Proof of Fourier Series: F(ax) = (1/a)f(k/a) with F(x) as Fourier Transform

In summary, the Fourier transform of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the Fourier transform is defined to have a factor of 1/2pi. This can be proven by making the change of variable \(x^{*}=ax\) in the Fourier transform equation. The particular form of the FT is not important, as long as the basic idea is understood.
  • #1
Poirot1
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Let f(k) be the Fourier transform of F(x). Prove that the Fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the Fourier transform is defined to have a factor of 1/2pi.
 
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  • #2
Poirot said:
Let f(k) be the Fourier transform of F(x). Prove that the Fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the Fourier transform is defined to have a factor of 1/2pi.

The particular form of the FT you are using is not that important, the basic idea is that:

\[\mathfrak{F} \big[ F(ax) \big] (k) = \int_{-\infty}^{\infty} F(ax) e^{-kx{\rm{i}}}\; dx\]

Now make the change of variable \(x^{*}=ax\) and the result drops out.

CB
 
  • #3
Poirot said:
Let f(k) be the Fourier transform of F(x). Prove that the Fourier transorm of F(ax) is $\frac{1}{a}f(\frac{k}{a})$ where a>0 and the Fourier transform is defined to have a factor of 1/2pi.

There is confusion here about the thread title (Fourier series) and content (Fourier transform). I have answered for the FT, is that what you intended?

CB
 

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function using a combination of sine and cosine functions. It is used to decompose a complex function into simpler components, making it easier to analyze and manipulate.

2. How is the Fourier transform related to the Fourier series?

The Fourier transform is a mathematical operation that transforms a function from the time or spatial domain to the frequency domain. The Fourier transform of a function is closely related to its Fourier series coefficients, which are used to construct the Fourier series.

3. What does the equation F(ax) = (1/a)f(k/a) mean in the context of Fourier series?

This equation is known as the scaling property of the Fourier transform and it states that the Fourier transform of a function F with a scaling factor a is equal to the Fourier transform of the function f, divided by the scaling factor a. This property is useful in simplifying the calculation of Fourier series coefficients.

4. Why is the Fourier transform important in signal processing?

The Fourier transform is an essential tool in signal processing because it allows us to analyze signals in the frequency domain. This helps in understanding the frequency components present in a signal and is useful in applications such as filtering, compression, and noise reduction.

5. What is the difference between a continuous and discrete Fourier transform?

A continuous Fourier transform is used for continuous-time signals, while a discrete Fourier transform is used for discrete-time signals. The continuous Fourier transform is a mathematical operation that transforms a continuous function into a continuous spectrum, while the discrete Fourier transform transforms a discrete sequence of values into a discrete spectrum. In practical applications, the discrete Fourier transform is used more often due to the discrete nature of digital signals.

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