Approximation by the nonlinear Fourier basis

Abstract

As a typical family of mono-component signals, the nonlinear Fourier basis \(\left\{ {e^{ik\theta _a \left( t \right)} } \right\}_{k \in \mathbb{Z}}\), defined by the nontangential boundary value of the Mobius transformation, has attracted much attention in the field of nonlinear and nonstationary signal processing in recent years. In this paper, we establish the Jackson’s and Bernstein’s theorems for the approximation of functions in \(X^p \left( \mathbb{T} \right)\) , 1 ⩽ p ⩽ ∞, by the nonlinear Fourier basis. Furthermore, the analogous theorems for the approximation of functions in Hardy spaces by the finite Blaschke products are established.