Fourier Series of Summable Functions

Given f ∈L₁(π,µ)the Fourier coefficients (c _n : n = 0, ±1, ±2,…) of f were introduced in Definition 8.1 by defining 1$${c_n} = {(2\pi )^{ - 1}}\int\limits_\Delta {f(x){e^{ - inx}}} dx,$$ where Δ is any interval of length 2π. To indicate that the Fourier coeffi­cients are those of the function f, the notation c _n (f) does sometimes occur. Frequently the notation fˆ(n) instead of c_n(f) is also used. The sequence (fˆ(n) : n = 0, ±1, ±2,…) is then denoted by fˆ. For any f ∈ L₁(ℝ,µ) there is an analogous notion, although now it is not a sequence of numbers but again a function defined on the whole of ℝ. Precisely formulated, for f ∈ L₁(ℝ,µ) the Fourier transformfˆ of f is the function, defined for any x ∈ ℝ by 2$${{f}^{{\left( x \right)}}} = \int\limits_{\mathbb{R}} {f(y){{e}^{{ - ixy}}}} dy.$$