A new class of Fourier series kernels

A class of weights is developed for minimizing the error in the approximation of a periodic function by a Fourier series with finitely many terms. The analysis is based on the following problem. Given a function
P(t)
, find the extrema of the integral
\int_{-T/2}^{T/2} P(t) | y(t) |^{2} dt
as
y(t)
ranges over all normalized trigonometric polynomials of specified order.