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A given set W = W X of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum _{\chi }\left({\int }_{{ℝ}^n}\nabla f·\nabla W_{\chi }^{*}\right)W_{\chi }$ converges to f with respect to the norm ${∥\nabla (·)∥}_{L^2\left({ℝ}^n\right)}$ . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = W x of compactly supported class C 2−ɛ functions on ℝn such that [...]...