@book{Shawe-Taylor:2004:KMP:975545, author = {Shawe-Taylor, John and Cristianini, Nello}, title = {Kernel Methods for Pattern Analysis}, year = {2004}, isbn = {0521813972}, publisher = {Cambridge University Press}, address = {New York, NY, USA}, } @article{fefferman2009fitting, title={Fitting a $C^{m}$-Smooth Function to Data \uppercase\expandafter{\romannumeral 1\relax}}, author={Fefferman, Charles and Klartag, Bo'az}, journal={{Annals of Mathematics}}, pages={315--346}, year={2009}, publisher={JSTOR} } @article{fefferman2009fitting2, title={Fitting a $ C^{m} $-smooth function to data \uppercase\expandafter{\romannumeral 2\relax}}, author={Fefferman, Charles and Klartag, Bo'az and others}, journal={Revista Matem{\'a}tica Iberoamericana}, volume={25}, number={1}, pages={49--273}, year={2009}, publisher={Departamento de Matem{\'a}ticas, Universidad Aut{\'o}noma de Madrid} } @article{Dachiraju2026PointDensityMeasure, author = {Rajesh Dachiraju}, title = {Point Density Measures: A Measure-Theoretic Framework with Applications to Geometric Probability}, journal = {Preprint}, year = {2026}, note = {Submitted}, } @misc{dachiraju2022approximationmultivariatefunctionbounded, title={Approximation of a Multivariate Function of Bounded Variation from its Scattered Data}, author={Rajesh Dachiraju}, year={2025}, eprint={2011.11258}, archivePrefix={arXiv}, primaryClass={math.NA}, url={https://arxiv.org/abs/2011.11258}, } @article{Dachiraju2026PointDensity, author = {Rajesh Dachiraju}, title = {Point Density Measure}, year = {2026}, note = {Preprint}, } @article{fefferman2009fitting3, title={Fitting a $C^{m}$-Smooth Function to Data \uppercase\expandafter{\romannumeral 3\relax}}, author={Fefferman, Charles and Klartag, Bo'az}, journal={{Annals of Mathematics}}, volume={170}, number={1}, pages={427--441}, year={2009}, publisher={JSTOR} } @article{fefferman2014fitting, title={Fitting a Sobolev function to data}, author={Fefferman, Charles L and Israel, Arie and Luli, Garving K}, journal={arXiv preprint arXiv:1411.1786}, year={2014} } @article{fefferman2016fitting, title={Fitting a Sobolev function to data \uppercase\expandafter{\romannumeral 1\relax}}, author={Fefferman, Charles and Israel, Arie and Luli, Garving K}, journal={Revista Matem{\'a}tica Iberoamericana}, volume={32}, number={1}, pages={275--376}, year={2016} } @article{fefferman2016fitting2, title={Fitting a Sobolev function to data \uppercase\expandafter{\romannumeral 3\relax}}, author={Fefferman, Charles and Israel, Arie and Luli, Garving K}, journal={Revista Matem{\'a}tica Iberoamericana}, volume={32}, number={3}, pages={1039--1126}, year={2016} } @inproceedings{carr2001reconstruction, title={Reconstruction and representation of 3D objects with radial basis functions}, author={Carr, Jonathan C and Beatson, Richard K and Cherrie, Jon B and Mitchell, Tim J and Fright, W Richard and McCallum, Bruce C and Evans, Tim R}, booktitle={Proceedings of the 28th annual conference on Computer graphics and interactive techniques}, pages={67--76}, year={2001}, organization={ACM} } @incollection{duchon1977splines, title={Splines minimizing rotation-invariant semi-norms in Sobolev spaces}, author={Duchon, Jean}, booktitle={{Constructive Theory of Functions of Several Variables}}, pages={85--100}, year={1977}, publisher={Springer} } @inproceedings{Carr:2001:RRO:383259.383266, author = {Carr, J. C. and Beatson, R. K. and Cherrie, J. B. and Mitchell, T. J. and Fright, W. R. and McCallum, B. C. and Evans, T. R.