Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Polynomial Approximation: Orthonormal Basis and Projection

Description

This lecture covers the concept of polynomial approximation using orthonormal bases and orthogonal projection. It starts with building an orthonormal basis from a naive basis, then introduces the Gram-Schmidt orthonormalization procedure. The lecture also discusses Legendre polynomials and demonstrates orthogonal projection over a specific interval. Finally, it compares the error of the approximation using orthogonal projection with Taylor's series.

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (30)

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for arises in this fashion.

Orthonormality

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis. The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces.

Legendre polynomials

In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions.

Gegenbauer polynomials

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer. File:Plot of the Gegenbauer polynomial C n^(m)(x) with n=10 and m=1 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.

Orthogonal basis

In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis. Any orthogonal basis can be used to define a system of orthogonal coordinates Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds.

Related lectures (33)

Orthogonal/Orthonormal Bases and Polynomials

Explores orthogonal and orthonormal bases, Gram-Schmidt process, and orthogonal polynomials in physics.

Orthogonal Projection on Vector Subspace MOOC: Linear Algebra (Part 3)

Explains orthogonal projection on a vector subspace in Euclidean space.

Orthogonal Projection: Vector Decomposition

Explains orthogonal projection and vector decomposition with examples in particle trajectory analysis.

Linear Regression: Absence or Presence of Covariates

Explores linear regression with and without covariates, covering models captured by independent distributions and tools like subspaces and orthogonal projections.

Orthogonal Projection: Example and Additional Remarks MOOC: Linear Algebra (Part 3)