Fourier Series and Martingale Transforms
Ferenc Schipp · Birkhäuser Basel eBooks · 1978
In this paper a theorem of D.L. Burkholder [2], [3] on martingale transforms is generalized for the case when the index set on the martingale is a directed set. Using a generalization of the notion of stopping time and applying an elementary lemma, the proof of the generalized theorem is similar to the original one. From the mentioned theorem there follows without difficulties a generalization of a theorem of P. Billard [1]: The product system of independent complex valued functions with zero expectation and module one is an a.e. convergence system. The results obtained show that Carleson’s method [4] is essentially a martingale theoretic method. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.