Wavelet bases and multiresolution

On $[0,1]$, let $V_j$ be the L² space of functions constant on each of the $2^j$ dyadic subintervals. These finite-dimensional spaces are nested, and their union is dense in L². At each scale, the finer space contains both old coarse information and new detail. Define $W_j$ as the orthogonal complement of $V_j$ inside $V_{j+1}$. Then every fine approximation splits uniquely into a coarse projection plus a detail component at that scale.

For Haar wavelets, the coarse scaling function is the indicator of the whole unit interval. A mother wavelet is plus one on the first half and minus one on the second, with value zero outside. Dilation and translation place normalized copies on dyadic subintervals. Endpoints of these pieces can be assigned consistently in any way because changing finitely many point values does not change an L² function. The normalization factor ensures each wavelet has norm one, not merely height one.

If two neighboring fine-scale scaling coefficients are a and b in an orthonormal basis, the next coarser scaling coefficient is $(a+b)/\sqrt2$ and the detail coefficient is $(a-b)/\sqrt2$. This is an orthogonal two-by-two transformation, so it preserves the sum of squares. Applying it recursively gives the Haar pyramid transform. Raw sample heights are not always the same as normalized scaling coefficients; interval lengths determine the conversion between them.

For a square-integrable function, the coefficient of each basis function is its inner product with that function. Orthogonality means the squared norm of a finite reconstruction equals the sum of squared coefficients, and the squared approximation error equals the omitted coefficient energy when the full basis expansion is used. Projection onto V_j replaces the function by its average on every dyadic cell. It therefore need not preserve a point value or reproduce a discontinuity whose location falls inside a cell.

Every Haar wavelet has zero integral, so its coefficient vanishes for a constant target on its support. This is one vanishing moment. More regular wavelet families can have several vanishing moments, canceling local polynomial behavior through higher degree. A small coefficient can then signal that the target is locally well represented at a coarser scale. The precise approximation rate depends on the function class, the wavelet's regularity and moments, and the norm in which error is measured.

Haar is compactly supported but discontinuous, so localization does not imply smooth basis functions. Global sine functions are smooth but not spatially localized. A finite discrete wavelet transform also requires decisions about boundary treatment and how samples represent an underlying function. Perfect algebraic reconstruction of the input samples is different from recovering unresolved continuous features. Compression introduces further choices: discarding coefficients gives an approximation, and quantizing retained coefficients adds an error beyond the discarded tail.