Know your brain!

Aside from mathematics, I have other interests. Ever since I started to use my brain for mathematics, I am interested about how thinking works. Since problem solving is my day job, I have plenty of opportunities to observe my cognitive processes and I have a passion for optimizing my brainworks. Thinking about thinking is a … More Know your brain!

Spectral theory, part I: The moment problem and the spectral measure

Let’s briefly recall what I am about to do here. If is a measure supported on the real line and the -th orthonormal polynomial is denoted with , then, as we have seen before, there are positive numbers and real numbers such that , which is called the three-term recursion. I asked the question that … More Spectral theory, part I: The moment problem and the spectral measure

Recurrence relations and a prelude to spectral theory

This time I want to talk about one of the most basic and most important tools in the theory of orthogonal polynomials: the recurrence relations. This topic also demonstrates that the theory of orthogonal polynomials on the real line and on the unit circle differ significantly. Let’s start with OPRL! (That is, orthogonal polynomials on … More Recurrence relations and a prelude to spectral theory

The zeros of orthogonal polynomials, part II: OPRL vs OPUC

I finished up the previous post with some questions. We studied the zero distributions of orthogonal polynomials on the real line, and I was curious about the possible generalizations of Theorem 2. The ultimate generalization lies deeper then I can dig in one post, but this motivated me to write about orthogonal polynomials on the … More The zeros of orthogonal polynomials, part II: OPRL vs OPUC