Take equation (234) from http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node31.html#eq:completeness_with_function and there, replace \hat{f}(k) by equation (233).
Then you almost have equation (235), except that you have to change
the order of integration.
pivoxa15 said:
Right, but why is the f(hat) function in terms of t?
f is in terms of x and k only.
The function \hat{f}(k) is indeed a function of k, but it is not the function f, whose variable is x. Maybe you are a little confused about the meaning of \hat{f}(k).
Think of the Fourier expansion as sort of linear combination of basis-vectors.
Let me try to explain:
Take the vectors in R^3.
Every vector can be represented as a linear combination of the basis-vectors (1,0,0), (0,1,0) and (0,0,1).
For example:
Take the vector v=(9,8,4). This vector can be written as a linear combination of the basisvectors vectors
u_1=(1,0,0)
u_2=(0,1,0)
u_3=(0,0,1)
in the following way:
(9,8,4) = 9 \cdot (1,0,0) + 8 \cdot (0,1,0) + 4 \cdot (0,0,1)
(9,8,4) = 9 \cdot u_1 + 8 \cdot u_2 + 4 \cdot u_3
Let's call the numbers 9,8,4 in front of the basis-vectors coefficients.
(9,8,4) = c_1 \cdot u_1 + c_2 \cdot u_2 + c_3 \cdot u_3
where c_1 = 9, c_2=8 and c_3=4
What if I choose different basis vectors, for example:
b_1=(3,0,0)
b_2=(0,2,0)
b_3=(0,0,2)
Then our vector v=(9,8,4) can be written as:
(9,8,4) = 3 \cdot b_1 + 4 \cdot b_2 + 2 \cdot b_3
Thus, the coefficients are 3,4,2 for our new basis-vectors
b_1,b_2 and b_3.
In general, you can write a vector v as a linear combination of
basis-vectors b_k, where in front of the basis-vectors you have the coefficients c_k.
v = \sum_{k=1}^{3} c_k b_k
In our last example we had
b_1=(3,0,0)
b_2=(0,2,0)
b_3=(0,0,2)
together with the coefficients:
c_1=3
c_2=4
c_3=2
Just check the formula
v = \sum_{k=1}^{3} c_k b_k
by plugging in the above values:
v = c_1 \cdot b_1 + c_2 \cdot b_2 + c_3 \cdot b3
v = 3 \cdot b_1 + 4 \cdot b_2 + 2 \cdot b_3
=3 \cdot (3,0,0) + 4 \cdot (0,2,0) + 2 \cdot (0,0,2)
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So, every vector can be represented by some basis vectors b_k
as
v = \sum_{k=1}^{3} c_k b_kMore general, if we have a vector with n entries instead of 3, we write
v = \sum_{k=1}^{n} c_k b_k = c_1 b_1 + c_2 b_2 + ... + c_n b_n
Now, let's make a step from the discrete to the continuous case.
Say we have a function f(x) and we also want to ask, whether it's possible to represent f(x) as a linear combination of ''basis-vectors''.
Question: Is it possible to write
f(x) = \sum_{k=1}^n c_k b_k
Let us be more specific and ask: Can I write f(x) as a sum
of b_k = e^{ikx}? So my new basis-vectors are b_k=e^{ikx}.
The question then becomes:
Question: Is it possible to write
f(x) = \sum_{k=1}^n c_k e^{ikx}
Indeed, it is possible, with a correction for the values of k.
Instead of going from k=1 to n, we use infinitely many basis-vectors, and
write k=- \infty to k=+ \infty.
Corrected version:
f(x) = \sum_{k=-\infty}^{+\infty} c_k e^{ikx}
The only question is, how do the coefficients
c_k look like?
Have a look at
Wikipedia
or
here on page 2 of the pdf. It shows how the coefficients can be calculated.
To finally get to your Fourier integral, we replace the sum by an integral
f(x) = \int_{-\infty}^{+\infty} c(k) e^{ikx} dk
Now, on your ohio-website, c(k) is \hat{f}(k), see equation (234) on the ohio-website.
Thus, \hat{f}(k) plays the role of the coefficients.
----------
Note 1 (on how to get from the Fourier series to the Fourier integral):
A Fourier series can sometimes be used to represent a function over an interval. If a function is defined over the entire real line, it may still have a Fourier series representation if it is periodic. If it is not periodic, then it cannot be represented by a Fourier series for all x. In such case we may still be able to represent the function in terms of sines and cosines, except that now the Fourier series becomes a Fourier integral.
The motivation comes from formally considering Fourier series for functions of period 2T and letting T tend to infinity.
The quote is taken from
http://209.85.135.104/search?q=cach...mes+fourier+integral&hl=de&ct=clnk&cd=4&gl=de.
Also see http://phy.syr.edu/~trodden/courses/mathmethods/chapter4.pdf
for the transition from the Fourier series to the Fourier integral.
Note 2: Summary:
In summary, consider the Fourier integral as a linear combination of basis vectors e^{ikx} with the
coefficients c(k) (or \hat{f}(k)).
Note 3 (on applications of the Fourier transform:
Have a look at
What is a Fourier Transform and what is it used for?.
http://grus.berkeley.edu/~jrg/ngst/fft/applicns.html
in Communications, Astronomy, Geology and Optics
Note 4 (functions as basis-vectors?)
You might ask "Why can I consider the functions e^{ikx} as
basis-vectors?"
The functions e^{ikx} fulfill some properties similar to those of
basis vectors (1,0,0),(0,1,0),(0,0,1)from R^3.
The functions are orthonormal, that is they are
orthogonal to each other and they are normalized to 1 (delta-function?).
See
Fourier Analysis on page 3.
http://web.mit.edu/8.05/handouts/SupplementarynotesonDiracNotation,QuantumStatesEtc.pdf on page 8.
Orthonormal functions: Definition on Wolfram mathworld
http://mathworld.wolfram.com/OrthonormalBasis.html
Note 5: Some examples of coefficients
Examples of Fourier transforms can be found http://www.mpipks-dresden.mpg.de/~jochen/methoden/topics/ft_ex.html
Note 6: Java Applets
Approximation of a function by a Fourier transform
This applet shows you how you can approximate a function by a Fourier transform. The more
coefficients you use, the better the approximation becomes.
Applet: Rectangular pulse approximation by Fourier Transform
This applet shows how you approximate a rectangular shaped pulse. If you don't use enough
"basis-vectors" (bandwidth is limited), then the pulse will not look rectangular anymore. This has applications in
electronics where you want to transmit a signal but the bandwidth is limited.
