This table summarizes how the symmetry of a function \( f(x) \) relates to the symmetry of its Fourier Transform \( F(s) \).
| f(x) ⇔ F(s) | Definitions |
|---|---|
| even ⇔ even | real: \( f(x) = f^*(x) \) |
| odd ⇔ odd | imaginary: \( f(x) = -f^*(x) \) |
| real, even ⇔ real, even | even: \( f(x) = f(-x) \) |
| real, odd ⇔ real, odd | odd: \( f(x) = -f(-x) \) |
| imaginary, even ⇔ imaginary, even | |
| complex, even ⇔ complex, even | |
| complex, odd ⇔ complex, odd | |
| real, asymmetric ⇔ complex, Hermitian | Hermitian: \( f(x) = f^*(-x) \) |
| imaginary, asymmetric ⇔ complex, anti-Hermitian | anti-Hermitian: \( f(x) = -f^*(-x) \) |