The ring of real trigonometric polynomials
The ring
Throughout we consider the polynomial ring
This ring has a lot of non-trivial properties which give us a good chance to study commutative ring theory.
Characters in Analysis and Algebra
In this post, we study the concept of character, what it is about in abstract harmonic analysis and how to use it Galois theory.The Banach Algebra of Borel Measures on Euclidean Space
This blog post is intended to deliver a quick explanation of the algebra of Borel measures on $\mathbb{R}^n$. It will be broken into pieces. All complex-valued complex Borel measures $M(\mathbb{R}^n)$ clearly form a vector space over $\mathbb{C}$. The main goal of this post is to show that this is a Banach space and also a Banach algebra.
In fact, the $\mathbb{R}^n$ case can be generalised into any locally compact abelian group (see any abstract harmonic analysis books), this is because what really matters here is being locally compact and abelian. But at this moment we stick to Euclidean spaces. Note since $\mathbb{R}^n$ is $\sigma$-compact, all Borel measures are regular.
To read this post you need to be familiar with some basic properties of Banach algebra, complex Borel measures, and the most important, Fubini’s theorem.
The concept of generalised functions (distributions) and derivatives
In this post, we study the concept of generalised functions (a.k.a. distributions), and let's see how to evaluate the derivative no matter the function is differentiable or not.Curse of Knowledge
You may have known what it is…
Let us say you are a programmer who has been working in big companies for a decade. How does it feel when you want to help someone who starts studying programming from scratch? You may find it makes no sense that he or she cannot understand that, by copying several lines of code on the book, they has successfully made a programme printing “Hello, world!” on the screen. You know what I am talking about - the curse of knowledge.
Elementary Properties of Cesàro Operator in L^2
We study the average of sum, in the sense of integral.Left Shift Semigroup and Its Infinitesimal Generator
Left shift operator
Throughout we consider the Hilbert space $L^2=L^2(\mathbb{R})$, the space of all complex-valued functions with real variable such that $f \in L^2$ if and only if
where $m$ denotes the ordinary Lebesgue measure (in fact it’s legitimate to consider Riemann integral in this context).
For each $t \geq 0$, we assign an bounded linear operator $Q(t)$ such that
This is indeed bounded since we have $\lVert Q(t)f \rVert_2 = \lVert f \rVert_2$ as the Lebesgue measure is translate-invariant. This is a left translation operator with a single step $t$.
Quasi-analytic Vectors and Hamburger Moment Problem (Operator Theory)
Analytic and quasi-analytic vectors
Guided by researches in function theory, operator theorists gave the analogue to quasi-analytic classes. Let $A$ be an operator in a Banach space $X$. $A$ is not necessarily bounded hence the domain $D(A)$ is not necessarily to be the whole space. We say $x \in X$ is a $C^\infty$ vector if $x \in \bigcap_{n \geq 1}D(A^n)$. This is quite intuitive if we consider the differential operator. A vector is analytic if the series
has a positive radius of convergence. Finally, we say $x$ is quasi-analytic for $A$ provided that
or equivalently its nondecreasing majorant. Interestingly, if $A$ is symmetric, then $\lVert{A^nx}\rVert$ is log convex.
Based on the density of quasi-analytic vectors, we have an interesting result.
(Theorem) Let $A$ be a symmetric operator in a Hilbert space $\mathscr{H}$. If the set of quasi-analytic vectors spans a dense subset, then $A$ is essentially self-adjoint.
This theorem can be considered as a corollary to the fundamental theorem of quasi-analytic classes, by applying suitable Banach space techniques in lieu.