Potpourri, 5

In this note, we study a certain class of trigonometric series which is important in many problems. An unproved statement in Zygmund's book [5] will be proved and generalized. Further discussions based on this problem will also be made here.

A number of topics in analysis are discussed, with emphasis on basic principles. There is some overlap with "Elements of linear and real analysis" (arXiv:math/0108030), with numerous changes in content and presentation since then.

Trigonometry is the study of circular functions, which are functions defined on the unit circle $x^2+y^2 =1$, where distances are measured using the Euclidean norm. When distances are measured using the $L_p$-norm, we get generalized trigonometric functions. These are parametrizations of the unit $p$-circle $|x|^p+|y|^p =1$. Investigating these new functions leads to interesting connections involving double angle formulas, norms induced by inner products, Stirling numbers, Be...

It is shown that quasi all continuous functions on the unit circle have the property that, for many small subsets E of the circle, the partial sums of their Fourier series considered as functions restricted to E exhibit certain universality properties.

We announce numerous new results in the theory of orthogonal polynomials on the unit circle.

These informal notes deal with Fourier series in one or more variables, Fourier transforms in one variable, and related matters.

These notes are a chapter in Real Analysis. While primarily standard, the reader will find a discussion of certain topics that are ordinarily not covered in the usual accounts. For example, the notion of bounded variation in the sense of Cesari is introduced. This is Volume 1 of 4, to be followed by Curves and Length, Functions of Several Variables, and Surfaces and Area.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation and let $f^{n}$ be its $n$-th iterate. Let $\{a_{n}\}$ be a sequence of complex numbers. We prove that the series $\sum a_{n}f^{n}(\xi)$ converges at almost every point $\xi$ of the unit circle if and only if $\sum |a_n|^2 < \infty$. The main step in the proof is to show that under this assumption, the function $F= \sum a_n f^n$ has bounded mean oscillation. We also prove that $F$ is bounded on the unit disc i...

These informal notes consider Fourier transforms on a simple class of nice functions and some basic properties of the Fourier transform.

This paper establishes necessary and sufficient conditions for the products of freely independent unitary operators to converge in distribution to the uniform law on the unit circle.