# Introduction to (construction) of bases.

• Dec 5th 2010, 10:37 PM
dervast
Introduction to (construction) of bases.
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.

Best Regards
Alex
• Dec 6th 2010, 08:17 PM
dwsmith
1
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

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

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

$\displaystyle =\frac{1}{\pi}\int_{-\pi}^{\pi}sin^2(x)dx=1$

2
http://en.wikipedia.org/wiki/Basis_function
• Dec 7th 2010, 01:00 AM
dervast
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.

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.
• Dec 7th 2010, 03:27 PM
dwsmith
I, myself, have never seen the Legendre Polynomials in two variables.
• Jan 13th 2011, 05:41 AM
dervast
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
• Jan 17th 2011, 04:46 AM
dervast
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?

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.