Fourier coefficient is zero. For $r=0$ it is one. These are exactly the Fourier coefficients of normalized Lebesgue measure $m$. By the Stone-Weierstrass theorem, trigonometric polynomials are uniformly dense in the continuous functions on the circle group; hence $\nu$ and $m$ integrate every continuous function equally and are the same Borel probability measure. Since $m$ is invariant, the irrational rotation of the circle is uniquely ergodic, with unique measure $m$.

For the irrational skew shift on the two-dimensional torus, writeThe map is invertible, with $T^{-1}(x,y)=(x-\alpha ,y-x+\alpha )$ modulo one, and it preserves $m_2$ as allowed in the question. We first prove the ergodic transformation property by the invariant-function characterization of ergodicity.

Let $F\in L^2(m_2)$ satisfy $F\circ T=F$. Its expansion in the Fourier basis is $F={\sum }_{r,s}{c}_{r,s}{e}_{r,s}$ in $L^2$. Direct calculation of the Koopman operator givesUniqueness of the Fourier coefficients therefore impliesFor $s\ne 0$, the magnitudes of the Fourier coefficients along all distinct indices $(r+js,s)$, $j\in \mathbb{Z}$, are equal. By the Bessel inequality they are square summable, so every such coefficient must be zero. When $s=0$, the relation becomes $c_{r,0}=e^{2\pi ir\alpha }{c}_{r,0}$, which forces $c_{r,0}=0$ for $r\ne 0$. Only $c_{0,0}$ remains. Thus every invariant $L^2$ function is constant, and