The guest editors discuss the highlights of the 5-th Dolomites Workshop on Constructive Approximation and Applications, and briefly introduce the papers included in this special issue

Marchetti F.

Perracchione E.

Santin G.

Sommariva A.

English

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Special Issue dedicated to Robert Schaback on the occasion of his 75th birthday, Volume 15 · 2022 · Pages i–iii5-th Dolomites Workshop on Constructive Approximation andApplications – Special Issue dedicated to Robert Schaback on theoccasion of his 75th birthdayFrancesco Marchetti a · Emma Perracchione b · Gabriele Santin c · Alvise Sommariva aAbstractThe guest editors discuss the highlights of the 5-th Dolomites Workshop on Constructive Approximationand Applications, and briefly introduce the papers included in this special issue.1 Report on the conferenceThe virtual conference 5-th Dolomites Workshop on Constructive Approximation and Applications DWCAA21 was held on September6–10 2021. The main scope of the meeting was to provide a forum for researchers to present and discuss ideas, theories, andapplications of constructive approximation.Throughout this five-days event we had a total of 55 speakers, and many topics in different fields of numerical analysis havebeen discussed. Among them we mention approximation theory, multiresolution techniques and numerical integration.The complete details of the workshop are available on the website:https://events.math.unipd.it/dwcaa21/.The conference featured five invited presentations:1. M. Buhmann, University of Giessen, Germany:Celebrating Robert Schaback’s birthday.2. A. Iske, University of Hamburg, Germany:Greedy update strategies for kernel-based approximation algorithms.3. L. Romani, Alma Mater Studiorum - University of Bologna, Italy:Pythagorean-Hodograph curves in exponential polynomial spaces.4. R. Schaback, Göttingen Georg-August University, Germany:My Failures, past and present.5. H. Wendland, University of Bayreuth, Germany:A meshfree method for a PDE-constrained optimisation problem.Moreover, a poster session with 9 participants, and 5 parallel sessions were organized on the following topics: Approximationtheory in imaging science; Meshless methods; Multivariate polynomial approximation; Multiresolution techniques; Numericalintegration, integral equations and transforms.At the end of the conference two awards (both amounting to 500 euros) were assigned to the best talk and the best poster.The winners have been1. I. Tominec, Uppsala University, Sweden: An oversampled RBF-FD method for elliptic and hyperbolic PDEs (best talk).2. N. Siar: University of Calabria, Italy: Numerical differentiation on scattered data through multivariate polynomial interpolation(best poster).2 Introduction to the special issueDuring the workshop the speakers were invited to submit contributions for a special volume of the Dolomites Research Notes onApproximation (DRNA). After peer-revision, we are pleased to announce that the following papers have been included in thevolume:• Interpolating sequences of 3D-data with C2 quintic PH B-spline curves, by G. Albrecht, C. V. Beccari and L. Romani [1]. Inthis work, PH quintic B-spline curves are employed in solving the C2 interpolation problem, dealing with sequences of3D-data. The proposed scheme is efficient and improves the relevant state of the art.aDepartment of Mathematics “Tullio Levi-Civita”, University of Padova, ItalybDipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, ItalycDigital Society Center (DIGIS), Bruno Kessler Foundation, Trento, ItalyMarchetti · Perracchione · Santin · Sommariva ii• Nonlinear multivariate sampling Kantorovich operators: quantitative estimates in functional spaces, by N. Çetin, D. Costarelli,M. Natale, and G. Vinti [2]. The work deals with quantitative estimates for the nonlinear sampling Kantorovich operatorsin the multivariate setting using the modulus of smoothness of Lp(Rn). The above results have been then extended to thegeneral case of Orlicz spaces Lϕ(Rn).• Solving surface interpolation problems stochastically and greedily, by M. Chen, L. Ling, and Y. Su [3]. The task of the paper isto introduce efficient techniques for the optimization of the shape parameter of a Radial Basis Function kernel when solvinginterpolation problems on a surface. These novel methods are based on (stochastic) Leave One Out Cross Validation, andon a greedy algorithm for the solution of the interpolation linear system.• Reconstruction of volatility surfaces: a first computational study, by S. Cuomo, A. De Rossi, L. Rizzo and F. Sica [4]. Inthis work, the authors tackle the implied volatility surface interpolation problem by considering a Radial Basis Functiontechnique. By combining both global and a local approach via the partition of unity strategy, this computational studyshows the effectiveness of the proposed method both in interpolation and extrapolation tasks.• Numerical method for hypersingular integrals of highly oscillatory functions on the positive semiaxis, by M. C. De Bonis and V.Sagaria [5]. In this paper, the authors analyze and then prove the stability and the convergence of a proposed quadraturerule for the numerical evaluation of hypersingular integrals of highly oscillatory functions on the positive semiaxis.