  There is proposed a method for improving the convergence of Fourier series by
function systems, orthogonal at the segment, the application of which allows
for smooth functions to receive uniformly convergent series. There is also
proposed the method of phantom nodes improving the convergence of interpolation
polynomials on systems of orthogonal functions, the application of which in
many cases can significantly reduce the interpolation errors of these
polynomials. The results of calculations are given at test cases using the
proposed methods for trigonometric Fourier series; these calculations
illustrate the high efficiency of these methods. Undoubtedly, the proposed
method of phantom knots requires further theoretical studies.
