The Toeplitz theorem and its applications to approximation theory and linear PDEs
Rong Qing Jia · Transactions of the American Mathematical Society · 1995
We take an algebraic approach to the problem of approximation by dilated shifts of basis functions. Given a finite collection Φ \Phi of compactly supported functions in L p ( R s ) ( 1 ⩽ p ⩽ ∞ ) {L_p}({\mathbb {R}^s})\quad (1 \leqslant p \leqslant \infty ) , we consider the shift-invariant space S S generated by Φ \Phi and the family ( S h : h > 0 ) ({S^h}:h > 0) , where S h {S^h} is the h h -dilate of S S . We prove that ( S h : h > 0 ) ({S^h}:h > 0) provides L p {L_p} -approximation order r r only if S S