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 , their detail-space orthogonal complements have 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.
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