Chebyshev approximation by non-Chebyshev systems

We address the problem of uniform approximation by linear combinations of a finite system of functions (possibly non-Chebyshev) under arbitrary linear constraints. The proposed solution is based on the modified concept of alternance and of the Remez iterative method. A criterion for the best approximation is proved and the full set of polynomials of best approximation (which may be non-singleton in the non-Chebyshev case) is characterized. We derive a regularization procedure that makes the generalized Remez algorithm avoid cycling. The efficiency is demonstrated in examples with systems of complex exponents, Gaussian functions, and lacunar polynomials. As an application, the Markov-Bernstein type inequality is obtained for systems of complex exponents. Applications to signal processing, linear ODEs, switching dynamical systems are considered.