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A self-adjoint operator $T$ satisfies $⟨Tx,y⟩=⟨x,Ty⟩$. If $Tx=\lambda x$, then $\lambda ⟨x,x⟩=⟨Tx,x⟩=⟨x,Tx⟩=\overline{\lambda }⟨x,x⟩$, so $\lambda \in \mathbb{R}$. An eigenvector exists over $\mathbb{C}$; its orthogonal complement is $T$-invariant. Induction on dimension produces an orthonormal basis of eigenvectors, proving the finite-dimensional spectral theorem.

A self-adjoint $T$ is positive definite exactly when every eigenvalue is positive, since $⟨Tx,x⟩={\sum }_i{\lambda }_i|x_i{|}^2$ in an eigenbasis. The quadratic forms of two positive-definite operators add, so their sum is positive definite.

The same eigenbasis shows that $\mathrm{spec}T\subset [a,b]$ exactly when $T-\lambda I$ is positive definite for every $\lambda <a$ and negative definite for every $\lambda >b$. Finally, if $v$ is a unit eigenvector of $\alpha +\beta$ with eigenvalue $\gamma$, the Rayleigh quotient bounds giveThus every eigenvalue of $\alpha +\beta$ lies in $[a+c,b+d]$.

First construct the Dirichlet eigensystem. By part (i) and the Lax-Milgram theorem, the inverse $S=A_D^{-1}$ exists from $L^2$ into $H_0^1$. Elliptic regularity gives $Sf\in H^2\cap H_0^1$, and the Rellich-Kondrachov compactness theorem makes $S$ a compact operator on $L^2$. Symmetry givesso $S$ is a self-adjoint operator. It is injective, and $(Sf,f)=B[Sf,Sf]>0$ for $f\ne 0$. The spectral theorem for compact Hermitian operators supplies an orthonormal basis $(w_m)$ with $S{w}_m={\mu }_m{w}_m$, ${\mu }_m>0$, and ${\mu }_m\to 0$. Put ${\lambda }_m={\mu }_m^{-1}$. ThenRepeated elliptic regularity makes each $w_m$ smooth up to the boundary.

As with part (ii), the stated characterization omits boundary compatibility. At $k=0$, its right-hand side is finite for every $u\in L^2$ by Parseval identity, whereas its left-hand side would require $u\in H_0^1$. At higher orders, the preceding polynomial example is also decisive for the Dirichlet eigensystem: on $(0,\pi )$,Although this function belongs to every ordinary $H^k$, the weighted sum diverges at $k=3$.

The correct spectral characterization of elliptic Dirichlet domains iswith $X_k$ defined in part (ii). For finite eigenfunction sums the squared norm $((u,u){)}_k$ is exactly the weighted sum: even orders follow from $L^l{w}_m={\lambda }_m^l{w}_m$, and odd orders also use $B[w_m,w_j]={\lambda }_m{\delta }_{mj}$. These sums are dense in $X_1$: if $v$ is $B$-orthogonal to every $w_m$, then $0=B[v,w_m]={\lambda }_m(v,w_m)$, so $v=0$. They are dense in $X_0$ by construction, and the isomorphisms $A_D:X_{k+2}\to X_k$ propagate density to every $X_k$. Completion in the equivalent norms from part (ii) proves both directions of the corrected equivalence. In particular, the printed characterization is correct for $k=1,2$.

Choose an orthonormal basis $(e_n)$ of the Dirichlet energy space and independent variables $({\xi }_n)$ with the standard normal distribution. The zero-boundary Gaussian free field is the formal seriesIts precise energy-indexed interpretation is the isonormal Gaussian processFor every fixed $f$, this sum converges in $L^2$ of the probability space by Parseval identity, and it is a centered Gaussian random variable. For $f,g\in H_0^1(D)$,This covariance characterizes its law independently of the chosen basis. The series does not define an $H_0^1$-valued random element: ${\sum }_n{\xi }_n^2=\infty$ almost surely. It is realized as a random distribution, and its pairing with ordinary test functions is constructed in part (b).

