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A boundary wavelet replaces a translated interior wavelet whose support of a function meets a domain endpoint. Compatible finite boundary refinement matrices preserve orthogonality and localization across scales. If the coarse boundary spaces reproduce polynomials of degree less than $q$, their detail-space orthogonal complements have $q$ vanishing moments.

A Daubechies wavelet of order $q$ is a compact support orthonormal wavelet with $q$ vanishing moments and a minimal-length finite refinement filter. The minimal filter has $2q$ taps, giving support length $2q-1$ in the standard dyadic normalization. Its low-pass symbol has a zero of order $q$ at $\pi$. Increasing order improves polynomial cancellation and, for sufficiently large order, regularity, at the price of a wider support.

The Haar wavelet has one vanishing moment and compact support of unit length. Its scaling function is ${\chi }_{[0,1)}$, and its refinement coefficients are $h_0=h_1=1/\sqrt{2}$. The approximation spaces consist of functions constant on dyadic cells. Its discontinuity makes it effective for jump localization but limits smooth-function approximation.

An interval-adapted wavelet basis is an orthonormal basis of $L^2[0,1]$ made of a finite coarse scaling function family and resolution-indexed wavelets. A localized construction modifies only a bounded number of functions near each endpoint at every level. Boundary modification must preserve nested refinement spaces and the required vanishing moments; simple restriction of a whole-line basis does not do so.

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.

The factor $1+e^{-i\xi }$ has a simple zero at $\pi$, and $L(\pi )\ne 0$ makes the zero of $m$ there exactly order $N$. The quadrature mirror filter construction gives a high-pass symbol $m_1(\xi )=e^{-i\xi }\overline{m(\xi +\pi )}$, up to a constant phase, andThe normalized scaling function has $\hat{\phi }(0)=1$, so $\hat{\psi }$ has a zero of exactly order $N$ at zero. Compact support of the Daubechies wavelet permits differentiation under its Fourier transform:HenceThere are exactly $N$ vanishing moments, rather than just at least $N$.

Use the usual localized, compact support construction of an interval-adapted wavelet basis, including boundary wavelets with the stated vanishing moments, and order the linear N-term approximation by increasing resolution. Also interpret a piecewise polynomial function as having finitely many pieces. These conventions matter: regularity and vanishing moments alone, or an arbitrary enumeration, do not establish the asserted rates.

At scale $j$, a wavelet whose support lies in one polynomial piece has zero coefficient because $p<q$. Only a bounded number $B$ of wavelets per scale can meet a partition point. Their $L^1$ norms are bounded by $C{2}^{-j/2}$, and $f$ is bounded. Thus $|c_{j,k}|\le C_f{2}^{-j/2}$ andRetain the fixed number of coarse scaling function coefficients and every nonzero coefficient through level $J$. This uses at most $B(J+1)+B_0$ terms, leaving squared error at most $C{2}^{-J}$. The optimal best N-term approximation is no worse; choose $J$ proportional to $N$ to obtain $O({\omega }^N)$ for some $\omega \in (0,1)$. In contrast, retaining all wavelets through level $J$ costs $\asymp 2^J$ terms. Choosing the last complete level before $N$ givesThese are squared errors; the corresponding $L^2$ errors are $O({\omega }^{N/2})$ and $O(N^{-1/2})$.

Use localized tensor-product wavelets and finitely many bounded polynomial pieces, with the boundary wavelets adapted as in part (b). A rectifiable smooth curve of length $\ell$ meets $O(1+\ell 2^j)$ dyadic squares of side $2^{-j}$: subdividing an arclength parametrization into pieces of length at most $2^{-j}$ covers it by that many balls, each meeting only a bounded number of squares. Enlarging squares by the fixed support diameter preserves the count.

Each normalized two-dimensional wavelet has $L^1$ norm $O(2^{-j})$. A coefficient meeting the curve is therefore $O(2^{-j})$, and the total squared energy of these coefficients at level $j$ is $O(2^{-j})$. If $p<q$, all other coefficients vanish by the vanishing moments. Keeping the curve coefficients through level $J$ costs $O(2^J)$ and leaves squared error $O(2^{-J})$.

The printed part (e) does not repeat $p<q$. The bound still holds for any fixed polynomial degree when $q\ge 1$: on a smooth piece, a Taylor polynomial in the variable carrying a wavelet gives coefficient size $O(2^{-(q+1)j})$. There are $O(4^j)$ such coefficients, so their squared energy is $O(2^{-2qj})$. Retain all coefficients through $K=\lfloor J/2\rfloor$, and curve coefficients through $J$. The cost is $O(4^K+2^J)=O(2^J)$ and the omitted squared energy isThe best N-term approximation is at least as good as this selection, provingThis argument covers the unqualified finite-degree clause without adding an unnecessary restriction $p<q$.

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.

A piecewise polynomial function agrees with a polynomial on each member of a partition. Approximation claims usually require finitely many pieces with controlled interfaces. In one dimension, finitely many partition points and localized wavelets with more vanishing moments than the polynomial degree leave only a bounded number of nonzero coefficients per scale, yielding exponential squared best N-term approximation error.