Faber systems and their use in sampling, discrepancy, numerical integration.

*(English)*Zbl 1246.46001
EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-107-1/pbk). viii, 107 p. (2012).

In recent years the study of the efficiency of numerical approximation based on function evaluations (sampling) turned out to be important. The monograph [H. Triebel, Bases in function spaces, sampling, discrepancy, numerical integration. EMS Tracts in Mathematics 11 (2010; Zbl 1202.46002)] gives a good account of this problem. The present monograph continues the study from the one above with focus restricted to the use of Faber systems. The Faber system on \((0,1)\) consists of piecewise linear (hat) functions with respect to a dyadic partition. This function system has remarkable properties as being a (conditional) basis in the space of continuous functions, and becoming an unconditional basis in certain smoothness spaces of Sobolev type. Similar assertions hold true when turning to product Faber systems and appropriate function spaces. For the convenience of the reader the present monograph recalls the corresponding results from the above-mentioned monograph.

The objective in the present text can be summarized as follows. One feature of the Faber system is that only a finite number of point evaluations is required to compute a coefficient in the Faber system. The question arises whether all (natural candidate) Besov spaces have such a basis. The affirmative answer to this question would have impact on the sampling numbers in such spaces, say \(X\), given by \[ g_{k}(\text{id}) := \inf\left\{\sup\left\{\| f- S_{k}(f) \|_{Y},\;f\in B_{X}\right\},\;\phi, \;x_{1},\dots,x_{k}\right\}, \] where the supremum is taken over the unit ball \(B_{X}\subset X\), and the infimum is taken over all choices of at most \(k\) points in the domain \(\Omega\), and reconstructions \(\phi\).

A second motivation is the extension of previous results to weighted spaces on \(\mathbb R\) or \(\mathbb R^{2}\), respectively. Specifically weights of the form \(w^{\alpha}(x):= (1 + \left| x\right|^{2})^{\alpha/2}\) are considered, or products thereof in the multivariate case. What can be said about the sampling numbers, numerical integration (with its integral numbers), or the discrepancy (with discrepancy numbers) in weighted spaces? The goal is not to extend to most general cases, but to clarify the influence of the weights on these questions. Finally, results for numerical discrepancy are given, clarifying some open problems in the affirmative.

To accomplish this goal the book starts with a detailed introduction in § 1 (Introduction, definitions, basic assertions), culminating in § 1.4 which discusses the objective of this study, as outlined before. In order to lay the foundations for the subsequent analysis, § 2 (Spaces on intervals) deals with the constructions and basic results for spaces on intervals. Here, the emphasis in on the discrepancy problem mentioned before. § 3 (Spaces on the real line) adapts the construction of the Faber system, originally given on \((0,1)\). Once this is accomplished one may ask similar questions as in the interval case, as in [loc. cit.], concerning the behavior of the sampling, integral and discrepancy numbers. The final § 4 (Spaces on the plane) extends the previous results to (anisotropic) weighted spaces. The basic results, Theorem 3.7 and Theorem 4.5, establish the asymptotics of the sampling numbers in weighted spaces. It is beyond the scope of this review to give a detailed account of this, but in loose terms the results asserts that for large (depending on smoothness and other parameters involved) weights \(\alpha\) the situation in the weighted cases is as in the unweighted ones, whereas small weights \(\alpha\) will have impact on the decay rate.

The monograph adds important information to problems of sampling, discrepancy and numerical integration in function spaces, and thus it contributes to a better understanding of sampling.

The objective in the present text can be summarized as follows. One feature of the Faber system is that only a finite number of point evaluations is required to compute a coefficient in the Faber system. The question arises whether all (natural candidate) Besov spaces have such a basis. The affirmative answer to this question would have impact on the sampling numbers in such spaces, say \(X\), given by \[ g_{k}(\text{id}) := \inf\left\{\sup\left\{\| f- S_{k}(f) \|_{Y},\;f\in B_{X}\right\},\;\phi, \;x_{1},\dots,x_{k}\right\}, \] where the supremum is taken over the unit ball \(B_{X}\subset X\), and the infimum is taken over all choices of at most \(k\) points in the domain \(\Omega\), and reconstructions \(\phi\).

A second motivation is the extension of previous results to weighted spaces on \(\mathbb R\) or \(\mathbb R^{2}\), respectively. Specifically weights of the form \(w^{\alpha}(x):= (1 + \left| x\right|^{2})^{\alpha/2}\) are considered, or products thereof in the multivariate case. What can be said about the sampling numbers, numerical integration (with its integral numbers), or the discrepancy (with discrepancy numbers) in weighted spaces? The goal is not to extend to most general cases, but to clarify the influence of the weights on these questions. Finally, results for numerical discrepancy are given, clarifying some open problems in the affirmative.

To accomplish this goal the book starts with a detailed introduction in § 1 (Introduction, definitions, basic assertions), culminating in § 1.4 which discusses the objective of this study, as outlined before. In order to lay the foundations for the subsequent analysis, § 2 (Spaces on intervals) deals with the constructions and basic results for spaces on intervals. Here, the emphasis in on the discrepancy problem mentioned before. § 3 (Spaces on the real line) adapts the construction of the Faber system, originally given on \((0,1)\). Once this is accomplished one may ask similar questions as in the interval case, as in [loc. cit.], concerning the behavior of the sampling, integral and discrepancy numbers. The final § 4 (Spaces on the plane) extends the previous results to (anisotropic) weighted spaces. The basic results, Theorem 3.7 and Theorem 4.5, establish the asymptotics of the sampling numbers in weighted spaces. It is beyond the scope of this review to give a detailed account of this, but in loose terms the results asserts that for large (depending on smoothness and other parameters involved) weights \(\alpha\) the situation in the weighted cases is as in the unweighted ones, whereas small weights \(\alpha\) will have impact on the decay rate.

The monograph adds important information to problems of sampling, discrepancy and numerical integration in function spaces, and thus it contributes to a better understanding of sampling.

Reviewer: Peter Mathé (Berlin)

##### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |

42C15 | General harmonic expansions, frames |

41A63 | Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX) |

41A55 | Approximate quadratures |

41A30 | Approximation by other special function classes |

65D30 | Numerical integration |