On wavelets in \(L_ 1\) (Q1327824)

The paper investigates the \(L^ 1\)-approximation and \(L^ 1\)-smoothness characterization of the function \(f\) by \(r\)-regular multiresolution analysis. More precisely, let \(\varphi(x)\) be the scaling function of an \(r\)-regular multiresolution analysis, and \[ P_ n f(x)= \sum_ k \langle f,\varphi_{n,k}\rangle\varphi_{n,k}(x),\quad \varphi_{n,k}(x)= 2^{{n\over 2} \varphi(2^ n x-k)},\quad n,k\in \mathbb{Z}, \] be the associated projection. Then basing on the properties of \(\{P_ n\}\) shown in \textit{Y. Meyer's} ``Ondelettes et opérateurs'' (I: 1990; Zbl 0694.41037, II: 1990; Zbl 0745.42011), and the \(K\)-functional estimates given in the paper, following results are obtained: (1) \(\| P_ n f- f\|_ 1\leq c \omega_ r(f,2^{-n})\), \(\forall f\in L^ 1\), with \(\omega_ r(f,t)\) the \(r\)-modulus of smoothness; (2) For \(0< \alpha< r\), \(f\in L^ 1(\mathbb{R}^ 1)\), the following conditions are equivalent: \[ \begin{aligned}(\text{a})\quad \| (P_ n f)^{(r)}\|_ 1= O(2^{n(r-\alpha)});\qquad & (\text{b})\quad \| P_ n f- f\|_ 1= O(2^{-n\alpha});\\ (\text{c})\quad \omega_ r(f,t)= O(t^ \alpha);\qquad & (\text{d})\quad K_ r(f,t^ r)_ 1= O(t^ \alpha),\end{aligned} \] where \(K_ r(f,t)_ 1= \inf_{g\in W^{r,1}}\{\| f- g\|_ 1+ t\| g\|_{W^{r,1}}\}\), \(W^{r,1}= \{g\in L^ 1: g(r)\in L^ 1\}\).