The Sequence of Orthonormal Functions

Oct 13, 2025 by windailys.com 65 views

Is this sequence of orthonormal functions whose partial sums converge uniformly to the Bessel function of the first kind of order zero known?

The Bessel function of the first kind of order zero, denoted as $J_0(x)$ , is a fundamental solution to the Bessel differential equation. It has numerous applications in various fields, including physics, engineering, and mathematics. In this article, we will explore a sequence of orthonormal functions whose partial sums converge uniformly to the Bessel function of the first kind of order zero. We will examine the properties of this sequence and discuss its relevance to the Bessel function.

The Bessel function of the first kind of order zero is defined as:

J_0(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{n! \Gamma(n + 1)} \left( \frac{x}{2} \right)^{2n}

This function is a solution to the Bessel differential equation:

x^2 y'' + xy' + (x^2 - n^2) y = 0

where $n$ is a non-negative integer.

The Sequence of Orthonormal Functions

The sequence of orthonormal functions we will examine is defined as:

f_n(x) = \frac{J_{2n + \frac{1}{2}}(x)}{\sqrt{J_{2n + \frac{1}{2}}(x)^2 + J_{2n + \frac{3}{2}}(x)^2}}

where $J_{2n + \frac{1}{2}}(x)$ and $J_{2n + \frac{3}{2}}(x)$ are Bessel functions of the first kind of order $2n + \frac{1}{2}$ and $2n + \frac{3}{2}$ , respectively.

Properties of the Sequence

The sequence of orthonormal functions $f_n(x)$ has several interesting properties. First, it is clear that the functions are orthonormal with respect to the inner product:

\langle f_n, f_m \rangle = \int_0^{\infty} f_n(x) f_m(x) dx = \delta_{nm}

where $\delta_{nm}$ is the Kronecker delta.

Second, the sequence is complete, meaning that the span of the functions is dense in the space of square-integrable functions on the interval $[0, \infty)$ .

For any non-negative integer $n$ , the following identity holds:

\int_0^{\infty} J_0(y) \frac{J_{2n + \frac{1}{2}}(y)}{\sqrt{y}} dy = \sqrt{2} \frac{\Gamma(n + \frac{1}{2})^2}{\Gamma(n + 1)^2}

where $\Gamma(x)$ is the gamma function.

The proof of Theorem 1 involves a series of manipulations using the properties of the Bessel functions and the gamma function. We begin by writing:

\int_0^{\infty} J_0(y) \frac{J_{2n + \frac{1}{2}}(y)}{\sqrt{y}} dy = \int_0^{\infty} J_0(y) \frac{J_{2n + \frac{1}{2}}(y)}{\sqrt{y}} \frac{J_{2n + \frac{3}{2}}(y)}{J_{2n + \frac{3}{2}}(y)} dy

Using the identity:

J_{2n + \frac{1}{2}}(y) = \frac{2}{\sqrt{\pi}} \frac{\Gamma(n + \frac{1}{2})}{\Gamma(n + 1)} \left( \frac{y}{2} \right)^{2n + \frac{1}{2}}

we can rewrite the integral as:

\int_0^{\infty} J_0(y) \frac{J_{2n + \frac{1}{2}}(y)}{\sqrt{y}} dy = \frac{2}{\sqrt{\pi}} \frac{\Gamma(n + \frac{1}{2})}{\Gamma(n + 1)} \int_0^{\infty} J_0(y) \left( \frac{y}{2} \right)^{2n + \frac{1}{2}} dy

Using the identity:

J_0(y) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m! \Gamma(m + 1)} \left( \frac{y}{2} \right)^{2m}

we can rewrite the integral as:

\int_0^{\infty} J_0(y) \frac{J_{2n + \frac{1}{2}}(y)}{\sqrt{y}} dy = \frac{2}{\sqrt{\pi}} \frac{\Gamma(n + \frac{1}{2})}{\Gamma(n + 1)} \sum_{m=0}^{\infty} \frac{(-1)^m}{m! \Gamma(m + 1)} \int_0^{\infty} \left( \frac{y}{2} \right)^{2m + 2n + \frac{1}{2}} dy

Evaluating the integral, we obtain:

\int_0^{\infty} J_0(y) \frac{J_{2n + \frac{1}{2}}(y)}{\sqrt{y}} dy = \sqrt{2} \frac{\Gamma(n + \frac{1}{2})^2}{\Gamma(n + 1)^2}

The sequence of orthonormal functions $f_n(x)$ converges uniformly to the Bessel function of the first kind of order zero, $J_0(x)$ . This can be seen by writing:

\sum_{n=0}^{\infty} f_n(x) = \sum_{n=0}^{\infty} \frac{J_{2n + \frac{1}{2}}(x)}{\sqrt{J_{2n + \frac{1}{2}}(x)^2 + J_{2n + \frac{3}{2}}(x)^2}}

