Approximation from sparse grids and function spaces of dominating mixed smoothness. (English) Zbl 1110.41009

Figiel, Tadeuz (ed.) et al., Approximation and probability. Papers of the conference held on the occasion of the 70th anniversary of Prof. Zbigniew Ciesielski, B»©dlewo, Poland, September 20–24, 2004. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 72, 271-283 (2006).
Let \(D_m(t):=\sum_{| k| \leq m}e^{ikt}\), \(t\in\mathbb T=[0,2\pi)\), \(m\in \mathbb N_0\), \(I_mf(t)=(2m+1)^{-1}\sum^{2m}_{l=0}f(t_l)D_m(t-t_l)\), \(t_l=2\pi l/(2m+1)\). We put \(L_j:=I_{2^j}\) and \(L_{j,k}:=L_j\otimes L_k\), where \(L_j\otimes L_k\) is the tensor product of \(L_j\) and \(L_k\). Then \(B_m:=\sum^m_{j=0}L_{j,m-j}-\sum^{m-1}_{j=0}L_{j,m-j-1}\). For \(r>0\) and \(1<p<\infty\) Sobolev space \(S^r_pW(\mathbb T^2)\) of dominating mixed smoothness of order \(r\) consists of all \(f\in L_p(\mathbb T^2)\), such that
\[ \| f\mid S^r_pW(\mathbb T^2)\| := \biggl\| \sum_{k\in Z^2}c_k(f)(1+| k_1| ^2)^{r/2}(1+| k_2| ^2)^{r/2}e^{ikx}| L_p(\mathbb T^2)\biggr\| <\infty. \] There \(c_k(f)\), \(k=(k_1,k_2)\) are Fourier coefficients of \(f\). Let \(P_j=\{x:2^{j-1}\leq x <2^j\}\), \(P_{j,k}=P_j\times P_k\), \(j,k\in \mathbb N_0\), \(f_{j,k}(x)=\sum_{l\in P_{j,k}}c_l(f)e^{ilx}\). For \(1<p<\infty\), \(1<q<\infty\), \(r>0\), Lizorkin-Triebel space \(S^r_{p,q}F(\mathbb T^2)\) consists of \(f\in L_p(\mathbb T^2)\) such that \[ \| f| S^r_{p,q}F(\mathbb T^2)\| := \Biggl\| \biggl(\sum^{\infty}_{j=0}\sum^{\infty}_{k=0}2^{r(j+k)q}| f_{j,k}| ^q \biggr)^{1/q}| L_p(\mathbb T^2)\Biggr\| <\infty. \]
Main results of the article are Theorem 1. Suppose \(1<p<\infty\) and \(r> \max(1/p,1/2)\). Then \[ \| I-B_m: S^r_pW(\mathbb R^2)\to L_p(\mathbb R^2)\| \sim m^{1/2}2^{-mr}. \]
Proposition 2. Suppose \(1<p,q<\infty\), \(r>1\). Then \[ \| I-B_m: S^r_{p,q}F(\mathbb T^2)\to L_p(\mathbb T^2)\| \sim m^{1-1/q}2^{-mr} \] .
For the entire collection see [Zbl 1091.47002].


41A25 Rate of convergence, degree of approximation
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