Convergence of Fourier Series

The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.

The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities in the signal when approximated by a truncated Fourier series. These high-frequency ripples do not vanish with an increasing number of terms; instead, they become compressed towards discontinuity. Gibbs first observed this effect, noting the characteristic ripples and overshoots that persist regardless of how many terms are included in the partial sum.

One strategy to mitigate the impact of the Gibbs phenomenon is to increase the number of terms in the partial sum. While the ripples' amplitude near the discontinuity remains, their total energy becomes negligible with a sufficiently large number of terms. Consequently, the overall energy in the approximation error decreases, allowing the Fourier series to effectively represent discontinuous signals. Truncating the Fourier series to a specific number of terms provides the best possible approximation under the given constraints, minimizing the error. As the number of terms increases, the error reduces, approaching zero if the signal is ideally represented by a Fourier series. This characteristic is particularly important in applications such as image processing, where minimizing error is crucial to avoid visual artifacts. In image signal approximation, reducing the Fourier series truncation error ensures higher fidelity and better visual quality.

As a result, while the Gibbs phenomenon presents a challenge in signal approximation using the Fourier series, increasing the number of terms and understanding the energy distribution in the approximation error can significantly mitigate its effects, allowing for accurate representations of even discontinuous signals.