Potpourri, 5

In the present paper we introduce analogs of almost periodic functions on the unit circle. We study certain uniform algebras generated by such functions, prove corona theorems for them and describe their maximal ideal spaces.

This paper is meant to be a gentle introduction to Carleson's Theorem on pointwise convergence of Fourier series.

We give a new lower bound for the discrete norm of a polynomial on the circle

The main result of this paper is a description of the space of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. This space coincides with the space of operator Lipschitz functions on the unit circle.

We study best approximation to a given function, in the least square sense on a subset of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary subset. We show that the solution to this problem, as the degree goes large, converges to the solution of a bounded extremal problem for analytic functions which is instrumental in system identification. We provide a numerical example on real data from a hyperfrequency filter.

Let $(C(t))\in\mathbb{R}}$ be a cosine function in a unital Banach algebra. We give a simple proof of the fact that if lim sup$\_{t\to 0}\vert C(t)-1\_A\vert\textless{}2,$ then $lim sup\_{t\to 0}\Vert C(t)-1\_A\Vert=0.$

We improve on the inequality $\displaystyle{\frac{1}{\pi}\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt\leq \frac{1}{\sqrt p}, {0.2 cm}p\geq 1,}$ showing that $\displaystyle{\frac{1}{\pi}\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt\leq C(p) \frac{\sqrt{3/\pi}}{\sqrt p},}$ with $\displaystyle{\lim_{p\longrightarrow \infty} C(p)=1,}$ and indeed that {align*} \displaystyle{\lim_{p\longrightarrow \infty}\frac{1}{\pi}\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt/ \f...

This paper has been withdrawn

In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts to force limit discussion into the language of individual real numbers and equality. The system of near-numbers properly fit the context of limits, allowing precise and unambiguous notation in limit computation. More importantly, the near-...

These are the unpublished notes on functional analysis, given by Professor Jaime Chica, School of Mathematics, University of Antioquia.