Unispherical windows

In this paper the author discusses a new class of window functions based on the orthogonal polynomials known as the Gegenbauer or ultraspherical polynomials. These functions have a close relationship with the Jacobi polynomials and with the well known Chebyshev polynomials which are a special case. The window functions derived from these polynomials have the interesting property that the rolloff of the sidelobes with frequency is controlled by a parameter, leading to the design of a whole class of windows, including some unique ones where the sidelobes increase in value with frequency from some minimum at the first sidelobe. The author shows that other window functions can be approximated by this new class, and also indicate some interesting applications in spectral analysis and filter design