Let $Z_{ir}=1_{\{\text{observation }i\text{ belongs to rat }r\}}$. After integrating out the random intercepts,The fixed-effect design matrix has four independent columns, so the given $A$ has dimensions $176\times 172$, with $A^{T}A=I_{172}$ and $A^{T}X=0$. Consequently the error contrasts $U=A^{T}Y$ have a multivariate normal distribution with mean zero and covariance matrix $W=A^{T}VA$.

The restricted likelihood from orthogonal error contrasts is their density. Equivalently, the estimates maximizeThis is restricted maximum likelihood, not maximization over individual rat effects. The unknown $\beta$ disappears because $A^{T}X=0$. Replacing $A$ by $AQ$ for an orthogonal $Q$ rotates $U$ and conjugates $W$, preserving the determinant and quadratic form; hence the objective is independent of the chosen orthonormal basis. The software's integrated REML log-likelihood can differ by an additive constant depending only on $X$, which has no effect on these variance estimates.

Use the gnomonic projection associated with an undistorted focal plane. Write $\Delta =\alpha -A$ and rotate the celestial Cartesian coordinate system so that the pointing meridian has longitude zero. The stellar direction and an orthonormal basis adapted to the optical axis areHere $\mathbf{e}$ points east and $\mathbf{n}$ points north. Intersect the ray $t\mathbf{s}$ with the plane $\mathbf{x}\cdot \mathbf{b}=f$. It gives $t=f/(\mathbf{s}\cdot \mathbf{b})$, so the detector coordinates areThese equations also avoid spurious singularities caused by writing individual tangents or cotangents.

To put this gnomonic projection in the desired form, let $\rho =({\cos }^2\delta {\cos }^2\Delta +{\sin }^2\delta {)}^{1/2}$ and choose $q$ locally byThus $\cot q=\cot \delta \cos \Delta$. The denominator becomes $\rho \cos (q-D)$, the numerator for $\eta$ becomes $\rho \sin (q-D)$, and $\cos \delta \sin \Delta =\rho \cos q\tan \Delta$. ConsequentlyUse the local meridian chart $\cos \Delta >0$, with $q\in [-\pi /2,\pi /2]$ chosen continuously near $D$ and $q=D$ at the image centre. A visible gnomonic projection requires $\mathbf{s}\cdot \mathbf{b}>0$, but this front-hemisphere condition alone does not select that meridian chart. A narrow field near a celestial pole can cross the opposite meridian; for such fields use the Cartesian expressions above and distinguish the oriented great circle parameter from ordinary declination. In particular, the small-field limit is $\xi \simeq f\cos D(\alpha -A)$ and $\eta \simeq f(\delta -D)$, with angles in radians. The factor $\cos D$ is the shrinking angular distance per unit right ascension near the pole.

An orthonormal wavelet is a function whose integer translates and dyadic dilatesform an orthonormal basis of the Hilbert space $L^2(\mathbb{R})$. Thus their pairwise inner products are ${\delta }_{j,j^{\prime }}{\delta }_{k,k^{\prime }}$, and their closed linear span is the whole space. Equivalently, every $f\in L^2(\mathbb{R})$ has the norm-convergent expansionThe factor $2^{j/2}$ preserves the $L^2$ norm under dilation.

Let $P_j$ be the orthogonal projection onto $V_j$. The dilation property and the orthonormal basis property give the orthonormal basis ${\phi }_{j,k}(x)=2^{j/2}\phi (2^{j}x-k)$ of $V_j$. The periodized nonnegative functionhas period one and ${\int }_0^{1}F=\Vert \phi {\Vert }_2^2=1$, by Tonelli theorem. For a function $g\in L^2(\mathbb{R})$ with compact support contained in a bounded interval $I$, Cauchy-Schwarz inequality and the Parseval identity giveas $j\to -\infty$. The last step uses absolute continuity of the integral of a locally $L^1$ function, since the interval's length tends to zero. If $f\in {\bigcap }_j{V}_j$, thenFunctions with compact support are dense in $L^2$, so $f=0$. ConsequentlyThis proof needs neither the density axiom nor any compact support assumption on the scaling function.

