Why Boundaries Require a Different Construction
Wavelets on the real line are built from translates of a scaling function and its associated wavelet. On a finite interval, simply retaining translates whose supports fit inside the interval leaves gaps near the endpoints. Wrapping the interval around, or periodically extending a function, avoids the gaps but imposes an additional assumption: the function should join smoothly from one endpoint to the other. Regression functions on an interval need not satisfy that condition.
An interval-adapted wavelet basis handles the endpoints directly. Most basis functions, those supported away from the endpoints, are the usual interior wavelets. A fixed number of functions near each endpoint are replaced by specially constructed boundary scaling functions and boundary wavelets. The replacements remain supported inside the interval and are chosen to preserve orthogonality and polynomial cancellation. This makes it possible to approximate functions on an interval without pretending that the interval is periodic.
Write the scaling functions as \(\phi_{j,k}\) and the wavelets as \(\psi_{j,k}\), with \(k\) indexing the functions present at level \(j\). At each level there are order \(2^j\) functions in total, but only a bounded number are boundary functions. Their supports have length at most \(C2^{-j}\), for a constant \(C\) independent of \(j\). In an order-\(r\) construction, each wavelet, including each boundary wavelet, has \(r\) vanishing moments:
This condition says that the wavelet is orthogonal to polynomials of degree less than \(r\). Boundary correction does not mean relaxing cancellation at the edges; it means designing endpoint functions that retain the required cancellation while staying inside the interval.
How the Endpoint Functions Are Adapted
A standard interval construction starts with a compactly supported real-line wavelet system. At a chosen coarse level, interior scaling functions whose supports lie fully inside \([0,1]\) are retained. The translates that would cross an endpoint are not wrapped around to the other side. Instead, boundary scaling functions are constructed on the left and right. Their refinement relations are selected so that the scaling spaces remain nested and the boundary functions reproduce polynomials to the required order.
At each finer level, interior wavelets are retained wherever their supports fit. Near each endpoint, boundary wavelets are obtained as an orthonormal complement to the finer scaling space, subject to support and moment constraints. In constructions such as the Cohen–Daubechies–Vial interval wavelets, only finitely many functions per endpoint require modified filters; the interior filters and wavelets are unchanged. The resulting basis has no support extending outside the interval and no identification of the two endpoints.
The construction is designed to preserve three properties together: orthogonality (or, in some variants, stable Riesz-basis bounds), short support, and approximation order. The moment conditions are especially important for estimation: they ensure that smooth trends do not create large wavelet coefficients merely because their support touches an endpoint.
Use the original translates and dilates wherever their supports lie within the interval.
Construct left and right scaling functions and wavelets with support contained in the interval.
Choose the boundary functions to preserve the nested spaces, orthogonality or stability, and the required polynomial moments.
Expand functions on \([0,1]\) directly, without extending them periodically across the endpoints.
Worked Example: A Linear Function Is Not Periodic
Consider \(f(x)=x\) on \([0,1]\). It is smooth on the interval, but its endpoint values differ: \(f(0)=0\) and \(f(1)=1\). The periodic extension therefore has a jump at each integer.
For the Fourier coefficient \(c_k=\int_0^1 x e^{-2\pi i kx}\,dx\), where \(k\ne0\), integration by parts gives
The integral in the second term is zero because \(e^{-2\pi i k}=1\). Thus \(|c_k|^2=1/(4\pi^2 k^2)\), so the squared \(L^2\) error from omitting frequencies \(|k|\le J\) is
The slow tail reflects the jump in the periodic extension, not a lack of smoothness of \(x\) on \([0,1]\). In an interval wavelet basis with at least two vanishing moments, every wavelet coefficient of \(x\) is zero, including at the boundary, because the wavelets annihilate constants and linear functions. At any level \(j\) for which the basis has this polynomial reproduction property, \(x\) lies in \(V_j\) and is represented exactly. The endpoint correction avoids creating a jump that was absent from the regression problem.
Approximation Order from Vanishing Moments
The approximation guarantee can be derived directly from the wavelet expansion. Assume an orthonormal interval wavelet basis with nested spaces \(V_j\), a complete expansion in \(L^2([0,1])\), at most \(C_0 2^\ell\) wavelets at level \(\ell\), and supports of length at most \(C_1 2^{-\ell}\). Assume also that every wavelet has \(r\) vanishing moments. For an integer \(r\ge1\), let \(f\in C^r([0,1])\), with \(\|f^{(r)}\|_\infty<\infty\). Let \(P_j f\) be the \(L^2\) orthogonal projection onto \(V_j\).
