NoteNextra-origin/content/Math401/Extending_thesis/Math401_S4.md

Math 401, Fall 2025: Thesis notes, S4, Bargmann space

Bargmann space (original)

Also known as Segal-Bargmann space or Bargmann-Fock space.

It is the space of holomorphic functions that is square-integrable over the complex plane.

Section belows use Remarks on a Hilbert Space of Analytic Functions as the reference.

A family of Hilbert spaces, \mathfrak{F}_n(n=1,2,3,\cdots), is defined as follows:

The element of \mathfrak{F}_n are entire analytic functions in complex Euclidean space \mathbb{C}^n. f:\mathbb{C}^n\to \mathbb{C}\in \mathfrak{F}_n

Let f,g\in \mathfrak{F}_n. The inner product is defined by

\langle f,g\rangle=\int_{\mathbb{C}^n} \overline{f(z)}g(z) d\mu_n(z)

Let z_k=x_k+iy_k be the complex coordinates of z\in \mathbb{C}^n.

The measure \mu_n is the defined by

d\mu_n(z)=\pi^{-n}\exp(-\sum_{i=1}^n |z_i|^2)\prod_{k=1}^n dx_k dy_k

Example

For n=2,

\mathfrak{F}_2=\text{ space of entire analytic functions on } \mathbb{C}^2\to \mathbb{C}

\langle f,g\rangle=\int_{\mathbb{C}^2} \overline{f(z)}g(z) d\mu(z),z=(z_1,z_2)

d\mu_2(z)=\frac{1}{\pi^2}\exp(-|z|^2)dx_1 dy_1 dx_2 dy_2

so that f belongs to \mathfrak{F}_n if and only if \langle f,f\rangle<\infty.

This is absolutely terrible early texts, we will try to formulate it in a more modern way.

The section belows are from the lecture notes Holomorphic method in analysis and mathematical physics

Complex function spaces

Holomorphic spaces

Let U be a non-empty open set in \mathbb{C}^d. Let \mathcal{H}(U) be the space of holomorphic (or analytic) functions on U.

Let f\in \mathcal{H}(U), note that by definition of holomorphic on several complex variables, f is continuous and holomorphic in each variable with the other variables fixed.

Let \alpha be a continuous, strictly positive function on U.

\mathcal{H}L^2(U,\alpha)=\left\{F\in \mathcal{H}(U): \int_U |F(z)|^2 \alpha(z) d\mu(z)<\infty\right\},

where \mu is the Lebesgue measure on \mathbb{C}^d=\mathbb{R}^{2d}.

Theorem of holomorphic spaces

1. For all z\in U, there exists a constant c_z such that

   
   |F(z)|^2\le c_z \|F\|^2_{L^2(U,\alpha)}
   

   for all F\in \mathcal{H}L^2(U,\alpha).
2. \mathcal{H}L^2(U,\alpha) is a closed subspace of L^2(U,\alpha), and therefore a Hilbert space.

Proof

First we check part 1.

Let z=(z_1,z_2,\cdots,z_d)\in U, z_k\in \mathbb{C}. Let P_s(z) be the "polydisk"of radius s centered at z defined as

P_s(z)=\{v\in \mathbb{C}^d: |v_k-z_k|<s, k=1,2,\cdots,d\}

If z\in U, we cha choose s small enough such that P_s(z)\subset U.

Tip

[1.] states that point-wise evaluation of F on U is continuous. That is, for each z\in U, the map \varphi: \mathcal{H}L^2(U,\alpha)\to \mathbb{C} that takes F\in \mathcal{H}L^2(U,\alpha) to F(z) is a continuous linear functional on \mathcal{H}L^2(U,\alpha). This is false for ordinary non-holomorphic functions, e.g. L^2 spaces.

Reproducing kernel

Let \mathcal{H}L^2(U,\alpha) be a holomorphic space. The reproducing kernel of \mathcal{H}L^2(U,\alpha) is a function K:U\times U\to \mathbb{C}, K(z,w),z,w\in U with the following properties:

1. K(z,w) is holomorphic in z and anti-holomorphic in w.

   
   K(w,z)=\overline{K(z,w)}
   

2. For each fixed z\in U, K(z,w) is a square integrable d\alpha(w). For all F\in \mathcal{H}L^2(U,\alpha),

   
   F(z)=\int_U K(z,w)F(w) \alpha(w) dw
   

3. If F\in L^2(U,\alpha), let PF denote the orthogonal projection of F onto closed subspace \mathcal{H}L^2(U,\alpha). Then

   
   PF(z)=\int_U K(z,w)F(w) \alpha(w) dw
   

4. For all z,u\in U,

   
   \int_U K(z,w)K(w,u) \alpha(w) dw=K(z,u)
   

5. For all z\in U,

   
   |F(z)|^2\leq K(z,z) \|F\|^2_{L^2(U,\alpha)}
   

Bargmann space

The Bargmann spaces are the holomorphic spaces

\mathcal{H}L^2(\mathbb{C}^d,\mu_t)

where

\mu_t(z)=\text{ CONTINUE HERE }