Nikolskii-type Inequalities for Shift Invariant Function Spaces
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چکیده
Let Vn be a vectorspace of complex-valued functions defined on R of dimension n + 1 over C. We say that Vn is shift invariant (on R) if f ∈ Vn implies that fa ∈ Vn for every a ∈ R, where fa(x) := f(x− a) on R. In this note we prove the following. Theorem. Let Vn ⊂ C[a, b] be a shift invariant vectorspace of complex-valued functions defined on R of dimension n+ 1 over C. Let p ∈ (0, 2]. Then ‖f‖L∞[a+δ,b−δ] ≤ 2 2/p ( n+ 1 δ )1/p ‖f‖Lp[a,b] for every f ∈ Vn and δ ∈ ( 0, 1 2 (b− a) )
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تاریخ انتشار 2013