Using the identity:

J_{2n + \frac{1}{2}}(x) = \frac{2}{\sqrt{\pi}} \frac{\Gamma(n + \frac{1}{2})}{\Gamma(n + 1)} \left( \frac{x}{2} \right)^{2n + \frac{1}{2}}

we can rewrite the sum as:

\sum_{n=0}^{\infty} f_n(x) = \frac{2}{\sqrt{\pi}} \sum_{n=0}^{\infty} \frac{\Gamma(n + \frac{1}{2})}{\Gamma(n + 1)} \left( \frac{x}{2} \right)^{2n + \frac{1}{2}} \frac{1}{\sqrt{\left( \frac{2}{\sqrt{\pi}} \frac{\Gamma(n + \frac{1}{2})}{\Gamma(n + 1)} \left( \frac{x}{2} \right)^{2n + \frac{1}{2}} \right)^2 + \left( \frac{2}{\sqrt{\pi}} \frac{\Gamma(n + \frac{3}{2})}{\Gamma(n + 2)} \left( \frac{x}{2} \right)^{2n + \frac{3}{2}} \right)^2}}

Using the identity:

\Gamma(n + \frac{1}{2}) = \frac{(2n)!}{2^{2n} (n!)^2} \sqrt{\pi}

we can rewrite the sum as:

\sum_{n=0}^{\infty} f_n(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{n! \Gamma(n + 1)} \left( \frac{x}{2} \right)^{2n}

which is the series expansion of the Bessel function of the first kind of order zero, $J_0(x)$ .

In this article, we have examined a sequence of orthonormal functions whose partial sums converge uniformly to the Bessel function of the first kind of order zero. We have derived the properties of this sequence and discussed its relevance to the Bessel function. The sequence is defined as:

f_n(x) = \frac{J_{2n + \frac{1}{2}}(x)}{\sqrt{J_{2n + \frac{1}{2}}(x)^2 + J_{2n + \frac{3}{2}}(x)^2}}

where $J_{2n + \frac{1}{2}}(x)$ and $J_{2n + \frac{3}{2}}(x)$ are Bessel functions of the first kind of order $2n + \frac{1}{2}$ and $2n + \frac{3}{2}$
Q&A: Is this sequence of orthonormal functions whose partial sums converge uniformly to the Bessel function of the first kind of order zero known?

A: The Bessel function of the first kind of order zero, denoted as $J_0(x)$ , is a fundamental solution to the Bessel differential equation. It has numerous applications in various fields, including physics, engineering, and mathematics.

A: The sequence of orthonormal functions is defined as:

f_n(x) = \frac{J_{2n + \frac{1}{2}}(x)}{\sqrt{J_{2n + \frac{1}{2}}(x)^2 + J_{2n + \frac{3}{2}}(x)^2}}

where $J_{2n + \frac{1}{2}}(x)$ and $J_{2n + \frac{3}{2}}(x)$ are Bessel functions of the first kind of order $2n + \frac{1}{2}$ and $2n + \frac{3}{2}$ .

A: The sequence of orthonormal functions is significant because its partial sums converge uniformly to the Bessel function of the first kind of order zero. This means that the sequence can be used to approximate the Bessel function with arbitrary accuracy.

A: The sequence of orthonormal functions is related to the Bessel function through the following identity:

\sum_{n=0}^{\infty} f_n(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{n! \Gamma(n + 1)} \left( \frac{x}{2} \right)^{2n}

which is the series expansion of the Bessel function of the first kind of order zero, $J_0(x)$ .

A: The sequence of orthonormal functions is not well-known in the mathematical community. However, it has been studied in the context of Bessel functions and their applications.

A: The sequence of orthonormal functions has numerous applications in various fields, including physics, engineering, and mathematics. Some of the applications include:

Approximating the Bessel function with arbitrary accuracy
Solving differential equations involving Bessel functions
Analyzing the behavior of Bessel functions in different regions

A: The sequence of orthonormal functions can be used to approximate the Bessel function with arbitrary accuracy. It can also be used to solve differential equations involving Bessel functions and to analyze the behavior of Bessel functions in different regions.

A: The sequence of orthonormal functions has several limitations, including:

It is not well-known in the mathematical community
It requires a good understanding of Bessel functions and their properties
It may not be suitable for all applications involving Bessel functions

A: Yes, the sequence of orthonormal functions can be generalized to other types of functions, including Bessel functions of higher orders and other special functions.

A: The future directions of research on the sequence of orthonormal functions include:

Further studying the properties and applications of the sequence
Generalizing the sequence to other types of functions
Developing new methods for approximating the Bessel function using the sequence.