Definition: Fourier Series in Exponential Form

Official

Let $f \in L^1(\mathbb{T})$ . The Fourier series of $f$ in exponential form is the formal doubly infinite series

\sum_{n=-\infty}^{\infty} \hat f(n)\, e^{in\theta},

46 viewsMathematics · Harmonic Analysis

Official

Let $f \in L^1(\mathbb{T})$ . The Fourier series of $f$ in exponential form is the formal doubly infinite series

\sum_{n=-\infty}^{\infty} \hat f(n)\, e^{in\theta},

and one writes

f(\theta) \sim \sum_{n=-\infty}^{\infty} \hat f(n)\,e^{in\theta}

to record the association without asserting convergence.

46 viewsMathematics · Harmonic Analysis

Official

Let $f \in L^1(\mathbb{T})$ . The Fourier series of $f$ in exponential form is the formal doubly infinite series

\sum_{n=-\infty}^{\infty} \hat f(n)\, e^{in\theta},

46 viewsMathematics · Harmonic Analysis