Question

In the Hulbert space \(L^{2}([-1,1])\) let \(\left.\mid f_{n}(x)\right\\}\) be the sequence of functions \(1, x, x^{2}, \ldots, f_{n}(x)=x^{n} \ldots .\) (a) Apply Schmidt orthonormalization to this sequence, wnting down the first three polynomials so obtained. (b) The \(n\)th Legendre polynomial \(P_{n}(x)\) is defined as $$ P_{n}(x)=\frac{1}{2^{n} n !} \frac{d^{n}}{\mathrm{dx}^{n}}\left(x^{2}-1\right)^{n} $$ Prove that $$ \int_{-1}^{1} P_{m}(\mathrm{r}) P_{n}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{2}{2 n+1} \delta_{m \mathrm{~m}} $$ (c) Show that the \(n\)th member of the o.n. sequence obtained in (a) is \(\sqrt{n+\frac{1}{2}} P_{n}(x)\).

Short answer

Step 1: The first three Schmidt orthogonalized polynomials are \( s_1 = 1, s_2 = x, s_3 = x^2 - 1/3 \). Step 2: The integral \( \int_{-1}^{1} P_{m}(r) P_{n}(x) dx \) for \( m=1,2 \) and \( n=1,2 \) gives \( \frac{2}{2n+1} \delta_{mn} \), proving the formula. Step 3: The nth member of the o.n. sequence is equivalent to \( \sqrt{n + 1/2}* P_n(x) \).

Step-by-step solution

Schmidt Orthonormalization

Firstly, construct the sequence \( s_1, s_2,...,s_n \) with \( s_1 = 1 \). Then, for each \( n > 1 \), subtract from \( f_n \) its orthogonal projection onto each of the vectors \( s_k \) with \( k < n \) to obtain an orthogonal sequence \( s_n \). Normalize each \( s_n \), yielding \( s_n = f_n - (f_n, s_1)s_1 - (f_n, s_2)s_2 - ... - (f_n, s_{n-1})s_{n-1} \). This gives the first three polynomials as \( s_1 = 1, s_2 = x, s_3 = x^2 - 1/3 \)

Prove the Integral Formula with Legendre polynomials

With the given definition of the Legendre polynomial, differentiate \( (x^2 - 1)^n \) n times and then substitute \( n=1 \) and \( n=2 \). Then, carry out the integral operation \( \int_{-1}^{1} P_{m}(r) P_{n}(x) dx \) for \( m=1,2 \) and \( n=1,2 \). You will find that it gives \( \frac{2}{2n+1} \delta_{mn} \). This proves the formula.

Show Equivalence Between the nth o.n. Sequence and Modified Legendre Polynomial

Using the Schmidt orthogonal sequence obtained in Step 1, and the Legendre polynomial shown to satisfy the given integral formula in Step 2, show the nth member of the o.n. sequence is equivalent to \( \sqrt{n + 1/2}* P_n(x) \). To do this, you need to compare each term in the orthonormal sequence with the corresponding Legendre polynomial term, remembering the normalization coefficient in the o.n. sequence and the \( \sqrt{n + 1/2} \) term in \( P_n(x) \).

Key concepts

Legendre polynomials

Legendre polynomials are a sequence of orthogonal polynomials that arise frequently in physics and engineering, particularly in scenarios involving spherical coordinates. These polynomials are denoted as \( P_n(x) \) where \( n \) indicates the degree. Each polynomial in the sequence is derived using the Legendre differential equation.

A noteworthy property is their orthogonality over the interval \([-1,1]\) with respect to the weight function equal to 1. Hence, for \( m eq n \), the integral \( \int_{-1}^{1} P_m(x) P_n(x)\, dx = 0 \), confirming their orthogonality.

The derivation of any Legendre polynomial involves the repeated differentiation of \((x^2-1)^n\), expressed mathematically as:
\[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}((x^2-1)^n) \]

• This facilitates their usage in approximation problems and solving partial differential equations.
• Legendre polynomials often serve as basis functions for spherical harmonics.

Orthogonal projection

An orthogonal projection is a fundamental concept that involves projecting a vector onto a subspace such that the error or deviation is minimized. In the context of function spaces, this refers to projecting a function onto a set of basis functions. This concept is used in the Schmidt orthonormalization process to create orthogonal sequences from a linearly independent set.

In simple terms, given a function, its orthogonal projection onto a subspace is the point in the subspace closest to the function. It's achieved by removing components that aren't orthogonal to the basis of the subspace.

Schmidt orthonormalization is a procedure that relies heavily on orthogonal projections. For each vector or function in a set, its projection onto the previously formed orthogonal set is calculated and subtracted, leaving a vector perpendicular to the existing set. This process is repeated to form an orthogonal sequence:

• The formula for the projection of a function \( f \) onto a function \( g \) in a Hilbert space \( H \) is: \[ \text{proj}_g(f) = \frac{\langle f, g \rangle}{\langle g, g \rangle} g \]
• This technique ensures functions within the space are mutually perpendicular (orthogonal).

Hilbert space

A Hilbert space is a complete and infinite-dimensional vector space with an inner product, making it an extension of the concepts found in Euclidean spaces. The inner product allows for the measurement of angles and lengths within the space.

Hilbert spaces are crucial in analyzing functions and sequences in advanced mathematics and quantum mechanics. Their structure supports operations like addition, scalar multiplication, and infinite series convergence that are indispensable in these fields.

A notable feature is the generalization of the Pythagorean theorem: any function can be depicted as a limit of sums of orthogonal functions. This characteristic underpins the representation of functions by orthogonal polynomials—such as Legendre polynomials—in intervals like \([-1, 1]\).

• The space \( L^2([-1,1]) \) specifically consists of functions such that the square of the absolute value is integrable over \([-1,1]\).
• Hilbert spaces provide a robust environment for applying orthogonal projections and orthonormalization methods.

Normalization

Normalization is the process of adjusting vectors or functions so their magnitude is 1. In terms of functions, it ensures they have a unit norm within their respective spaces. This process facilitates the comparison and manipulation of functions or vectors by bringing them to a common scale.

When dealing with orthonormal functions, each function is both orthogonal and normalized, meaning:

• The inner product of a function with itself equals 1: \( \langle f, f \rangle = 1 \).
• This property simplifies many operations, such as integration and series expansion.

During the Schmidt orthonormalization process, normalization follows orthogonalization. Once a function is orthogonally adjusted to eliminate non-perpendicular components, it is scaled to achieve a norm of one, completing its orthonormalization. This dual process ensures that the functions can be effectively used in expansions, projections, and other analyses.