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!
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
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
There is a phenomenon in physics and mathematics which has been captivating the minds of scientists for a long time and it is called simply “universality”. In brief, a phenomenon exhibits universality if, be as wild and diverse in microscopic scale as possible, a clear pattern emerges if one looks at it from macrosopic scale. … More Universality and orthogonal polynomials
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
Let be a Borel measure on the real line. Usually we impose two conditions on this measure: (i) the support contains at least infinitely many points, (ii) for all we have (This is called the finite moment condition.) We shall see in a minute why these restrictions are needed. The following theorem says that there … More The zeros of orthogonal polynomials, part I