Nikolskii-type inequalities for shift invariant function spaces
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Let V n be a vectorspace of complex-valued functions defined on R of dimension n + 1 over We say that V n is shift invariant (on ) if f V n implies that f a V n for every a , where f a(x) := f(x - a) on R. In this note we prove the following. Theorem. Let V n C[a, b] be a shift invariant vectorspace of complex-valued functions defined on of dimension n + 1 over . Let p (0, 2]. Then f L[a+,b-] 2 2/p2 (n+1/) 1/p f Lp[a,b]for every f V n and (0, 1/2 (b - a)).