In the paper, expansions of the identity operator on the space L p (I d ) are constructed into the multiple series from the orthoprojective operators on the mutually orthogonal subspaces of the piecewise polynomial functions, the special case of which is the Haar series. Estimates of the norms of the corresponding projections in L q (I d ) are established. With their help, estimates of the Kolmogorov n-diameter are obtained in the space L 2(I d ) for the unit balls of the spaces of the Nikol’skiĭ-Besov functions, which satisfy mixed Hölder conditions giving the order of magnitude of this quantity.