On functions defined by its Fourier transform (Q2890570)

In this paper the Gibbs overshoot of a wavelet expansion at a discontinuity is given by using the Fourier transform of the scaling function. The overshoot of the approximation of a function near a discontinuity is known as the Gibbs phenomenon. This phenomenon was classically explained by the finite bound of the truncated trigonometric series especially for the Fourier series. However, some authors have shown that this phenomenon is not restricted to Fourier series, in fact it is observed also in some other approximations (by splines, piecewise functions, Lagrange polynomials and others). In particular, Kelly showed that the Gibbs phenomenon appears also for truncated wavelet expansions. The overshoot is measured by the integral of a 2nd degree product of translated scaling functions. However this result was given only in the time/space domain. The equivalent condition in the Fourier domain was unknown.NEWLINENEWLINEIn this paper the authors give this condition in terms of Fourier series so that the Gibbs overshooting at discontinuity can be characterized also by the Fourier transform of the scaling function. The importance of this result is due to the fact that some family of wavelets are defined only in the Fourier domain, by the Fourier transform either of the scaling or the wavelet function, because the analytical form in the space of variables is unknown. Theorem 4 proven in this paper, enables us to evaluate the Gibbs phenomenon for wavelet expansions by knowing only the Fourier transform of the scaling function. Thus allowing us to study the overshooting at discontinuity also for those families of wavelets where the analytical form of the inverse Fourier transform is unavailable.