Define Fourier transform pair

A Fourier transform pair refers to a mathematical relationship between a function and its Fourier transform. Specifically, it consists of two functions—one in the time or spatial domain and the other in the frequency domain—that are related through the Fourier transform operation.

### Fourier Transform Pair Definition:

1. **Time Domain Function**:
   - Let \( f(t) \) be a function defined in the time domain (or spatial domain in some contexts).

2. **Frequency Domain Function**:
   - The Fourier transform of \( f(t) \), denoted as \( F(\omega) \) or \( \mathcal{F}\{f(t)\} \), represents the function in the frequency domain.

3. **Mathematical Relationship**:
   - The Fourier transform pair is defined as:
     \[
     F(\omega) = \mathcal{F}\{f(t)\} = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} \, dt
     \]
     \[
     f(t) = \mathcal{F}^{-1}\{F(\omega)\} = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} \, d\omega
     \]
     where \( j \) denotes the imaginary unit \( \sqrt{-1} \), \( \omega \) is the angular frequency variable, and \( \mathcal{F}^{-1} \) denotes the inverse Fourier transform.

### Properties and Usage:

- **Duality**: The Fourier transform pair exhibits duality, meaning that transforming \( f(t) \) gives \( F(\omega) \), and transforming \( F(\omega) \) gives back \( f(t) \).

- **Applications**: Fourier transform pairs are fundamental in signal processing, communications, physics (particularly in optics and quantum mechanics), and many other fields where analysis of signals in both time and frequency domains is crucial.

- **Integral Transform**: The Fourier transform pair is an example of an integral transform that converts a function between time (or spatial) domain and frequency domain representations.

In summary, a Fourier transform pair refers to a pair of functions—one in the time domain and its corresponding representation in the frequency domain—that are related through the Fourier transform operation, facilitating the analysis and manipulation of signals and functions across different domains.