Fourier_transform

Definition

The Fourier transform is an integral transform that decomposes a function into its constituent frequencies. For a function $f \in L^1(\mathbb{R}^n)$, the Fourier transform $\hat{f}$ (or $\mathcal{F}\{f\}$) is defined by:

$$ \hat{f}(\xi) = \int_{\mathbb{R}^n} f(x) e^{-2\pi i \xi \cdot x} \, dx $$

The inverse Fourier transform recovers the original function:

$$ f(x) = \int_{\mathbb{R}^n} \hat{f}(\xi) e^{2\pi i \xi \cdot x} \, d\xi $$

Properties

1. Linearity: $\mathcal{F}\{af + bg\} = a\mathcal{F}\{f\} + b\mathcal{F}\{g\}$

2. Convolution Theorem: $\mathcal{F}\{f * g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}$

3. Shift Theorem: $\mathcal{F}\{f(x - x_0)\} = e^{-2\pi i \xi \cdot x_0} \hat{f}(\xi)$

4. Scaling Theorem: $\mathcal{F}\{f(ax)\} = \frac{1}{|a|}\hat{f}\left(\frac{\xi}{a}\right)$

5. Parseval’s Theorem: $\int |f(x)|^2 \, dx = \int |\hat{f}(\xi)|^2 \, d\xi$

Remarks

• The Fourier transform is an automorphism on the Schwartz_space $\mathcal{S}(\mathbb{R}^n)$.
• The transform extends to tempered_distributions via duality.
• The convolution in one domain corresponds to multiplication in the other domain.

Related Articles

• Schwartz_space
• Schwartz_function
• convolution