Question

We define an inner product \(\star\) on the vector space \(V\) of continuous real- valued functions on the real interval \([-1,1]\) by \(f * g=\int_{-1}^1 f(y) g(y) \sqrt{1-y^2} d y\). (i) Convince yourself that \(\mathrm{i}\) is in fact an inner product. (ii) Compute the Gram-Schmidt orthogonal basis of \(f_0, f_1, f_2, f_3\), where \(f_i(x)=x^i\) for \(-1 \leq x \leq 1\). (The resulting polynomials are the monic associates of the first four Chebyshev polynomials of the second kind.).

Short answer

(i) Verify by showing positivity, linearity, and symmetry. (ii) Orthogonal basis: 1, x, x^2 - 1/2, x^3 - 3/4x.

Step-by-step solution

Verify Inner Product Properties - Positivity

To verify that \(\star \) is an inner product, we need to confirm it satisfies properties such as positivity. For any function \(f(y)\) in \(V\), the positivity condition is \(f \star f = \int_{-1}^1 f(y)^2 \sqrt{1-y^2} dy \geq 0\). Since \(\sqrt{1-y^2}\) is always non-negative and assuming \(f(y)^2\) is also non-negative, this integral will be non-negative. It equals zero if and only if \(f(y) = 0\) almost everywhere.

Verify Inner Product Properties - Linearity and Symmetry

Linearity in the first argument is satisfied if \((af + bg) \star h = a(f \star h) + b(g \star h)\) for all continuous functions \(f\), \(g\), and \(h\), and scalars \(a\), \(b\). This holds true based on properties of the integral. Symmetry is confirmed by showing that \(f \star g = g \star f\), which is true because the integral is commutative, i.e., \(\int f(y)g(y)\sqrt{1-y^2} dy = \int g(y)f(y)\sqrt{1-y^2} dy\). All properties are satisfied, confirming \(\star\) is an inner product.

Gram-Schmidt Process - Starting Vector

Start with the first polynomial, \(f_0(x) = 1\). It is already orthogonal since it's the only vector; normalization isn't needed because we'll express everything as monic polynomials. \(u_0 = f_0 = 1\).

Gram-Schmidt Process - Orthogonalizing f_1

For \(f_1(x) = x\), compute the projection onto \(u_0\):\[\text{proj}_{u_0}(f_1) = \frac{f_1 \star u_0}{u_0 \star u_0} u_0 = \frac{\int_{-1}^1 x \cdot 1 \cdot \sqrt{1-y^2} dy}{\int_{-1}^1 1^2 \cdot \sqrt{1-y^2} dy} \cdot 1 = 0\]So, \(u_1(x) = f_1(x) - 0 = x\).

Gram-Schmidt Process - Orthogonalizing f_2

For \(f_2(x) = x^2\), calculate the projections:\[\text{proj}_{u_0}(f_2) = \frac{f_2 \star u_0}{u_0 \star u_0} u_0 \quad \text{and}\quad \text{proj}_{u_1}(f_2) = \frac{f_2 \star u_1}{u_1 \star u_1} u_1\]Calculating these, we find \( \text{proj}_{u_0}(f_2) = \frac{1}{2} \) and \( \text{proj}_{u_1}(f_2) = 0 \). So, \( u_2(x) = x^2 - \frac{1}{2} \).

Gram-Schmidt Process - Orthogonalizing f_3

For \(f_3(x) = x^3\), calculate:\[\text{proj}_{u_0}(f_3) = \frac{f_3 \star u_0}{u_0 \star u_0} u_0, \quad \text{proj}_{u_1}(f_3) = \frac{f_3 \star u_1}{u_1 \star u_1} u_1, \quad \text{and}\,\text{proj}_{u_2}(f_3) = \frac{f_3 \star u_2}{u_2 \star u_2} u_2 \]Calculations yield \( \text{proj}_{u_0}(f_3) = 0 \), \( \text{proj}_{u_1}(f_3) = \frac{3}{4}x \), and \( \text{proj}_{u_2}(f_3) = 0 \). Thus, \( u_3(x) = x^3 - \frac{3}{4}x \).

Final Orthogonal Basis

The orthogonal basis, expressed as monic polynomials, is:\[ 1, \quad x, \quad x^2 - \frac{1}{2}, \quad x^3 - \frac{3}{4}x\] These correspond to the monic associates of the first four Chebyshev polynomials of the second kind.

Key concepts

Inner Product

An inner product is a way to multiply two functions or vectors to get a single number, promising some geometric meaning. For it to be considered an inner product, it must obey specific properties:

• Positivity: The inner product of a function with itself, say \(f \star f\), should always be non-negative. It equals zero if and only if the function itself is zero almost everywhere.
• Linearity: It should be linear in the first argument. That means if we have two functions \(f\) and \(g\), and scalars \(a\) and \(b\), the relation \((af + bg) \star h = a(f \star h) + b(g \star h)\) should hold.
• Symmetry: The inner product is symmetric if \(f \star g = g \star f\) for all functions \(f\) and \(g\) in the space. This means switching the order of the functions in the product doesn't change the result.

In this problem, the specified inner product \(f * g = \int_{-1}^1 f(y) g(y) \sqrt{1-y^2} d y\) satisfies these properties, making it a legitimate inner product for the vector space of continuous functions over the interval \([-1, 1]\). Using this, we can find orthogonal bases, geometric angles between functions, and more.

Chebyshev Polynomials

Chebyshev polynomials are a set of orthogonal polynomials that arise in approximation theory and are defined over specific interval \([-1, 1]\). For the Chebyshev polynomials of the second kind used here, they are often denoted as \(U_n(x)\). These polynomials have several significant properties and applications:

• Orthogonality: The integral of the product of any two distinct Chebyshev polynomials over \([-1, 1]\), weighted by \(\sqrt{1-y^2}\), is zero. This orthogonality is central in many numerical analysis techniques.
• Recurrence Relation: These polynomials can be generated through recurrence relations, offering a systematic way to construct higher-degree polynomials from lower-degree ones.

By applying the Gram-Schmidt process on the functions \(f_0, f_1, f_2, f_3\) (where \(f_i(x) = x^i\)), we derive the monic polynomials corresponding to the Chebyshev polynomials of the second kind. These polynomials are beneficial in various computational methods, particularly where minimization of errors over an interval is essential.

Vector Space

A vector space is a collection of elements known as vectors, which can be added together and multiplied by scalars, typically forming a systematic structure. The role of a vector space in mathematics is pivotal because it offers a framework for vector arithmetic.

• Function Spaces: In this context, the vector space \(V\) consists of all continuous real-valued functions defined on the interval \([-1, 1]\). Each continuous function acts like an infinite-dimensional vector in this space.
• Basis and Dimension: A basis in a vector space is a set of vectors that are linearly independent and span the entire space. The dimension is the number of vectors in any basis for the space. Using the Gram-Schmidt process, we find a specific basis that is orthogonal, which helps in simplifying many computations in the space.

Vector spaces provide the foundation for concepts like eigenvalues in linear algebra, which have applications in various fields like physics, engineering, computer science, and more. Understanding vector spaces helps to unlock the deeper elements of these subjects.