Lecture 2 Introduction to Some Convergence theorems Friday 14 , 2005 Lecturer : Nati Linial

r∈Z f̂(r)e In the last lecture, we proved Fejér’s theorem f ∗ kn → f where the ∗ denotes convolution and kn (Fejér kernels) are trignometric polynomials that satisfy 1. kn ≥ 0 2. ∫ T kn = 1 3. kn(s) → 0 uniformly as n→∞ outside [−δ, δ] for any δ > 0. If X is a finite abelian group, then the space of all functions f : X → C forms an algebra with the operations (+, ∗) where + is the usual pointwise sum and ∗ is convolution. If instead of a finite abelian group, we take X to be T then there is no unit in this algebra (i.e., no element h with the property that h ∗ f = f for all f ). However the kn behave as approximate units and play an important role in this theory. If we let Sn(f, t) = n ∑