}, title = {Reconstruction and Representation of 3D Objects with Radial Basis Functions}, booktitle = {Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques}, series = {SIGGRAPH '01}, year = {2001}, isbn = {1-58113-374-X}, pages = {67--76}, numpages = {10}, url = {http://doi.acm.org/10.1145/383259.383266}, doi = {10.1145/383259.383266}, acmid = {383266}, publisher = {ACM}, address = {New York, NY, USA}, keywords = {RBF, Radial Basis Function, geometry compression, mesh repair, point-cloud surfacing, solid modeling, surface reconstruction, variational implicit surfaces}, } @article{graphlaplacian, title={The game theoretic p-Laplacian and semi-supervised learning with few labels}, author={Jeff Calder}, journal={arXiv preprint arXiv:1711.10144v4}, year={2018} } @article{DBLP:journals/nn/Hornik91, author = {Kurt Hornik}, title = {Approximation capabilities of multilayer feedforward networks}, journal = {Neural Networks}, volume = {4}, number = {2}, pages = {251--257}, year = {1991}, url = {https://doi.org/10.1016/0893-6080(91)90009-T}, doi = {10.1016/0893-6080(91)90009-T}, timestamp = {Wed, 14 Nov 2018 10:30:18 +0100}, biburl = {https://dblp.org/rec/bib/journals/nn/Hornik91}, bibsource = {dblp computer science bibliography, https://dblp.org} } @article{DBLP:journals/mcss/Cybenko89, author = {George Cybenko}, title = {Approximation by superpositions of a sigmoidal function}, journal = {{MCSS}}, volume = {2}, number = {4}, pages = {303--314}, year = {1989}, url = {https://doi.org/10.1007/BF02551274}, doi = {10.1007/BF02551274}, timestamp = {Wed, 14 Nov 2018 10:49:33 +0100}, biburl = {https://dblp.org/rec/bib/journals/mcss/Cybenko89}, bibsource = {dblp computer science bibliography, https://dblp.org} } @article{10.2307/25050572, ISSN = {0581572X}, URL = {http://www.jstor.org/stable/25050572}, abstract = {This article deals with minimum distance estimators and minimum penalized distance estimators which are based on the Kolmogorov-Smirnov distance. It is proved that such estimators achieve the optimum rate of convergence within families of density functions which have a bounded m-th derivative.}, author = {R.-D. Reiss}, journal = {Sankhyā: The Indian Journal of Statistics, Series A (1961-2002)}, number = {1}, pages = {59--68}, publisher = {Springer}, title = {Sharp Rates of Convergence of Minimum Penalized Distance Estimators}, volume = {48}, year = {1986} } @Article{Gajek1989, author="Gajek, Les{\l}aw", title="Estimating a density and its derivatives via the minimum distance method", journal="Probability Theory and Related Fields", year="1989", month="Feb", day="01", volume="80", number="4", pages="601--617", abstract="This paper deals with minimum distance (MD) estimators and minimum penalized distance (MPD) estimators which are based on the Lpdistance. Rates of strong consistency of MPD density estimators are established within the family of density functions which have a bounded m-th derivative. For the case p=2, it is also proved that the MPD density estimator achieves the optimum rate of decrease of the mean integrated square error and the L1 error. Estimation of derivatives of the density is considered as well.", issn="1432-2064", doi="10.1007/BF00318908", url="https://doi.org/10.1007/BF00318908" } @book{Vapnik:1982:EDB:1098680, author = {Vapnik, Vladimir}, title = {{Estimation of Dependences Based on Empirical Data : Springer Series in Statistics}}, year = {1982}, isbn = {0387907335}, publisher = {Springer-Verlag}, address = {Berlin, Heidelberg}, } @article {Afunctionfittingmethod, author = "Rajesh Dachiraju", title = "A function fitting method", journal = "Journal of Applied Analysis", year = "2020", publisher = "De Gruyter", address = "Berlin, Boston", volume = "26", number = "1", pages = "59--65", url = "https://www.degruyter.com/view/journals/jaa/26/1/article-p59.xml" } @book{buhmann_2003, place={Cambridge}, series={Cambridge Monographs on Applied and Computational Mathematics}, title={Radial Basis Functions: Theory and Implementations}, DOI={10.1017/CBO9780511543241}, publisher={Cambridge University Press}, author={Buhmann, Martin D.