Put $W_j=V_{j+1}\cap V_j^{\perp }$, the detail space of the multiresolution analysis. The Meyer-Mallat theorem gives a function $\psi \in W_0$ whose integer translates form an orthonormal basis of $W_0$. Dilations then give an orthonormal basis of each $W_j$, and the density and trivial-intersection axioms implyFor an explicit filter construction, write the scaling refinement equation $\phi (x)=\sqrt{2}{\sum }_k{h}_k\phi (2x-k)$. With a quadrature mirror filter one may take $g_k=(-1{)}^k\overline{h_{1-k}}$ and $\psi (x)=\sqrt{2}{\sum }_k{g}_k\phi (2x-k)$. A harmless constant sign or integer translation gives an equivalent orthonormal wavelet.

The answer is no with the integer-translation and dilation conventions of this multiresolution analysis. The orthonormal basis axiom forces $V_0$ to consist of $L^2$ functions constant on the cells $[k-1/2,k+1/2)$. Dilation forces $V_1$ to consist of functions constant on $[k/2-1/4,k/2+1/4)$. However, the proposed scaling function changes value at $1/2$, in the interior of the $V_1$ cell $[1/4,3/4)$. It is therefore not in $V_1$, contradicting nesting:Changing values at endpoints makes no difference in $L^2$. The usual Haar wavelet instead uses the scaling function ${\chi }_{[0,1)}$; an arbitrary half-unit translation does not preserve the required refinement grid.

With $N$ vanishing moments, a wavelet annihilates every polynomial of degree below $N$. A Taylor polynomial then explains small wavelet coefficients on smooth parts of a signal, and efficient best N-term approximation. Compact support localizes coefficients near a feature, limits the number of boundary interactions, and permits a finite filter implementation. Increasing the number of vanishing moments while keeping an orthonormal basis generally requires a larger support of a function: a finite orthonormal filter with $N$ vanishing moments needs at least $2N$ taps, and the minimal-support Daubechies wavelet has support length $2N-1$ in the standard normalization.

The Haar wavelet, $\psi ={\chi }_{[0,1/2)}-{\chi }_{[1/2,1)}$, has one vanishing moment, unit support length and discontinuities. It is inexpensive and particularly suitable for piecewise constant data with sharp jumps. A Daubechies wavelet of a larger order has more vanishing moments and a longer finite filter; sufficiently large orders also provide greater regularity. It is useful when smooth trends should yield small coefficients, although the wider support of a function can spread a jump across more coefficients. Choose Haar for compact jump localization; choose a higher-order Daubechies wavelet for smooth polynomial structure. More vanishing moments alone does not make every low-order wavelet highly differentiable.

Use the Fourier transform convention $\hat{\phi }(\xi )=\int \phi (x)e^{-ix\xi }dx$. Nesting and the orthonormal basis of $V_1$ give the scaling refinement equationHere $h_k=⟨\phi ,\sqrt{2}\phi (2\cdot -k)⟩$. Compact support makes these inner products zero for all but finitely many $k$, so $m$ is a trigonometric polynomial. Expanding the orthogonality of the integer translates of $\phi$ in the orthonormal basis of $V_1$ yieldsConsequentlyThis holds everywhere because $m$ is continuous. The invoked properties are nesting, dyadic dilation, orthonormal integer translates, and compact support; mere finite-energy refinement would not imply the quadrature mirror filter identity.

