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Math Help - Introduction to (construction) of bases.

  1. #1
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    Introduction to (construction) of bases.

    Hello everyone. I would like to start reading about
    1.orthonormal bases and
    2. bases functions

    My intention is first to understand what are these two topics about (I recall some lectures since my bachelor studies.. but it would be great to understand them more in depth) and then to try to understand how descriptive some basis-expansion coefficients can be.

    Of course it is not very easy for you to give me one link that will contain everything.. but few links might be a good starting point for me. I already googled but I would prefer to have some links for people that have studies this topic before.

    I would like to thank you in advance for your help.

    Best Regards
    Alex
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  2. #2
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    To be an orthonormal base, you need a finite vector space defined on an inner product space. The vectors then need to be orthogonal unit vectors.

    Let \displaystyle cos(x),sin(x)\in V and cos(x) and sin(x) are orthonormal on an inner product space if

    \displaystyle <cos(x),cos(x)>=\frac{1}{\pi}\int_{-\pi}^{\pi}cos^2(x)dx=1

    \displaystyle <cos(x),sin(x)>=\frac{1}{\pi}\int_{-\pi}^{\pi}cos(x)*sin(x)dx=0

    \displaystyle <cos(x),cos(x)>=\frac{1}{\pi}\int_{-\pi}^{\pi}sin^2(x)dx=1

    2
    http://en.wikipedia.org/wiki/Basis_function
    Last edited by dwsmith; December 6th 2010 at 08:33 PM.
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  3. #3
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    Hello dwsmthith

    Now I am trying to understand the following text I found on some research paper.

    The text mentions that "the Ψ (can u see this capital char?)

    is formed using pre-selected basis functions. One such basis can be constructed easily for a rectangular area A using the set of Legendre polynomials in two variables."

    If I got it right the legendre polynomials are used to expand a function of two variables. Do you agree with that?

    I have a book that discusses one variable legendre polynomials but still I can not fill the gap between

    legendre polynomials of one variable-->legendre polynomials of two variables-->how to use them to construct a basis.

    I would like to thank you in advance for your help

    Regards
    Alex.

    P.S Do you might have some web site with many tutorials/lessons about basis Linear Algebra (usually taught at first or second semester) at bachelor studies. Might be useful to remember few things.
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  4. #4
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    I, myself, have never seen the Legendre Polynomials in two variables.
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  5. #5
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    I would like to thank u again.
    I would like to ask you one more thing. I would like to find a list of functions that can be used to create a basis. Do you know any web site that might contain this type of information? Why sometimes we choose some set of funcions for a basis while other times we pick anothers?

    Regards
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  6. #6
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    I would like to thank you for your reply.
    I searched a little bit in the literature and I found that there are already thousand different basis proposed. For example legendre polynomials. Can you please explain me how one picks one proper set to build a base?

    I would like to thank you in advance for your help
    Best Regards
    Alex
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  7. #7
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    First, have you taken linear algebra and are comfortable with bases of finite dimensional vector spaces? That really should be a "pre-requisite" for infinite dimensional "function space", which is what you are dealing with here.

    You need to understand that any function space has an infinite number of bases, each equally good, so which you choose depends upon other things, such as why you are looking at the function space in the first place- what you hope to do with it. There is a fundamental theorem of differential equations that the set of all solutions to a linear, homogeneous, differential equation (whether ordinary or partial) form a vector space. Solutions to an ordinary differential equation form a finite dimesional vector space (dimension equal to the order of the equation) and solutions to partial differential equation form an infinite dimensional vector space (the "function spaces" of functional analysis).

    So typically, you are looking at function spaces in connection with solutions to partial differential equations or related operators (integral equations, operator equations, etc.) and the problem will be simplified if you can find a basis consisting of eigenfunctions of those operators.

    sin(nx), cos(nx) are, of course, "eigenfunctions" of the simple \frac{d^2 u}{dx^2} so simple linear equations that reduce to just the second derivative lead to using those as "bases"- leading to the Fourier series.

    Other equations that typically arise in applications are "Legendre's equation" and "Bessel's equation" for which we define (mathematicians being basically unimaginative) "Legendre polynomials" and "Bessel functions" as solutions and use those as "basis" functions.
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