Martingale theory and pointwise convergence of certain orthogonal series

Pointwise Convergence of Trigonometric Series

We establish two results in the pointwise convergence problem of a trigonometric series [An] £ cne inl with lim Hm £ I bTck | = 0 |n|< -x. * Jn-»oo \k\-n for some nonnegative integer m. These results not only generalize Hardy's theorem, the Jordan test theorem and Fatou's theorem, but also complement the results on pointwise convergence of those Fourier series associated with known 1}-convergen...

Pointwise convergence of Fourier series

In the early 19 century, J. Fourier was an impassioned advocate of the use of such sums, of course writing sines and cosines rather than complex exponentials. Euler, the Bernouillis, and others had used such sums in similar fashions and for similar ends, but Fourier made a claim extravagant for the time, namely that all functions could be expressed in such terms. Unfortunately, in those days th...

Pointwise Convergence of Multiwavelet Packet Series

CORRECTION ON CONVERGENCE OF SERIES OF RANDOM ELEMENTS VIA MAXIMAL MOMENT RELATIONS WITH APPLICATIONS TO MARTINGALE CONVERGENCE AND TO CONVERGENCE OF SERIES WITH p-ORTHOGONAL SUMMANDS

The authors in their article [2] misstated a result due to Móricz, Su, and Taylor [1]. This misstatement resulted in an invalid formulation and proof of a corollary presented in [2]. In this correction note, the needed result from [1] is correctly stated and a corrected version of the invalid corollary in [2] is proved. 2000 Mathematics Subject Classification: 60F05, 60B12, 60B11.

ON CONVERGENCE OF SERIES OF RANDOM ELEMENTS VIA MAXIMAL MOMENT RELATIONS WITH APPLICATIONS TO MARTINGALE CONVERGENCE AND TO CONVERGENCE OF SERIES WITH p-ORTHOGONAL SUMMANDS

The rate of convergence for an almost surely convergent series of Banach space valued random elements is studied in this paper. As special cases of the main result, known results are obtained for a sequence of independent random elements in a Rademacher type p Banach space, and new results are obtained for a martingale difference sequence of random elements in a martingale type p Banach space a...