Is there a way to obtain the Fourier transform of a function (e.g., sinc) without the Gibbs phenomenon?
The Gibbs phenomenon is a property of the Fourier transform. Any way to produce the correct Fourier coefficients will show the Gibbs phenomenon. You can produce a different series, whose partial sums are not the partial sums of the Fourier series, to reduce or eliminate the Gibbs phenomenon. One such method is to use the Lanczos sigma factor.