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Approximation of distribution spaces by means of kernel operators. (English) Zbl 0893.46030

Given a Schwartz kernel \(K\in{\mathcal D}'(\mathbb{R}^d\times \mathbb{R}^d)\), the author sets \(K_h(x,y)= h^{-d}K(x/h, y/h)\) for \(h>0\) and defines the operator \(P_h\) by \(\langle P_h\eta,\psi\rangle= \langle K_h, \eta\otimes\psi\rangle\) for \(\eta,\psi\in{\mathcal D}(\mathbb{R}^d)\). The operator \(P= P_1\) is said to provide an order of approximation \(\mu>0\) for the pair of homogeneous Triebel-Lizorkin spaces \((\dot F^{\alpha,q}_p,\dot F^{\alpha+\mu, q}_p)\) with \(\alpha\in\mathbb{R}\), \(0<p<\infty\), \(0<q\leq\infty\), if \(\| f-P_hf\|_{\dot F^{\alpha,q}_p}\leq Ch^\mu\| f\|_{\dot F^{\alpha+\mu, q}_p}\) for every \(f\in F^{\alpha+\mu,q}_p\). Let \(T+h= P- h- P_{h/2}\) and assume that for every \(f\in\dot F^{\alpha+\mu, q}_p\) the series \(\sum_{j\geq 0} T_{2^{-j}}f\) converges to \(Pf- f\) in the quotient of the space \({\mathcal S}'\) of tempered distributions modulo the space of polynomials. Then \(P\) provides an order of approximation \(\mu>0\) for \((\dot F^{\alpha,q}_p,\dot F^{\alpha+\mu, q}_p)\) if and only if \(T_1:\dot F^{\alpha+\mu, q}_p\to \dot F^{\alpha,q}_p\) is continuous. Motivated by this result, the author considers operators \(T\) of the form \(Tf(x)= \int_{\mathbb{R}^d}{\mathcal R}(x, y)f(y)dy\), and proves various sufficient conditions for their continuity using the atomic and the molecular decomposition of the elements of Triebel-Lizorkin spaces. These conditions involve the growth of \({\mathcal R}\) and its derivatives with respect to \(x\), and require that \(Ty^\gamma= 0\) and that \(T^* x^\gamma\) be a polynomial for certain \(\gamma\in\mathbb{N}^d\). Returning to the approximation problem, he considers kernels of the form \(K(x,y)= \sum_{j\in\mathbb{Z}^d} \overline{\zeta(y- j)}\varphi(y-j)\), where \(\zeta\) and \(\varphi\) are normal functions, i.e. such that \(\xi(x)= \lim_{\varepsilon\to 0}| B_\varepsilon(x)|^{-1} \int_{B_\varepsilon(x)}\zeta(y) dy\) for every \(x\in\mathbb{R}^d\). He then gives sufficient conditions for \(P\) to provide an order of approximation \(\mu\) for \((\dot F^{\alpha,q}_p, \dot F^{\alpha+\mu, q}_p)\). These conditions concern the growth of \(\zeta\) and \(\varphi\) and their derivatives but also include the Strang-Fix conditions \(\widehat\varphi(0)\neq 0\) and \(\partial^\beta\varphi(0)= 0\) for sufficiently many \(\beta\in\mathbb{N}^d\).

MSC:

46F05 Topological linear spaces of test functions, distributions and ultradistributions
41A35 Approximation by operators (in particular, by integral operators)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
41A25 Rate of convergence, degree of approximation
42B25 Maximal functions, Littlewood-Paley theory
42B99 Harmonic analysis in several variables
46F12 Integral transforms in distribution spaces
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