Approximation order of nonstationary tight wavelet frames in Sobolev spaces (Q2897021)

The author investigates the approximation order of nonstationary tight wavelet frames in Sobolev spaces. The obtained results are similar to Theorem 1.1 in [\textit{B. Han} and \textit{Z. Shen}, SIAM J. Math. Anal. 40, No. 3, 905--938 (2008; Zbl 1161.42312)], where the approximation error is measured in the \(L^2(\mathbb{R})\)-norm.NEWLINENEWLINEThe main result (Theorem 3.1) states that, under certain assumptions on the masks \(\hat{a}_j\) and \(\hat{b}_j^\ell\), \(j \in \mathbb{N}\), \(\ell \in \{1,\dots,L\}\), the frame operator truncated at level \(n\), \(Q_n(f)\), satisfies: NEWLINE\[NEWLINE \| f- Q_n (f) \|_{H^s(\mathbb{R})} \leq C 2^{(s+1/2)n}n^{\nu \alpha} 2^{-\nu(1-\beta)n} |f|_{H^\nu(\mathbb{R})}, \quad n \geq N, \forall f \in H^\nu(\mathbb{R}),NEWLINE\]NEWLINE where \(C>0\) is independent of \(f\) and \(n\). The precise values of the constants \(\nu, s, \alpha,\beta,N\) in the above estimate depend on the masks \(\hat{a}_j\), \(j \in \mathbb{N}\), but must be in the range: NEWLINE\[NEWLINE s \geq 0, \, \nu \leq s +\frac{1}{2}, \, \nu \in \frac{1}{2}\mathbb{N}, \, N \in \mathbb{N}, \, \alpha\geq 0, \, 0 \leq \beta \leq 1-\frac{s+1/2}{\nu}. NEWLINE\]NEWLINE The term \(2^{(s+1/2)n}\) in the above estimate is new compared to the estimate in Theorem 1.1 in [loc. cit.] in which the error \(f- Q_n (f)\) is measured in the \(L^2(\mathbb{R})\)-norm. Hence, there is a gap of one-half in the exponent, i.e., \(2^{n\frac{1}{2}}\) between the two estimates.NEWLINENEWLINENote that (1) and the first part of (3) in the main theorem (Theorem 3.1) are stated as conclusions, but they are, in fact, assumptions.