Proof. Completeness and orthogonality give the squared projection error as the sum of squared wavelet coefficients at levels \(\ell\ge j\). Fix such a level and a wavelet \(\psi_{\ell,k}\) supported on an interval of length at most \(C_1 2^{-\ell}\). Choose a point \(x_0\) in that support. Taylor’s theorem gives a polynomial \(T_{r-1}\) of degree at most \(r-1\) such that, on the support,
The moment conditions imply \(\langle T_{r-1},\psi_{\ell,k}\rangle=0\). Since \(\|\psi_{\ell,k}\|_2=1\), the Cauchy–Schwarz inequality bounds its \(L^1\) norm by the square root of its support length. Consequently,
There are at most \(C_0 2^\ell\) coefficients at level \(\ell\). Their squared sum is therefore at most \(C_4\|f^{(r)}\|_\infty^2 2^{-2\ell r}\). Summing the geometric series over all levels \(\ell\ge j\) yields
Taking square roots proves the result. The proof uses only support size, the number of coefficients per level, orthonormality, and vanishing moments; it does not distinguish interior from boundary wavelets. \(\square\)
Worked Example: The Rate for a Twice-Smooth Regression Function
Suppose the interval basis has \(r=2\) vanishing moments, and \(f(x)=x^2\) on \([0,1]\). Here \(f''(x)=2\), so \(\|f''\|_\infty=2\). At level \(j\), the theorem gives
Thus doubling the resolution from level \(j\) to \(j+1\) reduces this upper bound by a factor of \(2^2=4\). The squared error bound decreases by a factor of \(16\). The constant \(C\) depends on the chosen wavelet system; the rate follows from the two vanishing moments and does not require \(f(0)=f(1)\).
A related coefficient calculation makes the endpoint role explicit. For any \(C^r\) function and any level-\(\ell\) wavelet, subtract the Taylor polynomial of degree \(r-1\) on that wavelet’s support. The polynomial integrates to zero against the wavelet, and the remainder is at most a constant times \(2^{-\ell r}\). The coefficient is therefore at most a constant times \(2^{-\ell(r+1/2)}\), whether the support touches an endpoint or lies in the interior. This conclusion depends on the boundary wavelet having the same required moments; a truncated interior wavelet does not automatically retain them.
Worked Example: Checking Boundary Moment Cancellation
Consider the function \(\psi(t)=\sqrt{5}(6t^2-6t+1)\) on \([0,1]\). It illustrates a boundary-supported shape with two vanishing moments. Direct integration gives
For a support of width \(h=2^{-j}\) at the left endpoint, define \(\psi_j(x)=h^{-1/2}\psi(x/h)\) on \([0,h]\) and zero elsewhere. A change of variable \(t=x/h\) gives
So this shape cancels both constant and linear trends even though its support touches the boundary. It illustrates the moment calculation, not a full wavelet construction: a valid basis also needs the appropriate orthogonality, refinement, and completeness properties.
Use in Regression and a Common Pitfall
In regression on a fixed interval, the fitted function is defined on the design domain, not on a circle. Boundary-corrected wavelets let the basis represent endpoint behavior without forcing the fitted values near \(0\) to agree with those near \(1\). Their approximation guarantees are therefore appropriate for regression functions that are smooth on the interval but have unequal endpoint values or derivatives.
Common mistake: “The wavelet support reaches past an endpoint, so I can wrap the excess part around to the opposite endpoint; this merely fills in the missing support.” That reasoning changes the domain: wrapping identifies \(0\) with \(1\) and imposes periodic boundary behavior. For \(f(x)=x\), it creates the jump shown above and slow Fourier decay. The correction is to use boundary functions supported within \([0,1]\), with moments and stability designed for the interval. Boundary correction changes the basis near the endpoints; it does not alter the regression function to make it periodic.
Check Your Understanding
Use the definitions and calculations above to answer the following.
- Why can a periodic extension of a smooth function on \([0,1]\) have a discontinuity?
- What three structural properties are typically balanced when constructing boundary wavelets?
- How does a wavelet’s \(r\)th-order moment condition help bound its coefficient for a \(C^r\) function?
- Under the theorem’s assumptions, what is the \(L^2\) approximation rate when the resolution level increases from \(j\) to \(j+1\)?
- Why does truncating an interior wavelet at an endpoint not, by itself, establish that the resulting function is a valid boundary wavelet?