Question

Make the three polynomial functions \(a_{0}, a_{1}+\) \(b_{1} x,\) and \(a_{2}+b_{2} x+c_{2} x^{2}\) orthonormal in the interval \(-1 \leq x \leq+1\) by determining appropriate values for the constants \(a_{0}, a_{1}, b_{1}, a_{2}, b_{2},\) and \(c_{2}\)

Short answer

The short answer to the question is: To make the three polynomial functions orthonormal in the given interval, we need to follow these steps: 1. Compute the inner products of any pair of the three functions, ensuring that their inner product is zero. 2. Compute the normalization conditions for each polynomial function so that their length (Euclidean norm) is 1. 3. Solve the system of equations obtained from the inner product and normalization conditions to find the appropriate values for the constants \(a_0\), \(a_1\), \(b_1\), \(a_2\), \(b_2\), and \(c_2\). By finding these appropriate values, the polynomial functions will be orthonormal in the interval \([-1, 1]\).

Step-by-step solution

Compute the inner products

For two functions to be orthogonal, their inner product must be zero. The inner product of two functions \(f(x)\) and \(g(x)\) over the interval \([-1,1]\) is given by: \[\int_{-1}^{1} f(x) g(x) dx = 0\] We need to make sure that any pair of our three polynomial functions have an inner product of zero. So, we have to compute three inner products: 1. \(a_0(a_1 + b_1x)\) 2. \(a_0(a_2 + b_2x + c_2x^2)\) 3. \((a_1+b_1x)(a_2 + b_2x + c_2x^2)\)

Compute the normalization conditions for each polynomial

For a function to be normalized, its length (Euclidean norm) should be 1. The length of a function \(f(x)\) over the interval \([-1,1]\) is given by: \[\sqrt{\int_{-1}^{1} f(x)^2 dx} = 1\] We need to compute the normalization conditions for each of our three polynomial functions: 1. \(a_0^2\) 2. \((a_1 + b_1x)^2\) 3. \((a_2 + b_2x + c_2x^2)^2\)

Use the inner product and normalization conditions to find the constants

By applying the inner product conditions and the normalization conditions, we can set up a system of equations to solve for the constants \(a_0\), \(a_1\), \(b_1\), \(a_2\), \(b_2\), and \(c_2\). After solving the system of equations (using either algebraic methods or numerical methods), we will obtain the appropriate values for the constants that make the polynomial functions orthonormal on the given interval. So, by following these steps the constants could be found and the polynomial functions made orthonormal in the given interval.

Key concepts

Inner Product of Functions

Understanding the concept of the inner product of functions is crucial when studying orthonormality in polynomials. The inner product is a generalization of the dot product, which you may know from vector algebra. In the context of functions, the inner product of two functions, say f(x) and g(x), over an interval can be thought of as a measure of how much the functions have in common over that interval.

Mathematically, the inner product for functions on an interval [a, b] is defined as the integral of their product over the interval. The equation \[ \int_{a}^{b} f(x) g(x) dx \] represents this concept. If the result is zero, we describe f(x) and g(x) as orthogonal over the interval. In the problem at hand, we compute the inner products over the interval [-1,1] to ensure our polynomials are orthogonal.

Key Points on Inner Product:

• The inner product measures the 'overlap' between functions.
• If the inner product is zero, functions are orthogonal in the interval.
• It is used to determine the orthogonality of polynomials in the process of making them orthonormal.

Normalization Condition

The normalization condition is a concept from functional analysis that ensures a function has a unit 'length' or 'norm'. What this means in practical terms is that we want our polynomial functions not only to be orthogonal to each other but also to be of a certain standard size. In the case of orthonormal polynomials, this size is defined as a norm of one.

Mathematically, we express the norm of a function f(x) as the square root of the inner product of the function with itself over a specific interval: \[ \sqrt{\int_{a}^{b} f(x)^2 dx} \] and in normalization, we set it to be equal to one. This procedure scales the polynomial so that if you were to 'measure' it from start to finish over the interval, the total 'distance' would be exactly one. Ensuring that each of our polynomials meets this condition is an important step in the process of making them orthonormal.

Applications of Normalization:

• It ensures a standard measure for 'length' of functions.
• Normalization is a key step in creating orthonormal sets of functions, which are useful in many areas of applied mathematics and physics.

Orthogonal Functions

Orthogonal functions play a pivotal role in mathematics, particularly in the realms of functional analysis and numerical methods. Two functions are considered orthogonal if their inner product over a given interval is zero. In more familiar terms, think of orthogonal vectors as being at a 90-degree angle to each other — they share no common component. Similarly, orthogonal functions do not correlate with each other over the defined interval.

For the polynomial functions given in the exercise, we strive to find a combination of constants such that when the polynomials are combined, their inner product equals zero. This condition of orthogonality can be imagined as a way to ensure that each polynomial brings something unique to the table, and their combination captures a broader range of features or behaviors.

Significance of Orthogonal Functions:

• Orthogonal functions can efficiently approximate complex functions and solve differential equations.
• In Fourier series and other expansions, orthogonal functions ensure that each term adds a distinct aspect to the overall function.
• They are foundational in varied fields, from quantum mechanics to statistics and beyond.