• Error bounds and the asymptotic setting in kernel-based approximation, by T. Karvonen [6]. This work introduces two newcomputable error bounds for kernel-based interpolation, borrowing ideas from Gaussian process regression. Elementsof the theory of the so-called asymptotic setting are furthermore used to argue that these bounds are not conservative.Moreover, interesting aspects of the optimality of existing an new error bounds for kernel interpolation are discussed undera new point of view.• Compounded product integration rules on (0,+∞), by D. Mezzanotte and D. Occorsio [7]. The paper provides and analysesa new approach for the numerical approximation of certain integral transforms with an exponentially decaying weight anda quite general kernel. A new product rule leveraging a modified Lagrange interpolation technique is introduced, and it isshown that the resulting method is stable and convergent at a certain optimal rate. Moreover, the computational benefitsof the new method are demonstrated.• Filtered integration rules for finite weighted Hilbert transform II, by D. Occorsio, M. G. Russo and W. Themistoclakis [8]. Inthis work, the authors obtain further results concerning a class of quadrature rules for the evaluation of the finite Hilberttransform, which was introduced in a previous paper.• Yet another DE-Sinc indefinite integration formula, by T. Okayama and K. Tanaka [9]. The paper focuses on Sinc approximationcombined with the tanh transformation. The authors propose the replacement of the tanh transformation with the double-exponential transformation in Stenger’s second formula.• Approximation tools for detecting unforeseen sudden events, by A. Perlo and E. Venturino [10]. In this paper a splines-basedalgorithm is devised to detect the onset of unforeseen events, such as the opening of windows in a house, causing a suddendrop of the internal temperature.• Mean Field limits of trained weights in deep learning, A. Smirnov, B. Hamzi, and H. Owhadi [11]. The authors study thetraining of certain Neural Networks from the point of view of the dynamical evolution of the network’s weights, establishingthe existence of a mean-field limit of this dynamic. This is the first paper published in DRNA addressing the approximationtheory of deep Neural Networks, an interesting and emerging topic.3 AcknowledgementsThe workshop was co-organized by the Padova-Verona CAA research group, “Constructive Approximation and Applications” withinthe Rete ITaliana di Approssimazione (RITA), and the Gruppo UMI “Teoria dell’Approssimazione e Applicazioni (TAA)”. For thefinancial support of the workshop, we would like to thank the Department of Mathematics of the Universities of Padova, Verona,Milano-Bicocca and Torino. We would also like to express our gratitude to the editors of the DRNA journal, who enabled thefulfillment of this volume and to the Organizing and Scientific Committee, respectively composed by:• M. Caliari, R. Cavoretto, S. De Marchi, F. Marchetti, E. Perracchione, D. Poggiali, M. Rossini, G. Santin, A. Sommariva;• L. Białas-Cież, L. Bos, C. Campi, S. Cuomo, F. Dell’Accio, S. De Marchi, A. De Rossi, W. Erb, L. Fermo, E. Francomano, N.Levenberg, I. Notarangelo, D. Occorsio, E. Perracchione, L. Romani, M. Rossini, M. G. Russo, G. Santin, A. Sommariva, M.Vianello, A. Viscardi.Special thanks are also due to the reviewers for their relevant comments which helped to improve the quality and clarity of thepapers published in this issue.References[1] B. Albrecht, C. V. Beccari, and L. Romani. Interpolating sequences of 3d-data with c2 quintic ph b-spline curves. Dolomites Res. NotesApprox., 15:1–11, 2022.[2] N. Çetin, D. Costarelli, M. Natale, and G. Vinti. Nonlinear multivariate sampling kantorovich operators: quantitative estimates in functionalspaces. Dolomites Res. Notes Approx., 15:12–25, 2022.[3] M. Chen, L. Ling, and Y. Su. Solving surface interpolation problems stochastically and greedily. Dolomites Res. Notes Approx., 15:26–36,2022.Dolomites Research Notes on Approximation ISSN 2035-6803Marchetti · Perracchione · Santin · Sommariva iii[4] S. Cuomo, A. De Rossi, L. Rizzo, and F. Sica. Reconstruction of volatility surfaces: a first computational study. Dolomites Res. Notes Approx.,15:37–48, 2022.[5] M. C. De Bonis and V. Sagaria. Numerical method for hypersingular integrals of highly oscillatory functions on the positive semiaxis.Dolomites Res. Notes Approx., 15:49–64, 2022.[6] T. Karvonen. Error bounds and the asymptotic setting in kernel-based approximation. Dolomites Res. Notes Approx., 15:65–77, 2022.[7] D. Mezzanotte and D. Occorsio. Compounded product integration rules on (0,+∞). Dolomites Res. Notes Approx., 15:78–92, 2022.[8] D. Occorsio, M. G. Russo, and W. Themistoclakis. Filtered integration rules for finite weighted Hilbert transform ii. Dolomites Res. NotesApprox., 15:93–104, 2022.[9] T. Okayama and K. Tanaka. Yet another de-sinc indefinite integration formula. Dolomites Res. Notes Approx., 15:105–116, 2022.[10] A. Perlo and E. Venturino. Approximation tools for detecting unforeseen sudden events. Dolomites Res. Notes Approx., 15:117–124, 2022.[11] A. Smirnov, B. Hamzi, and H. Owhadi. Mean field limits of trained weights in deep learning. Dolomites Res. Notes Approx., 15:125–145,2022.Dolomites Research Notes on Approximation ISSN 2035-6803

5-th Dolomites Workshop on Constructive Approximation and Applications – Special Issue dedicated to Robert Schaback on the occasion of his 75th birthday

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5-th Dolomites Workshop on Constructive Approximation and Applications – Special Issue dedicated to Robert Schaback on the occasion of his 75th birthday

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