Fourier transform

The Fourier transform is a mathematical operation that produces a function in one domain by integrating a function from another domain.

As mentioned in the previous article, a Fourier series represents a function  within a specific interval, for example . Outside this interval, the function is assumed to repeat itself. Therefore, a non-periodic function cannot be accurately represented by a Fourier series. If we want a mathematical tool to represent both periodic and non-periodic functions, we need a more generalised version of the Fourier series that is defined on the interval , as . To express this mathematically, let and substitute eq113 for this new interval in eq112 to give

Since and ,

As , the discrete sum in the above equation becomes a continuous integral:

where we have replaced the discrete variable with the continuous variable .



Show that in eq115 is equal to the function .


The integral is essentially a function of , as there is a corresponding integral value (output) for every value of (input).



is known as the Fourier transform of , and vice versa. The two equations in eq117 are regarded as a Fourier transform pair, which allow one function to be transformed into the other function. We can also say that is the inverse Fourier transform of , and vice versa.



i) Show that is an even function if both and are even functions.

ii) Show that is an odd function if is an even function and is an odd function or vice versa.

iii) If is an odd function, show that , assuming that the integral is carried out over a symmetric interval.

iv) Is if is an even function?


i) An even function and an odd function are defined as and respectively. So, .

ii) .

iii) Using the Cauchy principal value, . Let ,  for . So,

iv) No. With , for ,


Special forms of eq117 arise according to whether is even or odd. From eq117,

is an even function, while is an odd function. So, if is even, the first integrand is an even function, while the second integrand is an odd function. As shown in parts iii and iv the Q&A above,

It follows that is also an even function because . From eq117,

Eq118 and eq119 are known as a pair of Fourier cosine transforms, which has applications in Fourier-transform infrared spectroscopy.


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