Sampling Discretization of the Uniform Norm

Discretization of the uniform norm functions from a given finite dimensional subspace continuous is studied. We pay special attention to case trigonometric polynomials with frequencies an arbitrary set fixed cardinality. give two different proofs fact that for any N-dimensional space it sufficient use $$e^{CN}$$ sample points accurate upper bound norm. Previous known results show one cannot improve on exponential growth number sampling good discretization theorem in Also, we prove general result, which connects best m-term bilinear approximation Dirichlet kernel associated subspace. illustrate application our technique example polynomials.