Pointwise convergence of Fourier series (I). On a conjecture of Konyagin

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 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...

Convergence of Fourier Series

The purpose of this paper is to explore the basic question of the convergence of Fourier series. This paper will not delve into the deeper questions of convergence that measure theory illuminates, but requires only the basic principles set out by introductory real and complex analysis.

Pointwise Convergence of Multiwavelet Packet Series

Convergence of Random Fourier Series

This paper will study Fourier Series with random coefficients. We begin with an introduction to Fourier series on the torus and give some of the most important results. We then give some important results from probability theory, and build on these to prove a variety of theorems that deal with the convergence or divergence of general random series. In the final section, the focus is placed on r...