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Fourier-type minimal extensions in real \(L_1\)-space. (English) Zbl 0983.46031

The authors consider the Fourier-type operator \(F_w\) from the space \(L_1([0,2\pi]^n)\) to a finite-dimensional, shift-invariant space \(V\), with fix \(w\in V\). Main results are the following theorems:
2) \(F_w\) is the only minimal extension of \(R_w\).
2) \(\dim(\text{span}[\text{Min}_{R_w}(L_1, V)- F_w])= \infty\).
3) The identity operator on \(L_\infty\) has the only element of best approximation in \((P_{R_w}(L_1, V))^*\).
The paper contains also eight very important lemmas, a few remarks and four interesting examples.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
41A35 Approximation by operators (in particular, by integral operators)
41A52 Uniqueness of best approximation
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References:

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