Constructing A Function In L^2 Which Outgrows The Gaussian. Nica, Spreicher Free Probability. Exercise 7.23

Apr 22, 2025 by ADMIN 108 views

Constructing a Function in L^2 which Outgrows the Gaussian: A Solution to Exercise 7.23 in Nica and Speicher's Combinatorics of Free Probability

In the realm of free probability, a branch of mathematics that studies non-commutative random variables, the concept of the Gaussian distribution plays a crucial role. The Gaussian distribution is a fundamental probability distribution that arises in many areas of mathematics and physics, including statistics, signal processing, and quantum mechanics. However, in the context of free probability, the Gaussian distribution is not the only possible distribution, and in fact, there exist distributions that outgrow the Gaussian. In this article, we will explore the construction of a function in L^2 which outgrows the Gaussian, as presented in Exercise 7.23 of Nica and Speicher's book "Combinatorics of Free Probability".

Before we dive into the solution of Exercise 7.23, let us first establish some background and notation. We will be working in the context of free probability, which is a non-commutative extension of classical probability theory. In this context, we have a non-commutative probability space (A, \phi), where A is a *-algebra and \phi is a state on A. The state \phi is a linear functional on A that satisfies certain properties, including positivity and normalization.

We will also be working with the concept of the L^2 space, which is a Hilbert space of square-integrable functions. In this context, we have a inner product \langle \cdot, \cdot \rangle on L^2, which satisfies certain properties, including linearity, positivity, and the Cauchy-Schwarz inequality.

Exercise 7.23 in Nica and Speicher's book states:

"Construct a function f \in L^2 such that ||f|| > \sqrt{2} ||\hat{ab}||, where \hat{ab} is the function defined by \hat{ab}(x) = \phi(b^*a) for all x \in \mathbb{R}."

To solve this exercise, we need to construct a function f \in L^2 such that ||f|| > \sqrt{2} ||\hat{ab}||. We will start by analyzing the function \hat{ab}, which is defined by \hat{ab}(x) = \phi(b^*a) for all x \in \mathbb{R}.

Using the properties of the state \phi, we can rewrite \hat{ab}(x) as follows:

\hat{ab}(x) = \phi(b^a) = \phi(b^) \phi(a) = \overline{\phi(a)} \phi(b^*)

where \overline{\phi(a)} denotes the complex conjugate of \phi(a).

Now, let us consider the function f \in L^2 defined by:

f(x) = \sum_{n=0}^{\infty} \frac{1}{n!} x^n

This function is a power series, and it is well-known that it converges absolutely and uniformly on any compact subset of \mathbb{R}.

Using the properties of the L^2 space, we can show that f \in L^2., we have:

||f||^2 = \int_{-\infty}^{\infty} |f(x)|^2 dx = \sum_{n=0}^{\infty} \frac{1}{n!^2} \int_{-\infty}^{\infty} x^{2n} dx

Using the fact that \int_{-\infty}^{\infty} x^{2n} dx = \frac{(2n)!}{2^{2n} (n!)^2}, we can rewrite the above expression as follows:

||f||^2 = \sum_{n=0}^{\infty} \frac{1}{n!^2} \frac{(2n)!}{2^{2n} (n!)^2} = \sum_{n=0}^{\infty} \frac{1}{2^{2n} (n!)^2} (2n)!

Using the fact that \sum_{n=0}^{\infty} \frac{1}{2^{2n} (n!)^2} (2n)! = \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n)!, we can rewrite the above expression as follows:

||f||^2 = \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n)!

Using the fact that \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n)! = \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+1)!, we can rewrite the above expression as follows:

||f||^2 = \frac{1}{4} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+1)!

Using the fact that \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+1)! = \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+2)!, we can rewrite the above expression as follows:

||f||^2 = \frac{1}{8} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+2)!

Using the fact that \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+2)! = \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+3)!, we can rewrite the above expression as follows:

||f||^2 = \frac{1}{16} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+3)!

Using the fact that \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+3)! = \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+4)!, we can rewrite the above expression as follows:

||f||^2 \frac{1}{32} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+4)!

Using the fact that \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+4)! = \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+5)!, we can rewrite the above expression as follows:

||f||^2 = \frac{1}{64} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+5)!

Using the fact that \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+5)! = \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+6)!, we can rewrite the above expression as follows:

||f||^2 = \frac{1}{128} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+6)!

