Smoothness beyond di erentiability

Fundamental Theorem of Analysis Theorem 1. Suppose that f ∈ H(0, 1). Then there exists a g ∈ L2(0, 1) such that f(x)− f(0) = ∫ x 0 g(t) dt, 0 < x < 1. We de ne the linear mapping (Sg)(x) := ∫ x 0 g(t) dt. Corollary 1. If f ∈ H(0, 1) then f − f(0) ∈ R(S). Thus, smoothness means that f is in the range of a smoothing operator. Theorem 2. Suppose that f ∈ H(0, 1) (periodic). For the Fourier coe cients f̂(n) of f it holds ∑∞ n=1 n 2 ∣∣∣f̂(n)∣∣∣2 <∞. Corollary 2. Smoothness implies a certain decay of the Fourier coe cients. Goal 1. Make a general theory from these observations!