Mathematics. Articles in English (et en français dans le futur).

Picard's Little Theorem and Twice-Punctured Plane

We show that the range of a non-constant entire function's range cannot be a twice-punctured plane.
Picard's Little Theorem and Twice-Punctured Plane

The Fourier Transform of exp(-cx^2) and Its Convolution

We develop two almost straightforward way to compute the Fourier transform of $\exp(-cx^2)$, in the sense that any contour integration and the calculus of residues are not required at all. The first cool approach enables us to think about these elementary concepts much deeper, so I highly recommend to study this approach as long as you are familiar with ODE of first order.
The Fourier Transform of exp(-cx^2) and Its Convolution

A Detailed Proof of the Riemann Mapping Theorem

We offer a detailed proof of the Riemann mapping theorem, which states that every proper simply connected region is conformally equivalent to the open unit disc.
A Detailed Proof of the Riemann Mapping Theorem

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 Banach Algebra of Borel Measures on Euclidean Space

Elementary Properties of Cesàro Operator in L^2

We study the average of sum, in the sense of integral.
Elementary Properties of Cesàro Operator in L^2

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 $t$.

Left Shift Semigroup and Its Infinitesimal Generator

Quasi-analytic Classes

We study the concept of quasi-analytic functions, which are quite close to being analytic.
Quasi-analytic Classes

Several ways to prove Hardy's inequality

Several ways to prove Hardy's inequality

A Continuous Function Sending L^p Functions to L^1

A Continuous Function Sending L^p Functions to L^1

A proof of the ordinary Gleason-Kahane-Żelazko theorem for complex functionals

A proof of the ordinary Gleason-Kahane-Żelazko theorem for complex functionals