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A wavelet is a localized analyzing function whose shifted and scaled copies examine a signal at different positions and resolutions. Its usefulness is to separate where a change occurs from how large a region that change spans. A brief feature can contribute to a few nearby wavelet coefficients instead of being described entirely through global oscillations.

The Haar wavelet is the simplest example: compare the first half of an interval with the second. A constant value contributes nothing to this contrast, while a difference between the halves survives. Moving the function changes which part of the signal is examined; shrinking it compares smaller neighbouring regions. In an orthonormal wavelet construction, these comparisons fit together without duplicating squared-norm contributions.

A scaling function retains the coarse information that such contrasts omit. The discrete wavelet transform computes both coarse values and details, and the wavelet series recombines them. This gives signal processing a representation that can expose local structure and support image compression. The transform itself is reversible in the complete orthonormal representation; discarding coefficients introduces approximation error. ^[ref]

At design sites $i/n$, let ${\tilde{Y}}_n$ be the step function taking response $Y_i$ on $((i-1)/n,i/n]$. Project it onto an interval-adapted wavelet basis through resolution $J$. Each coefficient is ${\hat{c}}_{\lambda }={\sum }_i{Y}_i{\int }_{I_i}{b}_{\lambda }(t)dt$. Its expectation is the corresponding coefficient of the step function of sampled regression means. For a Haar wavelet basis, dyadic sample size and resolution at most the sampling scale, this is exactly $n^{-1}{\sum }_i{Y}_i{b}_{\lambda }(i/n)$ with right-closed pointwise versions. Independent errors of variance ${\sigma }^2$ give coefficient variance at most ${\sigma }^2/n$: Cauchy-Schwarz inequality on each sampling cell yields ${\sum }_i({\int }_{I_i}{b}_{\lambda }{)}^2\le \Vert b_{\lambda }{\Vert }_2^2/n$.