Interval-adapted wavelet basis

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.

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.

This construction adapts compactly supported orthonormal wavelets to an interval by finite changes near its endpoints. It preserves local support, polynomial reproduction, nested approximation spaces and orthogonal detail spaces. The resulting basis contains coarse scaling functions, unchanged interior wavelets, and finitely many boundary wavelets per endpoint and per scale, without forcing periodic or zero boundary values.