}, year={2003}, collection={Cambridge Monographs on Applied and Computational Mathematics}} @incollection{franke1991scattered, title={Scattered data interpolation and applications: A tutorial and survey}, author={Franke, Richard and Nielson, Gregory M}, booktitle={Geometric Modeling}, pages={131--160}, year={1991}, publisher={Springer} } @article{duchon1978erreur, title={Sur l’erreur d’interpolation des fonctions de plusieurs variables par les-splines}, author={Duchon, Jean}, journal={RAIRO. Analyse num{\'e}rique}, volume={12}, number={4}, pages={325--334}, year={1978}, publisher={EDP Sciences} } @article{madych1990multivariate, title={Multivariate interpolation and conditionally positive definite functions. II}, author={Madych, WR and Nelson, SA}, journal={Mathematics of Computation}, volume={54}, number={189}, pages={211--230}, year={1990} } @article{schaback1999improved, title={Improved error bounds for scattered data interpolation by radial basis functions}, author={Schaback, Robert}, journal={Mathematics of Computation}, pages={201--216}, year={1999}, publisher={JSTOR} } @article{wu1993local, title={Local error estimates for radial basis function interpolation of scattered data}, author={Wu, Zong-min and Schaback, Robert}, journal={IMA journal of Numerical Analysis}, volume={13}, number={1}, pages={13--27}, year={1993}, publisher={Oxford University Press} } @ARTICLE{2005NuAlg..39..307N, author = {{Narcowich}, Francis J.}, title = "{Recent developments in error estimates for scattered-data interpolation via radial basis functions}", journal = {Numerical Algorithms}, year = 2005, month = jan, volume = {39}, pages = {307-315}, doi = {10.1007/s11075-004-3644-7}, adsurl = {https://ui.adsabs.harvard.edu/abs/2005NuAlg..39..307N}, adsnote = {Provided by the SAO/NASA Astrophysics Data System} } @article{duchon1976interpolation, title={Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces}, author={Duchon, Jean}, journal={Revue fran{\c{c}}aise d'automatique, informatique, recherche op{\'e}rationnelle. Analyse num{\'e}rique}, volume={10}, number={R3}, pages={5--12}, year={1976}, publisher={EDP Sciences} } @article{bejancu1997uniform, title={The uniform convergence of multivariate natural splines}, author={Bejancu, Aurelian}, journal={University of Cambridge Department of Applied Mathematics and Theoretical Physics-Report-DAMTP NA}, year={1997}, publisher={Citeseer} } @article{powell1994uniform, title={The uniform convergence of thin plate spline interpolation in two dimensions}, author={Powell, MJD}, journal={Numerische Mathematik}, volume={68}, number={1}, pages={107--128}, year={1994}, publisher={Springer} } @article{buhmann_2000, title={Radial basis functions}, volume={9}, DOI={10.1017/S0962492900000015}, journal={Acta Numerica}, publisher={Cambridge University Press}, author={Buhmann, M. D.}, year={2000}, pages={1–38}} @article{buhmann1990multivariate, title={Multivariate cardinal interpolation with radial-basis functions}, author={Buhmann, MD}, journal={Constructive Approximation}, volume={6}, number={3}, pages={225--255}, year={1990}, publisher={Springer} } @incollection{buhmann1993new, title={New developments in the theory of radial basis function interpolation}, author={Buhmann, Martin D}, booktitle={Multivariate approximation: from CAGD to wavelets}, pages={35--75}, year={1993}, publisher={World Scientific} } @article{de1994approximation, title={Approximation from shift-invariant subspaces of ��₂ (��\^{}$\{$��$\}$)}, author={De Boor, Carl and DeVore, Ronald A and Ron, Amos}, journal={Transactions of the American Mathematical Society}, volume={341}, number={2}, pages={787--806}, year={1994} } @article{yoon2001spectral, title={Spectral approximation orders of radial basis function interpolation on the Sobolev space}, author={Yoon, Jungho}, journal={SIAM journal on mathematical analysis}, volume={33}, number={4}, pages={946--958}, year={2001}, publisher={SIAM} } @article{brownlee2004approximation, title={Approximation orders for interpolation by surface splines to rough functions}, author={Brownlee, Rob and Light, Will}, journal={IMA journal of numerical analysis}, volume={24}, number={2}, pages={179--192}, year={2004}, publisher={OUP} } @article{narcowich2004scattered, title={Scattered-Data Interpolation on $\backslash$bbR\^{}$\backslash$protectn: Error Estimates for Radial Basis and Band-Limited Functions}, author={Narcowich, Francis J and Ward, Joseph D}, journal={SIAM journal on mathematical analysis}, volume={36}, number={1}, pages={284--300}, year={2004}, publisher={SIAM} } @article{narcowich2005sobolev, title={Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting}, author={Narcowich, Francis and Ward, Joseph and Wendland, Holger}, journal={Mathematics of Computation}, volume={74}, number={250}, pages={743--763}, year={2005} } @article{narcowich1991norms, title={Norms of inverses and condition numbers for matrices associated with scattered data}, author={Narcowich, Francis J and Ward, Joseph D}, journal={Journal of Approximation Theory}, volume={64}, number={1}, pages={69--94}, year={1991}, publisher={Elsevier} } @article{boyd2011numerical, title={Numerical experiments on the condition number of the interpolation matrices for radial basis functions}, author={Boyd, John P and Gildersleeve, Kenneth W}, journal={Applied Numerical Mathematics}, volume={61}, number={4}, pages={443--459}, year={2011}, publisher={Elsevier} } @article{narcowich2005recent, title={Recent developments in error estimates for scattered-data interpolation via radial basis functions}, author={Narcowich, Francis J}, journal={Numerical Algorithms}, volume={39}, number={1-3}, pages={307--315}, year={2005}, publisher={Springer} } @article{JMLR:v18:16-140, author = {Jianqing Fan and Weichen Wang and Yiqiao Zhong}, title = {An $\ell_{\infty}$ Eigenvector Perturbation Bound and Its Application}, journal = {Journal of Machine Learning Research}, year = {2018}, volume = {18}, number = {207}, pages = {1-42}, url = {http://jmlr.org/papers/v18/16-140.html} } @article{davis1970rotation, title={The rotation of eigenvectors by a perturbation. III}, author={Davis, Chandler and Kahan, William Morton}, journal={SIAM Journal on Numerical Analysis}, volume={7}, number={1}, pages={1--46}, year={1970}, publisher={SIAM} } @book{evans1998partial, title={Partial Differential Equations}, author={Evans, L.C. and American Mathematical Society}, isbn={9780821807729}, lccn={97041033}, series={Graduate studies in mathematics}, url={https://books.google.co.in/books?id=5Pv4LVB\_m8AC}, year={1998}, publisher={American Mathematical Society} } @book{kato2013perturbation, title={Perturbation theory for linear operators}, author={Kato, Tosio}, volume={132}, year={2013}, publisher={Springer Science \& Business Media} } @article{riesz1955b, title={B.(1990). Functional analysis}, author={Riesz, F and Nagy, Sz-}, journal={Dover Publications, Inc., New York. First published in}, volume={3}, number={6}, pages={35}, year={1955} } @inproceedings{10.1145/800186.810616, author = {Shepard, Donald}, title = {A Two-Dimensional Interpolation Function for Irregularly-Spaced Data}, year = {1968}, isbn = {9781450374866}, publisher = {Association for Computing Machinery}, address = {New York, NY, USA}, url = {https://doi.org/10.1145/800186.810616}, doi = {10.1145/800186.810616}, abstract = {In many fields using empirical areal data there arises a need for interpolating from irregularly-spaced data to produce a continuous surface. These irregularly-spaced locations, hence referred to as “data points,” may have diverse meanings: in meterology, weather observation stations; in geography, surveyed locations; in city and regional planning, centers of data-collection zones; in biology, observation locations. It is assumed that a unique number (such as rainfall in meteorology, or altitude in geography) is associated with each data point.In order to display these data in some type of contour map or perspective view, to compare them with data for the same region based on other data points, or to analyze them for extremes, gradients, or other purposes, it is extremely useful, if not essential, to define a continuous function fitting the given values exactly. Interpolated values over a fine grid may then be evaluated. In using such a function it is assumed that the original data are without error, or that compensation for error will be made after interpolation.