Fourier-Haar coefficients and Banach limits

Fourier-Haar coefficients and Banach limits in the experiment was reported. The Haar system is complete and orthonormal and sometimes its convenient to consider the one-index Haar system. The Haar system is not uniformly bounded, Mercer's theorem, which states that the Fourier coefficients of an integrable function tend to zero,cannot be applied to it. This assertion can be regarded as an analogue of Mercer's theorem for the space Banach and the Haar system. Many equations were given in support of the experiment. The two-index Haar system with natural ordering mentioned in the experiment was also used.