# On the mean summability of series by nonlinear basis

## TL;DR

This paper investigates the approximation and summability properties of series using nonlinear Fourier bases, demonstrating their effectiveness in signal processing and establishing Bernstein's inequalities for nonlinear trigonometric polynomials.

## Contribution

It introduces new approximation results and summability methods for nonlinear Fourier series, extending classical analysis to nonlinear bases in signal processing.

## Key findings

- Partial sums and Cesàro means are effective for nonlinear Fourier series in Lp spaces.
- Bernstein's inequalities are proved for nonlinear trigonometric polynomials.
- Results enhance understanding of nonlinear basis approximation in signal analysis.

## Abstract

The nonlinear signal processing has achieved a rapid process in the recent years. A family of nonlinear Fourier bases, as a typical family of mono-component signals, has been constructed and applied to signal processing. In this paper, the approximation properties of the partial sums and Ces?aro summability of series by the nonlinear Fourier basis are investigated in the Lp(T). Furthermore, these results are applied to the prove of Bernstein's inequalities for nonlinear trigonometric polynomials.