Curvelets and Curvilinear Integrals

Let C(t) : I 7→ R be a simple closed unit-speed C curve in R with normal ~n(t). The curve C generates a distribution Γ which acts on vector fields ~v(x1, x2) : R 2 7→ R by line integration according to Γ(~v) = ∫ ~v(C(t)) · ~n(t)dt. We consider the problem of efficiently approximating such functionals. Suppose we have a vector basis or frame Φ = (~ φμ); then an m-term approximation to Γ can be formed by selecting m terms (μi : 1 ≤ i ≤ m) and taking