Explicit Wavelet Approximation in Weighted Besov Spaces with Applications to Piecewise Regular Series Under Structural Breaks

We establish explicit direct and inverse approximation estimates for biorthogonal multiresolution projections on weighted Besov spaces over Muckenhoupt Ap weights. A Jackson-type direct estimate bounds the weighted Lp projection error by 2−Js times the weighted Besov norm, and a matched Bernstein-type inverse estimate bounds the weighted Besov seminorm of a resolution-space element by 2Js times its weighted Lp norm. Every constant is displayed in factorized form: each factor is either given in closed form or is the operator norm of the Hardy–Littlewood maximal operator on the weighted Lebesgue space, through which the entire dependence on the Muckenhoupt characteristic is routed. For a piecewise regular class combining a Besov-smooth component with finitely many net-zero jumps, the projection error separates into a smooth part decaying at 2−Js and a localized jump part carrying the weighted measure of a shrinking interval about each jump; when the weight is locally comparable to Lebesgue measure near the jumps, this yields the effective rate min(s,1/p). A deterministic numerical experiment confirms this rate within one fixed biorthogonal analysis, and the same projection is illustrated on an empirical higher-education finance series.