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Finite‐element wavelets on manifolds

We construct locally supported, continuous wavelets on manifolds Γ that are given as the closure of a disjoint union of general smooth parametric images of an n-simplex. The wavelets are proven to generate Riesz bases for Sobolev spaces H s (Γ) when s ∈ (-1, 3/2), if not limited by the global smoothness of F. These results generalize the findings from Dahmen & Stevenson (1999) SIAM J. Numer. Anal., 37, 319-352, where it was assumed that each parametrization has a constant Jacobian determinant. The wavelets can be arranged to satisfy the cancellation property of, in principle, any order, except for wavelets with supports that extend to different patches, which generally satisfy the cancellation property of only order 1.