At zero, the partial products are $m(0{)}^n$. Their assumed convergence and $|m(0)|=1$ force $m(0)=1$ and $\Theta (0)=1$. Uniform convergence on compact sets makes $\Theta$ continuous. The identity with $k=0$ gives $\Theta \in L^2$ with squared norm $2\pi$. Define $\phi$ by the inverse Fourier transform in $L^2$. The Plancherel theorem and the other given integral identities giveDefine $V_j$ as the closed span of ${\phi }_{j,k}(x)=2^{j/2}\phi (2^{j}x-k)$. These functions form an orthonormal basis of $V_j$, and dyadic dilation gives the scale axiom. Shifting the infinite product givesso the finite Fourier series of $m$ gives a scaling refinement equation and $V_j\subset V_{j+1}$. The coarse-scale projection argument of part 1(b) proves the trivial-intersection axiom.

For density, take a function $f$ with bounded Fourier support. The MRA projection Fourier identity gives, for large $j$,Uniform convergence of $\Theta (2^{-j}\xi )$ to one on that support proves the limit. Since $P_j$ is an orthogonal projection, $\Vert f-P_{j}f{\Vert }_2^2=\Vert f{\Vert }_2^2-\Vert P_{j}f{\Vert }_2^2\to 0$. Such functions are dense in $L^2$, givingThus all the multiresolution analysis axioms hold. Under the strong convergence and orthogonality assumptions supplied here, the additional nonvanishing condition is not needed in this last verification; it is useful when establishing those assumptions from a filter.

For the multiresolution analysis $V_j$ and its detail spaces $W_j$, take the completed tensor product ${\mathcal{V}}_j=V_j\otimes V_j$. Since $V_{j+1}=V_j\oplus W_j$,The three tensor-product wavelets areTaking products of members of the two one-dimensional families gives an orthonormal basis of each respective completed tensor product. Density and the coarse-scale trivial intersection yieldas an orthonormal basis of $L^2({\mathbb{R}}^2)$. The normalization is $2^j$ in two dimensions. On the square, tensorize the interval-adapted wavelet basis and include the finite coarse scaling function part.

Either essay option suffices; both are developed here to make the two mathematical constructions available.

Diffusion for images. Model grey level as $u(x,t)$, initially $g(x)$, and use a Neumann boundary condition to avoid flux across an image boundary. The linear heat equation $u_t=\Delta u$ smooths the data. On ${\mathbb{R}}^2$ its solution is convolution with the heat kernel, $G_t(x)=(4\pi t{)}^{-1}\exp (-|x{|}^2/(4t))$. Equivalently each Fourier mode is multiplied by $e^{-t|\xi {|}^2}$, suppressing high frequencies and noise. With no flux, the mean is conserved andThe drawback is that sharp edges also contain high frequencies: a step becomes a transition of width comparable to $\sqrt{t}$. Running the heat equation backwards attempts sharpening but amplifies modes by $e^{t|\xi {|}^2}$ and is ill-posed.

The Perona-Malik equation instead uses a decreasing diffusivity,Small gradients are smoothed strongly and large ones less strongly. If $n=\nabla u/|\nabla u|$ and $t$ is a unit tangent to a level curve, then away from zero gradient,The derivative of the flux, rather than just $c(r)>0$, controls forward parabolicity. Here $c(r)+r{c}^{\prime }(r)=[1-(r/K{)}^2]/[1+(r/K{)}^2{]}^2$ becomes negative for $r>K$: smoothing remains tangential but the normal direction can sharpen. This formal edge enhancement comes with forward-backward ill-posedness, so existence and stability of the unregularized continuum equation must not be presumed. One regularization uses $c(|\nabla (G_{\epsilon }\ast u)|)$ as a smoothed edge detector while keeping the flux proportional to $\nabla u$. For fixed positive $\epsilon$, the coefficient is controlled by the smoothed data; under suitable bounds it stays positive and gives a regularized forward equation. It preserves edges through reduced cross-edge transport, without the same local backward-diffusion calculation.

