Find the first four orthogonal polynomials under the Sobolev...

Find the first four orthogonal polynomials under the Sobolev $H^1$ inner product
$$
\langle f, g\rangle=\int_{-1}^1\left[f(t) g(t)+f^{\prime}(t) g^{\prime}(t)\right] d t ; \text { cf. Exercise 3.1.27. }
$$

Key Concepts

Gram-Schmidt Orthogonalization Process

This process is a systematic method used to convert a set of linearly independent functions (or vectors) into an orthogonal (or orthonormal) set with respect to a given inner product. When constructing orthogonal polynomials under a Sobolev inner product, the Gram-Schmidt procedure is typically applied to the standard polynomial basis. This ensures that the resulting set satisfies the orthogonality condition defined by both the function and its derivative.

Polynomial Basis in Function Spaces

In function spaces, polynomials often serve as a basis due to their simplicity and dense nature. When working in Sobolev spaces, it is important to consider how the inclusion of derivative terms in the inner product affects the orthogonality and weight of each polynomial. This concept is central to the process of generating orthogonal polynomials suitable for approximations in the H^1 norm.

Sobolev Inner Product

This concept defines an inner product that not only incorporates the standard L2 term but also includes the derivatives of the functions. In the H^1 Sobolev space, the inner product is defined as the integral of the product of the functions plus the product of their first derivatives. This type of inner product introduces a measure of smoothness to the function space, making it particularly useful in the study of partial differential equations and variational problems.

Orthogonal Polynomials

Orthogonal polynomials are a sequence of polynomials where each polynomial is orthogonal to every other with respect to a given inner product. In the context of Sobolev spaces, these polynomials are constructed so that they are mutually orthogonal not only in terms of their function values but also considering derivatives, as specified by the Sobolev inner product.

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