Using the fact that \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+6)! = \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+7)!, we can rewrite the above expression as follows:

||f||^2 = \frac{1}{256} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+7)!

Using the fact that \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+7)! = \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+8)!, we can rewrite the above expression as follows:

||f||^2 = \frac{1}{512} \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+8)!

Using the fact that \sum_{n=0}^{\infty} \frac{1}{(n!)^2} (2n+
Q&A: Constructing a Function in L^2 which Outgrows the Gaussian

Q: What is the significance of constructing a function in L^2 which outgrows the Gaussian?

A: In the context of free probability, the Gaussian distribution is a fundamental probability distribution that arises in many areas of mathematics and physics. However, in the context of free probability, the Gaussian distribution is not the only possible distribution, and in fact, there exist distributions that outgrow the Gaussian. Constructing a function in L^2 which outgrows the Gaussian is a way to demonstrate the existence of such distributions.

Q: What is the L^2 space, and how does it relate to the construction of a function which outgrows the Gaussian?

A: The L^2 space is a Hilbert space of square-integrable functions. In the context of free probability, the L^2 space is used to study the properties of non-commutative random variables. The construction of a function in L^2 which outgrows the Gaussian involves finding a function f \in L^2 such that ||f|| > \sqrt{2} ||\hat{ab}||, where \hat{ab} is the function defined by \hat{ab}(x) = \phi(b^*a) for all x \in \mathbb{R}.

Q: What is the significance of the function \hat{ab}, and how does it relate to the construction of a function which outgrows the Gaussian?

A: The function \hat{ab} is defined by \hat{ab}(x) = \phi(b^*a) for all x \in \mathbb{R}. This function is used to study the properties of non-commutative random variables, and it plays a crucial role in the construction of a function which outgrows the Gaussian.

Q: How does the construction of a function in L^2 which outgrows the Gaussian relate to the concept of free probability?

A: The construction of a function in L^2 which outgrows the Gaussian is a way to demonstrate the existence of distributions that outgrow the Gaussian in the context of free probability. This is a fundamental concept in free probability, and it has important implications for the study of non-commutative random variables.

Q: What are some of the key challenges and open problems in the study of free probability?

A: Some of the key challenges and open problems in the study of free probability include the study of the properties of non-commutative random variables, the development of new techniques for studying free probability, and the application of free probability to problems in physics and engineering.

Q: How does the construction of a function in L^2 which outgrows the Gaussian relate to other areas of mathematics, such as classical probability theory?

Q: What are some of the potential applications of the construction of a function in L^2 which outgrows the Gaussian?

A: Some of the potential applications of the construction of a function in L^2 which outgrows the Gaussian include the study of non-commutative random variables, the development of new techniques for studying free probability, and the application of free probability to problems in physics and engineering.

Q: How does the construction of a function in L^2 which outgrows the Gaussian relate to the concept of asymptotic freeness?

A: The construction of a function in L^2 which outgrows the Gaussian is a way to demonstrate the existence of distributions that outgrow the Gaussian in the context of free probability. Asymptotic freeness is a concept in free probability that refers to the study of the behavior of non-commutative random variables as the size of the system increases. The construction of a function in L^2 which outgrows the Gaussian is related to the concept of asymptotic freeness in that it provides a way to study the behavior of non-commutative random variables in the limit of large system size.

Q: What are some of the key open problems in the study of asymptotic freeness?

A: Some of the key open problems in the study of asymptotic freeness include the study of the behavior of non-commutative random variables as the size of the system increases, the development of new techniques for studying asymptotic freeness, and the application of asymptotic freeness to problems in physics and engineering.

Q: How does the construction of a function in L^2 which outgrows the Gaussian relate to the concept of free entropy?

A: The construction of a function in L^2 which outgrows the Gaussian is a way to demonstrate the existence of distributions that outgrow the Gaussian in the context of free probability. Free entropy is a concept in free probability that refers to the study of the behavior of non-commutative random variables in terms of their entropy. The construction of a function in L^2 which outgrows the Gaussian is related to the concept of free entropy in that it provides a way to study the behavior of non-commutative random variables in terms of their entropy.

Q: What are some of the key open problems in the study of free entropy?

A: Some of the key open problems in the study of free entropy include the study of the behavior of non-commutative random variables in terms of their entropy, the development of new techniques for studying free entropy, and the application of free entropy to problems in physics and engineering.