Media not available in the Digital Archive
Description
This programme looks at the idea of approximating a function by using a set of orthogonal functions.
Module code and title: M203, Introduction to pure mathematics M203; 13 15-05-1980 1980 00:23:23 This programme was re-edited in 1995 as FOUM474T + Show more... Martin Wright Norman Gowar; Robin Wilson BBC Open University Eigenvalue; Functions; Matrix symmetry; Orthogonal functions; Orthogonal vectors; Orthogonality Fourier analysis; Fourier series; Mathematical analysis The programme begins with a computer animation showing the results of adding functions together. Norman Gowar argues that this technique is useful when trying to approximate to a particular function using other functions drawn from a specified set. Functions can also be thought of in terms of vector space. Using this idea of vector space the programme is going to show how a point P specified by vector P can be expressed in the best possible way, using only vectors in the subspace of P. To clarify this Norman uses an animation of a two dimensional man trying to interpret a point in three dimensional space. The best approximation of the point available to him is a point in the sub space perpendicular to the point P. Using a model in the studio Norman now describes this problem in terms of vectors and vector sub space. The best approximation to point P is a vector q. This can be characterised by a vector orthogonal to the sub-space and all vectors in that sub-space. This allows him to express the vector q algebraically, as the dot product of two orthogonal vectors is O. Vector q can be expressed as a linear combination of vectors (...). These vectors are a basis for the sub space. The coefficients for these vectors are the Fourier coefficients. Robin Wilson demonstrates how these coefficients can be calculated using the orthogonality relationships. Norman Gowar then asks the students to think of the vector sub-space in terms of functions. In order "to use the ideas developed so far, the idea of orthogonal functions must be clarified, we must be able to find their dot product. Robin Wilson defines the dot product of two functions. When this product is zero, these functions are orthogonal. The two functions 1 and cos x are orthogonal, and can be used as a basis. Using this basis and mod x as function p, Robin works out the Fourier coefficient and in turn the value of the vector q. He then compares this to the function mod x to see if it is a good approximation. Norman Gowar argues that the value achieved was a poor approximation. A better approximation could be attained by using more than two functions. In order to justify the use of extra functions he asks the student to visualise the sub-space as having four dimensions, that is four orthogonal vectors. This allows the vector q to be expressed as a combination of the four orthogonal basis vectors. This idea in turn leads to the use of any number of dimensions and orthogonal vectors, giving a better and better approximation to the functions in question. Robin Wilson works through an example to demonstrate this result. Using a computer animation he shows how the resulting function closely matches the function mod x. Norman Gowar reviews the findings of the programme. HOU2952 FOUM014R 841 no