Question

Use the Gram-Schmidt process to find the first four orthogonal polynols satisfying the following: a. Interval: \((-\infty, \infty)\) Weight Function: \(e^{-x^{2}}\). b. Interval: \((0, \infty)\) Weight Function: \(e^{-x}\).

Short answer

The first four orthogonal polynomials satisfying the conditions for part (a) are \(P_{0}(x) = 1\), \(P_{1}(x) = x\), \(P_{2}(x) = x^2 - 1\), and \(P_{3}(x) = x^3 - 3x\). For part (b), those are \(Q_{0}(x) = 1\), \(Q_{1}(x) = x - 1\), \(Q_{2}(x) = x^2 - 4x + 2\), and \(Q_{3}(x) = x^3 - 9x^2 + 18x - 6\).

Step-by-step solution

Part (a): Gram-Schmidt process with interval: (-∞, ∞) and weight function: \(e^{-x^{2}}\)

Consider the first four monomials \(1,x,x^{2},x^{3}\). Use the Gram Schmidt Process:1. The first polynomial \(P_{0}(x) = 1\)2. Next, use the idea of Gram Schmidt Process to find \(P_{1}(x), P_{2}(x), P_{3}(x)\). i. Orthogonalize: Subtract off the part of \(x, x^{2}, x^{3}\) that is in the direction of \(P_{0}(x)\) ii. Normalize: Divide by its norm.3. Compute them using integrals, we get: \(P_{1}(x) = x\), \(P_{2}(x) = x^2 - 1\), \(P_{3}(x) = x^3 - 3x\)

Part (b): Gram-Schmidt process with interval (0, ∞) and weight function \(e^{-x}\)

Again, consider the first four monomials \(1, x, x^{2}, x^{3}\). 1. The first polynomial \(Q_{0}(x) = 1\)2. Then apply Gram Schmidt Process to find \(Q_{1}(x),Q_{2}(x),Q_{3}(x)\) following the same process as in part (a)3. Compute them using integrals, we get: \(Q_{1}(x) = x - 1\), \(Q_{2}(x) = x^2 - 4x + 2\), \(Q_{3}(x) = x^3 - 9x^2 + 18x - 6\)

Key concepts

Orthogonal Polynomials

Orthogonal polynomials are a cornerstone concept in many mathematical applications, especially in numerical analysis and approximation theory. The term "orthogonal" means that these polynomials are perpendicular to each other in a function space, similar to how perpendicular lines work in geometry. This concept can be imagined in a space of functions where the "dot product" of two functions is described by an integral operation.
This orthogonality is computed over a specific interval with respect to a weight function. An important property is that the inner product, or overlap, of any two different orthogonal polynomials is zero:

• This property helps in simplifying many problems, such as differential equations in physics, where orthogonal functions serve as a base set.
• In the context of polynomials, orthogonality ensures that the functions do not interfere with each other.

Orthogonal polynomial systems, like Legendre or Laguerre polynomials, have special significance depending on the weight function chosen and the interval considered. They serve as vital tools in areas like signal processing and statistics.

Weight Function

The weight function plays a critical role in the Gram-Schmidt process for generating orthogonal polynomials. It defines the "importance" or "weight" of different parts of the interval considered when computing the orthogonality of polynomials. In simpler terms, it's like giving different preferences or importance to different regions along the x-axis.
Here are some key points about weight functions:

• They affect how the integrals are computed when generating orthogonal polynomials using the Gram-Schmidt process.
• Not all weight functions are created equal; they should be positive in the interval to ensure proper function space weighting.
• Common weight functions include exponential functions that correspond to well-known polynomial solutions.

In the exercise, functions like \( e^{-x^2} \) and \( e^{-x} \) serve as weight functions corresponding to Hermite and Laguerre polynomials, respectively.

Continuous Functions

In mathematics, a continuous function is a type of function where small changes in the input result in small changes in the output. This concept is crucial when dealing with integrals and differential equations because it ensures stability and predictability of solutions.
Key aspects of continuous functions include:

• Their smooth nature, which means they have no jumps or breaks in their graph.
• Well-suited for modeling natural phenomena because they maintain consistency over an interval.
• Essentials for mathematical operations like integration and differentiation, which are fundamental in the development of orthogonal polynomials.

Continuous functions guarantee that the integral operations involved in the calculation of orthogonal polynomials are possible and meaningful, as seen in the exercise.

Mathematical Physics

Mathematical physics involves the development and use of mathematical methods to solve problems in physics. Orthogonal polynomials and tools like the Gram-Schmidt process are frequently used in this field. They help in areas involving waves, quantum mechanics, and many more applications due to their powerful approximation capabilities.
Some important insights into the role of mathematical physics include:

• Orthogonal polynomials are often used to solve partial differential equations found in physics.
• The choice of polynomial systems, influenced by weight functions, can tailor solutions to specific conditions or symmetries in a problem.
• They provide a way to decompose complex physical systems into simpler parts that are more manageable.

In our exercise context, the polynomials derived through the Gram-Schmidt process offer valuable ways to explore physical systems, leveraging their unique properties for modeling and computation.