Question

Let \(\phi_{1}, \phi_{2}, \ldots, \phi_{n}, \ldots\) be the normalized eigenfunctions of the Sturm-Liouville problem \((11),(12) .\) Show that if \(a_{n}\) is the \(n\) th Fourier coefficient of a square integrable function \(f,\) then \(\lim _{n \rightarrow \infty} a_{n}=0\) Hint: Use Bessel's inequality, Problem \(9(b)\).

Short answer

#Short Answer# For any square-integrable function f(x) with respect to the orthonormal set of functions φ_n(x), the nth Fourier coefficient is defined as a_n = ∫ f(x) φ_n(x) dx. Bessel's inequality states that the sum of the squared absolute values of the Fourier coefficients converges to a finite number: ∑|a_n|^2 ≤ ∫|f(x)|² dx. Since the series converges, the sequence of partial sums converges as well. As N approaches infinity, the limit of |a_N|^2 approaches zero. Therefore, taking the square root of both sides, we find that the limit of |a_n| as n approaches infinity is equal to zero. Consequently, it can be concluded that the limit of a_n as n approaches infinity is zero.

Step-by-step solution

Definition of the \(n\)th Fourier coefficient

The \(n\)th Fourier coefficient \(a_n\) of a square-integrable function \(f(x)\) with respect to an orthonormal set of functions \(\phi_n(x)\), for \(n=1,2,\ldots\), is defined by: \[a_n = \int f(x) \phi_n(x) dx\]

Bessel's inequality

Bessel's inequality states that for any square-integrable function \(f(x)\) and set of orthonormal functions \(\phi_n(x)\), the following inequality holds: \[\sum_{n=1}^{\infty} |a_n|^2 \leq \int |f(x)|^2 dx\] for any \(n \in \mathbb{N}\), where \(|a_n|^2\) is the squared absolute value of the \(n\)th Fourier coefficient.

Prove that \(\lim_{n\rightarrow \infty} a_n = 0\)

Since the sum of the squared absolute values of the Fourier coefficients converges, i.e., \(\sum_{n=1}^{\infty} |a_n|^2 < \infty\), we can use Bessel's inequality to show that the limit of \(a_n\) as \(n\) approaches infinity is equal to \(0\). Let's consider the partial sum: \[S_N = \sum_{n=1}^{N} |a_n|^2\] According to Bessel's inequality, for each positive integer \(N\), \(S_N \leq \int |f(x)|^2 dx\). As this inequality holds for all values of \(N \in \mathbb{N}\), it also holds in the limit as \(N\rightarrow \infty\): \[\lim_{N\rightarrow \infty} S_N \leq \int |f(x)|^2 dx\] Since the series \(\sum_{n=1}^{\infty} |a_n|^2\) converges, the sequence of partial sums \(S_N\) converges as well. Hence, \[\lim_{N\rightarrow \infty} |a_N|^2 = \lim_{N\rightarrow\infty} (S_N - S_{N - 1}) = 0\] Taking the square root of both sides, we obtain: \[\lim_{n\rightarrow \infty} |a_n| = \sqrt{\lim_{n\rightarrow\infty} |a_n|^2} = 0\] Finally, since \(|a_n|\) converges to \(0\), we conclude that: \[\lim_{n\rightarrow \infty} a_n = 0\]

Key concepts

Fourier Coefficients

The concept of Fourier coefficients is central to understanding how complex, periodic functions can be represented as a sum of simpler sine and cosine functions. When we speak of Fourier coefficients in the context of a Sturm-Liouville problem, we refer to the constants that multiply the series of functions—typically sinusoids in classical Fourier analysis or eigenfunctions in the Sturm-Liouville case—that collectively approximate another function.

The method works by projecting a square-integrable function onto a basis of orthonormal functions. In the case of the Sturm-Liouville problem, these projections are onto the set of eigenfunctions of the problem, denoted as \(\phi_n\). This projection is quantified by calculating the integral of the product of the function and each eigenfunction in the set, which gives us the nth Fourier coefficient, \(a_n\), directly related to the amplitude of each function in the series representing our original function.

In more formal terms, to find the nth Fourier coefficient, we calculate:
\[a_n = \int f(x) \phi_n(x) dx\]
Understanding Fourier coefficients is crucial because it allows us to deconstruct a function into its fundamental components, which is invaluable for solving differential equations and analyzing signals.

Bessel's Inequality

Bessel's inequality is a fundamental result in the field of Fourier analysis and functional analysis that places an upper bound on the sum of the squares of the Fourier coefficients when the series of orthonormal functions is used to approximate a function. The main takeaway from Bessel's inequality is that the energy of a function, captured by the square of its defining series, cannot exceed the function's own energy.

In mathematical terms, for a set of orthonormal functions \(\phi_n(x)\), Bessel's inequality is given by:
\[\sum_{n=1}^{\infty} |a_n|^2 \leq \int |f(x)|^2 dx\]
Why is this inequality important? It ensures that when we work with projections onto orthonormal bases, like those of the Sturm-Liouville problem, the process does not create any 'extra' energy or information not present in the original function. This inequality is also a stepping stone to proving more complex results like Parseval's identity and helps us to establish the convergence of Fourier coefficients, integral to solving Sturm-Liouville and other boundary value problems.

Orthonormal Functions

Orthonormal functions are the building blocks of many areas of mathematical analysis, particularly when working with series solutions to differential equations. When a set of functions is orthonormal, each pair of different functions is orthogonal—meaning their inner product or integral of their product over a specified range is zero—and each function is normalized—its inner product with itself is equal to one.

This concept is more than a mathematical curiosity; it has practical implications. For instance, the idea of orthonormality ensures that the functions used in series expansions like the Fourier series are independent of one another, which simplifies the computation of coefficients. Because these functions do not overlap, they can effectively represent different features or frequencies of the function being analyzed.

Mathematically, when we say that the functions \(\phi_n(x)\) are orthonormal, we mean:

\[\int \phi_m(x) \phi_n(x) dx = \delta_{mn}\]
The Kronecker delta, \(\delta_{mn}\), is 1 if \(m=n\) and 0 otherwise. The beautiful simplicity of orthonormal functions profoundly affects how we approach problems like the Sturm-Liouville problem, making it possible to isolate and solve for individual terms in a complex series.

Eigenfunctions

In the Sturm-Liouville problem, eigenfunctions play the role of the special set of functions with respect to which we express our solution. An eigenfunction of a Sturm-Liouville problem is a non-trivial solution of a differential equation that also satisfies certain boundary conditions associated with the problem.

Eigenfunctions are critical because they form an orthonormal basis that we can use to expand other functions—in this case, a square-integrable function we're studying. This relates back to the Fourier coefficients because each coefficient represents the 'weight' of the corresponding eigenfunction in the overall composition of the original function.

The power of eigenfunctions lies in their ability to diagonalize linear operators, which means that problems involving these operators become significantly easier to handle. Further, since they satisfy both a differential equation and boundary conditions, they carry a wealth of information about the behaviour of solutions not only at every point in a domain but also at its edges, which is extremely useful in both pure and applied sciences.

In the setting of the Sturm-Liouville problem, finding these eigenfunctions is often the primary goal, as they grant us the tools we need to expand and thus analyze the behavior of more complex functions under the scope of the problem.