Fourier-type minimal extensions in real \(L_1\)-space (Q5928722)

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:NEWLINENEWLINENEWLINE2) \(F_w\) is the only minimal extension of \(R_w\).NEWLINENEWLINENEWLINE2) \(\dim(\text{span}[\text{Min}_{R_w}(L_1, V)- F_w])= \infty\).NEWLINENEWLINENEWLINE3) The identity operator on \(L_\infty\) has the only element of best approximation in \((P_{R_w}(L_1, V))^*\).NEWLINENEWLINENEWLINEThe paper contains also eight very important lemmas, a few remarks and four interesting examples.