Diffusion from a gradient energy connects these equations with variational regularization. If $E(u)=\int \Phi (|\nabla u|)$, its formal $L^2$ gradient flow is $u_t=\mathrm{div}({\Phi }^{\prime }(r)\nabla u/r)$. Linear diffusion corresponds to $\Phi (r)=r^2/2$. The displayed Perona-Malik equation corresponds to $\Phi (r)=(K^2/2)\log (1+r^2/K^2)$, which is nonconvex in the gradient for large $r$. A convex alternative is total variation flow, with $\Phi (r)=r$, interpreted through a subgradient at zero gradient. Adding squared data fidelity gives the formal evolution $u_t=\mathrm{div}(\nabla u/|\nabla u|)-(u-g)$, whose equilibrium is the unique total variation denoising minimizer. Convex TV preserves sharp interfaces more effectively than the heat energy, but it may produce piecewise constant plateaux, known as staircasing in total variation denoising. A stopping time or fidelity weight controls the smoothing scale. Linear diffusion is stable but blurs edges; nonlinear diffusion must balance edge selectivity with parabolicity and regularization.

Wavelets on an interval. Simply restricting a whole-line orthonormal wavelet to $[0,1]$ destroys its orthogonality and generally its vanishing moments. Periodizing the basis restores orthogonality on the circle, but treats the two endpoints as neighbors. This is suitable for periodic data and can create an artificial endpoint jump for nonperiodic data.

A localized interval-adapted wavelet basis instead uses unchanged interior functions and finitely many special boundary wavelets at each endpoint. Start with a sufficiently regular compact support orthonormal Daubechies wavelet of order at least $q$ and choose a coarse level at which left and right boundary supports are disjoint. At each boundary, take appropriate finite combinations of the scaling functions that meet the endpoint, restricted to the interval. Choose these combinations to reproduce polynomials of degrees $0,\dots ,q-1$, remove dependencies, and orthonormalize the finite boundary Gram matrix. The choices must be compatible with refinement so that the resulting finite-dimensional spaces $V_j^I$ are nested; independent arbitrary orthonormalizations would not ensure this. Interior functions retain their whole-line filters, while the boundary functions use finite boundary refinement matrices.

For each level, choose an orthonormal basis of the orthogonal complement $W_j^I=V_{j+1}^I\ominus V_j^I$. It consists of interior wavelets and a bounded number of left and right boundary wavelets. Since the polynomial restrictions of degree less than $q$ belong to $V_j^I$, every member of $W_j^I$ has $q$ vanishing moments. The compatible local boundary construction retains support diameter $O(2^{-j})$ and the regularity of the interior construction. Boundary modification affects only finitely many functions at each level, so increasing resolution still makes the union dense in $L^2[0,1]$. Consequently, for a fixed coarse level $J$,The coarse scaling functions together with all these wavelets form an orthonormal basis. This construction is the Cohen-Daubechies-Vial interval wavelet construction. It preserves localization, polynomial cancellation and stable coefficient extraction without imposing periodic or zero boundary data. The finite boundary refinement matrices also permit a fast transform; a mere restriction followed by one unrelated Gram-Schmidt process at each scale does not establish all these properties.

Let $T$ be the $M\times M$ tridiagonal matrix with diagonal two and adjacent entries minus one, incorporating zero Dirichlet boundary conditions. The actual PDF scheme isAn orthonormal basis of eigenvectors of $T$ has components $\sqrt{2/(M+1)}\sin (jm\pi /(M+1))$, and substitution givesThey form an orthonormal basis by the spectral theorem for real symmetric matrices. Hence $I+\mu T$ is invertible for every $\mu >0$, and the amplification eigenvalues are $g_j=(1+\mu {\lambda }_j{)}^{-1}\in (0,1)$. For the discrete L2 norm $\Vert U{\Vert }_h^2=h{\sum }_m|U_m{|}^2$,This bound has constant one independent of time step and mesh, which is the required stability of a numerical method. Therefore every positive Courant number is stable. If a forcing or local-defect sequence is added, the discrete variation-of-constants formula gives a bound by the initial norm plus the sum of those perturbation norms; this also justifies the global error conclusion in part (a).