}, booktitle = {Proceedings of the 1968 23rd ACM National Conference}, pages = {517–524}, numpages = {8}, series = {ACM '68} } @article{lukaszyk2004new, title={A new concept of probability metric and its applications in approximation of scattered data sets}, author={{\L}ukaszyk, Szymon}, journal={Computational Mechanics}, volume={33}, number={4}, pages={299--304}, year={2004}, publisher={Springer} } @article{gordon1978shepard, title={Shepard’s method of “metric interpolation” to bivariate and multivariate interpolation}, author={Gordon, William J and Wixom, James A}, journal={Mathematics of computation}, volume={32}, number={141}, pages={253--264}, year={1978} } @article{de1983approximation, title={Approximation by smooth multivariate splines}, author={de Boor, Carl and DeVore, Ronald}, journal={Transactions of the American Mathematical Society}, volume={276}, number={2}, pages={775--788}, year={1983} } @article{gasca2000polynomial, title={Polynomial interpolation in several variables}, author={Gasca, Mariano and Sauer, Thomas}, journal={Advances in Computational Mathematics}, volume={12}, number={4}, pages={377}, year={2000}, publisher={Springer} } @article{de1990multivariate, title={On multivariate polynomial interpolation}, author={De Boor, Carl and Ron, Amos}, journal={Constructive Approximation}, volume={6}, number={3}, pages={287--302}, year={1990}, publisher={Springer} } @article{zheng2005friedrichs, title={On {F}riedrichs--{P}oincar{\'e}-type inequalities}, author={Zheng, Weiying and Qi, He}, journal={Journal of mathematical analysis and applications}, volume={304}, number={2}, pages={542--551}, year={2005}, publisher={Elsevier} } @book{kuipers2012uniform, title={Uniform distribution of sequences}, author={Kuipers, Lauwerens and Niederreiter, Harald}, year={2012}, publisher={Courier Corporation} } @book{book, author = {Devroye, Luc and Györfi, László and Lugosi, Gábor}, year = {1996}, month = {01}, pages = {}, title = {A Probablistic Theory of Pattern Recognition}, volume = {31}, isbn = {978-1-4612-6877-2}, journal = {NewYork: Springer Verlag}, doi = {10.1007/978-1-4612-0711-5} } @MISC {391340, TITLE = {A problem on rate of decay of fill distance?}, AUTHOR = {Iosif Pinelis (https://mathoverflow.net/users/36721/iosif-pinelis)}, HOWPUBLISHED = {MathOverflow}, NOTE = {URL:https://mathoverflow.net/q/391340 (version: 2021-05-20)}, EPRINT = {https://mathoverflow.net/q/391340}, URL = {https://mathoverflow.net/q/391340} } @book{10.5555/559923, author = {Scholkopf, Bernhard and Smola, Alexander J.}, title = {Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond}, year = {2001}, isbn = {0262194759}, publisher = {MIT Press}, address = {Cambridge, MA, USA}, abstract = {From the Publisher:In the 1990s, a new type of learning algorithm was developed, based on results from statistical learning theory: the Support Vector Machine (SVM). This gave rise to a new class of theoretically elegant learning machines that use a central concept of SVMs -kernels--for a number of learning tasks. Kernel machines provide a modular framework that can be adapted to different tasks and domains by the choice of the kernel function and the base algorithm. They are replacing neural networks in a variety of fields, including engineering, information retrieval, and bioinformatics. Learning with Kernels provides an introduction to SVMs and related kernel methods. Although the book begins with the basics, it also includes the latest research. It provides all of the concepts necessary to enable a reader equipped with some basic mathematical knowledge to enter the world of machine learning using theoretically well-founded yet easy-to-use kernel algorithms and to understand and apply the powerful algorithms that have been developed over the last few years.} }