Optimal interpolation in Hardy and Bergman spaces: A reproducing kernel Banach space approach
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- by Gilbert J. Groenewald, Sanne ter Horst and Hugo J. Woerdeman;
- Trans. Amer. Math. Soc. 378 (2025), 4303-4333
- DOI: https://doi.org/10.1090/tran/9389
- Published electronically: March 19, 2025
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Abstract:
After a review of the reproducing kernel Banach space framework and semi-inner products, we apply the techniques to the setting of Hardy spaces $H^p$ and Bergman spaces $A^p$, $1<p<\infty$, on the unit ball in $\mathbb {C}^n$, as well as the Hardy space on the polydisk and half-space. In particular, we show how the framework leads to a procedure to find a minimal norm element $f$ satisfying interpolation conditions $f(z_j)=w_j$, $j=1,\ldots , n$. We also explain the techniques in the setting of $\ell ^p$ spaces where the norm is defined via a change of variables and provide numerical examples.References
- M. A. Aizerman, E. M. Braverman, and L. I. Rozonoer, Theoretical foundations of the potential function method in pattern recognition learning, Automat. Remote Control 25 (1964), 821–837.
- N. Aronszajn, La théorie des noyaux reproduisants et ses applications. I, Proc. Cambridge Philos. Soc. 39 (1943), 133–153 (French). MR 8639, DOI 10.1017/S0305004100017813
- N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
- Kendall Atkinson and Weimin Han, Theoretical numerical analysis, 3rd ed., Texts in Applied Mathematics, vol. 39, Springer, Dordrecht, 2009. A functional analysis framework. MR 2511061, DOI 10.1007/978-1-4419-0458-4
- Sheldon Axler, Bergman spaces and their operators, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 1–50. MR 958569
- Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR 1184139, DOI 10.1007/b97238
- Catherine Bénéteau and Dmitry Khavinson, A survey of linear extremal problems in analytic function spaces, Complex analysis and potential theory, CRM Proc. Lecture Notes, vol. 55, Amer. Math. Soc., Providence, RI, 2012, pp. 33–46. MR 2986891, DOI 10.1090/crmp/055/03
- Claude Berge, Topological spaces, Dover Publications, Inc., Mineola, NY, 1997. Including a treatment of multi-valued functions, vector spaces and convexity; Translated from the French original by E. M. Patterson; Reprint of the 1963 translation. MR 1464690
- Garrett Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), no. 2, 169–172. MR 1545873, DOI 10.1215/S0012-7094-35-00115-6
- B. E. Boser, I. M. Guyon, and V. N. Vapnik, A training algorithm for optimal margin classifiers, Proceedings of the fifth annual workshop on Computational learning theory, 1992, pp. 144–152.
- Hernán Centeno and Juan Miguel Medina, A converse sampling theorem in reproducing kernel Banach spaces, Sampl. Theory Signal Process. Data Anal. 20 (2022), no. 2, Paper No. 8, 19. MR 4443619, DOI 10.1007/s43670-022-00026-6
- Raymond Cheng and Yuesheng Xu, Minimum norm interpolation in the $\ell _1(\Bbb N)$ space, Anal. Appl. (Singap.) 19 (2021), no. 1, 21–42. MR 4178411, DOI 10.1142/S0219530520400059
- J. A. Cima, T. H. MacGregor, and M. I. Stessin, Recapturing functions in $H^p$ spaces, Indiana Univ. Math. J. 43 (1994), no. 1, 205–220. MR 1275459, DOI 10.1512/iumj.1994.43.43010
- John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926, DOI 10.1007/978-1-4757-3828-5
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 268655
- Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762, DOI 10.1090/surv/100
- Samuel E. Ebenstein, Some $H^{p}$ spaces which are uncomplemented in $L^{p}$, Pacific J. Math. 43 (1972), 327–339. MR 318793, DOI 10.2140/pjm.1972.43.327
- Sam J. Elliott and Andrew Wynn, Composition operators on weighted Bergman spaces of a half-plane, Proc. Edinb. Math. Soc. (2) 54 (2011), no. 2, 373–379. MR 2794660, DOI 10.1017/S0013091509001412
- Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, and Václav Zizler, Banach space theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. The basis for linear and nonlinear analysis. MR 2766381, DOI 10.1007/978-1-4419-7515-7
- Gregory E. Fasshauer, Fred J. Hickernell, and Qi Ye, Solving support vector machines in reproducing kernel Banach spaces with positive definite functions, Appl. Comput. Harmon. Anal. 38 (2015), no. 1, 115–139. MR 3273290, DOI 10.1016/j.acha.2014.03.007
- Timothy Ferguson, Solution of extremal problems in Bergman spaces using the Bergman projection, Comput. Methods Funct. Theory 14 (2014), no. 1, 35–61. MR 3194312, DOI 10.1007/s40315-013-0046-7
- S. Fine and K. Scheinberg, Efficient SVM training using low-rank kernel representations, J. Mach. Learn. Res. 2 (2001), 243–264.
- Kenji Fukumizu, Francis R. Bach, and Michael I. Jordan, Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces, J. Mach. Learn. Res. 5 (2003/04), 73–99. MR 2247974
- J. R. Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc. 129 (1967), 436–446. MR 217574, DOI 10.1090/S0002-9947-1967-0217574-1
- Bas Harmsen, Interpolatieproblemen in $H_p$-ruimten, Doctoraalscriptie, Radboud Universiteit Nijmegen, The Netherlands, 2005.
- Arthur Gretton, Ralf Herbrich, Alexander Smola, Olivier Bousquet, and Bernhard Schölkopf, Kernel methods for measuring independence, J. Mach. Learn. Res. 6 (2005), 2075–2129. MR 2249882
- Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653, DOI 10.1007/978-1-4612-0497-8
- Henry Helson, Harmonic analysis, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. MR 729682
- Wolfgang Hensgen, On the dual space of $\textbf {H}^p(X),\;1<p<\infty$, J. Funct. Anal. 92 (1990), no. 2, 348–371. MR 1069250, DOI 10.1016/0022-1236(90)90055-P
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962. MR 133008
- Vasile I. Istrăţescu, Strict convexity and complex strict convexity, Lecture Notes in Pure and Applied Mathematics, vol. 89, Marcel Dekker, Inc., New York, 1984. Theory and applications. MR 728388
- Robert C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265–292. MR 21241, DOI 10.1090/S0002-9947-1947-0021241-4
- Dmitry Khavinson and Michael Stessin, