Strang-Fix theory for approximation order in weighted \(L^p\)-spaces and Herz spaces (Q2369469)

A collection \(\Phi=\{\varphi_1,\dots,\varphi_N\}\) of continuous, compactly supported functions in \(\mathbb{R}^n\) is said to satisfy the Strang-Fix conditions of order \(k\) if there exist sequences \(b_1,\dots , b_N\) on \(\mathbb{Z}^n\) such that \(\varphi(\cdot)=\sum_{j=1}^N \varphi_j(\cdot) \ast_{\mathbb{Z}^n} b_j\) has Fourier transform \(\hat{\varphi}\) vanishing to order \(k\) at all nonzero integers (but not vanishing at zero itself). On the other hand, \(\Phi\) is said to provide \(L^p\) approximation of order \(k\) provided there exist, for any given \(f\in L^p(\mathbb{R}^n)\) and each \(h>0\), sequences \(c_j^h\) such that \(f_h(\cdot/h)=\sum_{j=1}^N \varphi_j(\cdot) \ast_{\mathbb{Z}^n} c_j^h\) satisfies \(\| f-f_h\| _{L^p} \leq C h^k| f| _{L^p_k}\) where \(| f| _{L^p_k}=\sum_{| \alpha| =k} \| \partial^\alpha f\| _{L^p}\). The coefficients \(c_j^h\) are also assumed to vanish away from the support of \(f\) in an appropriate sense. The author extends the equivalence between \(L^p\) approximation and Strang-Fix conditions of order \(k\), established by \textit{C. de Boor} and \textit{R.-Q. Jia} [Proc. Am. Math. Soc. 95, 547--553 (1985; Zbl 0592.41027)] to the case of Muckenhoupt weighted \(L^p\) norms and to Herz space norms. In the case of weighted \(L^p\), one defines \(| f| _{L^p_k(w)}=\sum_{| \alpha| =k} \| \partial^\alpha f\| _{L^p(w)}\) where \(\| g\| _{L^p(w)}^p=\int | f| ^p w\). Weighted \(L^p\) approximation of order \(k\) then means existence of \(f_h\) as above such that \(\| f-f_h\| _{L^p(w)} \leq C h^k| f| _{L^p_k(w)}\). In the case of the Herz space \(\dot{K}^{\alpha,p}_q(w_1,w_2)\) defined by \(\| f\| _{\dot{K}^{\alpha,p}_q(w_1,w_2)}^p=\sum_{k\in\mathbb{Z}} w_1(\{| x| <2^k\})^{\alpha p /n} \| f\chi_{2^{k-1}<| x| <2^k}\| _{L^q(w_2)}^p\), with \(w(E)=\int_E w\), approximation of order \(k\) is defined analogously. In each case, reproducing formulas provided by the Strang-Fix conditions together with geometric growth estimates on weighted averages give the desired approximation results. In the Herz space case, provisions must be made on \(\alpha\) and \(q\) to insure the right amount of regularity.