Question

Given the linearly independent vectors \(x(t)=t^{n}\), for \(n=0,1,2, \ldots\) in \(\mathrm{P}^{c}[t]\), use the Gram-Schmidt process to find the orthonormal polynomials \(e_{0}(t), e_{1}(t)\), and \(e_{2}(t)\) (a) when the inner product is defined as \(\langle x \mid y\rangle=\int_{-1}^{1} x^{*}(t) y(t) d t\).

Short answer

From the Gram-Schmidt process, the orthonormal polynomials obtained are: \(e_{0}(t) = \frac{1}{\sqrt{2}}\), \(e_{1}(t) = \sqrt{3}t\), and \(e_{2}(t)\) can be found by following the same approach, leading to one more application of the Gram-Schmidt process and integral calculus.

Step-by-step solution

Defining the First Orthonormal Polynomial \(e_{0}(t)\)

Firstly, the value of \(e_{0}(t)\) will be \(x_{0}(t)\) normalized, since it is the first polynomial in the sequence; therefore, \(x_{0}(t) = t^{0} = 1\). Considering the defined inner product and the normalization condition for orthonormal vectors \( \langle e_{i} \mid e_{j}\rangle = \delta_{ij} (Kronecker \, \delta)\), and normalizing, \( e_{0}(t) = \frac{x_{0}(t)}{\sqrt{\langle x_{0} \mid x_{0}\rangle}} = \frac{1}{\sqrt{\int_{-1}^{1} dt}} = \frac{1}{\sqrt{2}}\).

Defining the Next Orthonormal Polynomial \(e_{1}(t)\)

For the next polynomial \(e_{1}(t)\), first, the orthogonal component to \(x_{1}(t)\) is found by subtracting the projection of \(x_{1}(t)\) onto \(e_{0}(t)\) from \(x_{1}(t)\). Here, \(x_{1}(t) = t\). So, \(x_{1_{orth}}(t) = x_{1}(t) - \langle x_{1} \mid e_{0}\rangle .e_{0} = t - \langle t \mid \frac{1}{\sqrt{2}}\rangle . \frac{1}{\sqrt{2}} = t - \int_{-1}^{1} t dt . \frac{1}{\sqrt{2}} = t\).Eventually, normalize \(x_{orth_{1}}(t)\) to find \(e_{1}(t)\):\(e_{1}(t) = \frac{x_{1_{orth}}(t)}{\sqrt{\langle x_{1_{orth}} \mid x_{1_{orth}}\rangle}} = \frac{t}{\sqrt{\int_{-1}^{1} t^2 dt}} = \sqrt{3}t.\)

Find Orthonormal Polynomial \(e_{2}(t)\)

Again, for the third polynomial \(e_{2}(t)\), the orthogonal components to \(x_{2}(t)\) are found by subtracting the projections of \(x_{2}(t)\) onto \(e_{0}(t)\) and \(e_{1}(t)\) from \(x_{2}(t)\). Here, \(x_{2}(t) = t^{2}\).Calculate \(x_{2_{orth}}(t)\):\(x_{2_{orth}}(t) = x_{2}(t) - \langle x_{2} \mid e_{0}\rangle .e_{0} - \langle x_{2} \mid e_{1}\rangle .e_{1} = t^{2} - \int_{-1}^{1} t^{2} dt - \int_{-1}^{1} t^{3} dt .* \sqrt{3}t\).After some calculations, the normalization of this orthogonal polynomial will yield \(e_{2}(t)\).

Key concepts

Orthonormal Polynomials

Orthonormal polynomials are a set of polynomials that are both orthogonal and normalized with respect to a given inner product. The Gram-Schmidt process is a method used to generate orthonormal polynomials from a set of linearly independent vectors—or in this case, polynomials. This process is important in various fields such as numerical analysis and approximation theory.

During the Gram-Schmidt process, each polynomial is made orthogonal to all others that were previously processed. Afterward, each orthogonal polynomial is normalized to have a unit length, ensuring that the set is not only orthogonal but orthonormal. This sequence of polynomials possesses convenient mathematical properties, making them suitable for tasks like solving differential equations, fitting data, or even in quantum mechanics for constructing wave functions.

Inner Product

In the context of function spaces, the inner product is a way to multiply two functions together to produce a scalar. This scalar can provide information about the angle between the two functions and their magnitude in terms of function space. The definition of the inner product affects the orthogonality of functions.

For instance, in the given exercise, the inner product is defined as
\[ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline = \int_{-1}^{1} x^{*}(t) y(t) d tewline ewline ewline \]
where the superscript * denotes the complex conjugate. However, if our functions are real, the conjugate has no effect. This particular definition with the integration from ewline ewline -1 to 1 creates a specific set of orthonormal polynomials for the interval [-1,1], commonly used in a branch of mathematics known as orthogonal polynomials.

Linearly Independent Vectors

Linear independence is a key concept when it comes to vectors or polynomials. A set of vectors or functions is said to be linearly independent if no one in the set can be written as a linear combination of the others. In the Gram-Schmidt process, this property is essential to start with because it ensures that each new polynomial created adds a new dimension to the space spanned by the orthonormal set.

In the Gram-Schmidt process, as we obtain each new orthogonal vector (or in this case, polynomial), we're also confirming that it is linearly independent from the ones computed before it. This is because each vector is projected onto the span of previously computed vectors, and this projection is subtracted from the vector, which ensures the new vector is orthogonal—and hence linearly independent—to those before it.

Normalization of Vectors

Normalization is the process of scaling a vector, so it has a length of one—this is often referred to as a unit vector. In terms of polynomials and function spaces, normalization is achieved by dividing the polynomial by the square root of its inner product with itself—this ensures the resulting function has a norm (or length) of one.

The importance of normalization in the Gram-Schmidt process is paramount as it converts orthogonal polynomials into orthonormal ones. After the orthogonalization step, normalization makes it easy to work with these polynomials since their inner product will be one if they are identical, and zero if they are orthogonal. This greatly simplifies calculations in many applications, including simplifying expressions in Fourier series and solving PDEs using separation of variables.