http://i50.tinypic.com/a9pkpe.gif
1.cant see how the input in *
gives the result
??
there is no mathmatical description
i cant get this expression
http://i50.tinypic.com/2aq6pu.jpg
why when we input this basis we get this expression
i cant see how its done mathematickly
The functions , , and form an "orthonormal basis". That means specifically, that that the "inner product", defined here as the integral of the product of two functions, is 0 for any two distinct functions of this set and 1 for the product of a function with itself. If f(x) is written as a combination of those functions, multiplying it by itself and then integrating gives the coefficient, squared, for each such function it self- that's the reason for the sum of " .
I don't understand your reference to "two sums". can as easily be written as the single sum [tex]\sum_{n=0}^\infty (a_n sin(nx)+ b_n cos(nx)). Similarly can be written as the two sums .
We don't divide the norm by pi, we divide each sine or cosine by . That way, when we multiply one by itself and integrate, to find the norm, we have the entire product multiplied by . Of course the integral of is which gives evaluated between 0 and . Since sin(2nx) is 0 at 0 and at , that integral is . A similar calculation gives the same result for cos(nx). Dividing each "basis" function by divides the integral of the square by and so gives an integral (norm) of 1.
If we start with the "constant" function, 1, the integral is so we divide that by .
That way our "basis" is an orthonormal basis.