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.

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.

We study the average of sum, in the sense of integral.

Left shift operatorThroughout 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 \lVert f \rVert_2^2=\int_{-\infty}^{\infty}|f(t)|^2dm(t)

Analytic and quasi-analytic vectorsGuided 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 \sum_{n=0}^{\infty}\lVert{A^n x}\rVert\frac{t^n}{n!}has a positive radius of convergence. Finally, we say $x$ is quasi-analytic for $A$ provided that \sum_{n=0}^{\infty}\left(\frac{1}{\lVert A^n x \rVert}\right)^{1/n} = \inftyor 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.

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