Some orthogonalities in approximation theory

In approximation theory, a number of quantities are interrelated via orthogonal transformations. The revelation of these orthogonalities allows one to obtain useful relations and, in particular, to use known theorems on the Fourier coefficients. The paper considers this approach in application to the Fourier transforms of finite functions, to the best $L_2$-approximations, and to trigonometric polynomials with coefficients in a vector space.