Question

If Fourier transform of \(f(x)=F(a)\), then Fourier Transform of \(f(2 x)\) is

Short answer

The Fourier transform of \(f(2x)\) is \(\frac{1}{2} F\left(\frac{a}{2}\right)\).

Step-by-step solution

Understanding the Problem

We are given a function \(f(x)\) whose Fourier transform is \(F(a)\). We need to find the Fourier transform of \(f(2x)\). This involves understanding how scaling the input of the function affects its Fourier transform.

Fourier Transform Scaling Property

The Fourier transform of \(f(x)\), denoted as \(F(a)\), is defined by \(F(a) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i ax} \, dx\). The scaling property of Fourier transforms states that if \(F(a)\) is the Fourier transform of \(f(x)\), then the Fourier transform of \(f(bx)\) is \(\frac{1}{|b|}F\left(\frac{a}{b}\right)\).

Apply Scaling Property

For the function \(f(2x)\), we apply the scaling property. Here, \(b = 2\), so the transform becomes \(\frac{1}{|2|}F\left(\frac{a}{2}\right)\). Thus, the Fourier transform of \(f(2x)\) is \(\frac{1}{2} F\left(\frac{a}{2}\right)\).

Key concepts

Fourier Transform Scaling Property

The Fourier Transform is a mathematical operation that converts a function from its original domain, often time or space, into the frequency domain. One particular property of the Fourier Transform that highlights its power is the scaling property. This property can help us understand how changes in a function, such as stretching or compressing it, will affect its Fourier transform.

The Scaling Property states that if you have a function, say \( f(x) \), and its Fourier transform is \( F(a) \), transforming the function by multiplying the independent variable by a constant \( b \), leads to a new Fourier transform. Specifically, if you transform \( f(x) \) to \( f(bx) \), the Fourier transform becomes \( \frac{1}{|b|} F\left(\frac{a}{b}\right) \).

This means:

• When you scale the function in its original domain, you inversely scale the frequency domain representation.
• The resulting transformation is also scaled by a factor \( \frac{1}{|b|} \).

Understanding this property allows us to predict how manipulations in the function's domain impact its frequency representation.

Function Scaling

Function scaling in the realm of mathematical transformations involves changing the input of a function by a scaling factor. Essentially, it means adjusting the function’s horizontal or vertical stretch or compression.

In our exercise, when the original function \( f(x) \) is replaced by \( f(2x) \), it indicates horizontal compression by a factor of 2. This is because:

• In function scaling, any number multiplying the input \( x \) modifies the graph horizontally.
• If the multiplying factor is greater than 1, as with \( 2 \), it compresses the function.
• If the factor is between 0 and 1, the function stretches.

It's crucial to note how the input scaling will alter the Fourier Transform, as the scaling property will take effect allowing us to find the new frequency representation.

Mathematical Analysis

Mathematical analysis provides the tools required to understand changes and transformations of functions through operations like the Fourier Transform. When faced with adjusting the function's form—like scaling—it helps to logically break down the steps using analytical thinking.

Analysis begins with identifying what the problem asks—understanding what transformation has been applied to the function and what result is desired, like finding the Fourier transform of a scaled function. Each transformation can be handled by applying corresponding mathematical properties, such as the scaling property for Fourier transforms. This property simplifies the calculation by connecting transformations to the properties of function changes.

By applying rigorous analysis, you can simplify complex transformations into manageable steps, making it clearer how the function’s alteration affects its Fourier representation.

Frequency Domain Analysis

Frequency domain analysis involves examining signals or functions in terms of their frequency components rather than time or space. This is crucial as it allows us to easily analyze the structure and behavior of complex signals.

When analyzing functions in the frequency domain, the Fourier Transform is our tool of choice. It reveals how different frequency components combine to form the original function. The scaling property is particularly useful here as it dictates that scaling a function in its time domain results in inversely scaling in the frequency domain.

For instance, in our problem:

• The transformation of \( f(x) \) to \( f(2x) \) results in the compression in the time domain, leads to expansion in the frequency domain.
• This means lower frequencies mapped from the scaled time signal.

By using frequency domain analysis, we are able to translate these changes and predict the frequency spectrum shape after modifications, much like we achieved with the scaled Fourier transform result: \( \frac{1}{2} F\left(\frac{a}{2}\right) \).