Abstract:
We obtain an explicit expression for the norms of the embedding operators of the periodic Sobolev spaces into the space of continuous functions (the norms of this type are usually called embedding constants). The corresponding formulas for the embedding constants express them in terms of the values of the well-known Epstein zeta function which depends on the smoothness exponent $s$ of the spaces under study and the dimension $n$ of the space of independent variables. We establish that the embeddings under consideration have the embedding functions coinciding up to an additive constant and a scalar factor with the values of the corresponding Epstein zeta function. We find the asymptotics of the embedding constants as $s\to n/2$.
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