Existence and explicit form of nonlinear Hermite–Chebyshev approximations

In this paper, sufficient conditions for the existence of trigonometric Hermite–Jacobi approximations of a system of functions that are sums of convergent Fourier series are found. Based on these results, sufficient conditions are established under which nonlinear Hermite–Chebyshev approximations of systems of functions representable by Fourier series in Chebyshev polynomials of the first and second kind exist. When the found conditions are met, explicit formulas are obtained for the numerators and denominators of trigonometric Hermite–Jacobi approximations and nonlinear Hermite–Chebyshev approximations of the first and second kind of the specified systems of functions.