Bernoulli numbers in the embedding constants of Sobolev spaces with different boundary conditions

We consider Sobolev spaces $W^n_2[0;1]$ with five different boundary conditions (periodic, antiperiodic, even and odd orders, and even-odd order). Exact estimates for derivatives of the order $k=0,1,\ldots, n-1$ are obtained, exact constants for the embedding of the spaces $W^n_2[0;1]$ into $W^k_\infty[0;1]$ are found, it is shown that they are rationally expressed in terms of Bernoulli numbers and, therefore, are rational. The exact embedding constants can also be expressed in terms of the Riemann $\zeta$-function. The reproducing kernels in these spaces are calculated.