On The Gibbs Phenomenon For Fractional Fourier Series

The Gibbs phenomenon originated from the partial sums of Fourier series for non-periodical piecewise continuous function, which affected seriously the series in engineering applications. Recently, to overcome the negative impact of the Gibbs phenomenon, many effective methods have been proposed to recover the original signal using the coefficients of the finite Fourier series expansions. However, with the development of object and spectrum of study, the traditional Fourier transforms expose the limitations. Therefore, it is necessary fractional Fourier transforms (FrFT) as a generation of Fourier transforms is proposed and has became a hot research field due to the unique characteristics of FrFT.The expression of Fourier transforms can be derived based on the Fourier series theory. In the same way, the fractional Fourier transforms also deduced the formulas of the fractional Fourier series (FrFS). Firstly, two-dimensional FrFS are proposed based on the definition of one-dimensional FrFS and the orthogonality of the basis. Similar to the Fourier series, FrFS of a function will exhibit Gibbs oscillation near the discontinuities and boundaries, and the oscillation will not diminish as the number of terms in the fractional Fourier series is increased. In this paper, one mainly focuses on the related characteristics of Gibbs phenomenon and attempts to obtain the relationship between the oscillation in theory and the jump value. In addition, Gibbs phenomenon is also observed in the series of other orthogonal polynomials for the simple step functions.In this paper, one mainly introduced briefly the spectral splitting method, the direct method and inverse polynomials reconstruction method (IPRM) and their basic principles. IPRM as one of resolution of Gibbs phenomenon has been more widely used. Therefore, IPRM is also applied to the elimination of Gibbs phenomenon for one-dimensional and two-dimensional fractional Fourier series, and the numerical experiments results show the effectiveness and accuracy of IPRM.