The Continuous Fourier Transform (CFT) converts a time-domain signal into its frequency-domain representation. It is defined as: \[ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt \] where \( F(\omega) \) is the frequency-domain representation, and \( f(t) \) is the time-domain signal.

The inverse Fourier Transform reconstructs a time-domain signal from its frequency-domain representation. It is given by: \[ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d\omega \]

The Discrete Fourier Transform (DFT) is used for discrete signals, and it transforms a sequence of complex numbers into another sequence representing the signal's frequency components. It is defined as: \[ F[k] = \sum_{n=0}^{N-1} f[n] e^{-i 2\pi kn / N} \] Unlike the Continuous Fourier Transform, the DFT operates on finite and discrete sequences rather than continuous signals.

The Fast Fourier Transform (FFT) is an efficient algorithm to compute the Discrete Fourier Transform (DFT) in \( O(N \log N) \) operations. It is widely used in signal processing, audio analysis, and image processing due to its speed, making real-time applications possible.

The Fourier Transform is linear, meaning the transform of a linear combination of functions is the same linear combination of their transforms: \[ \mathcal{F}\{af(t) + bg(t)\} = aF(\omega) + bG(\omega) \] where \( a \) and \( b \) are constants, and \( f(t) \) and \( g(t) \) are functions.

If a signal is shifted in time by \( t_0 \), the Fourier Transform introduces a phase shift. Mathematically: \[ \mathcal{F}\{f(t - t_0)\} = e^{-i\omega t_0} F(\omega) \]

The convolution theorem states that the Fourier Transform of the convolution of two signals is the product of their individual Fourier Transforms: \[ \mathcal{F}\{f(t) * g(t)\} = F(\omega) G(\omega) \] This theorem simplifies the computation of convolutions in the frequency domain.

Parseval's theorem relates the total energy of a signal in the time domain to the total energy in the frequency domain. It states: \[ \int_{-\infty}^{\infty} |f(t)|^2 dt = \int_{-\infty}^{\infty} |F(\omega)|^2 d\omega \]

The Fourier Transform is fundamental in signal processing because it allows for the decomposition of a signal into its frequency components. This enables tasks such as filtering, compression, noise reduction, and analysis of the frequency content of signals in various applications, including audio, image, and communication systems.

In image processing, the Fourier Transform is used to analyze the spatial frequencies within an image. It is useful in tasks such as image filtering, noise reduction, and compression. The 2D Fourier Transform is applied to convert an image from the spatial domain to the frequency domain, where different operations can be performed before converting the image back to the spatial domain.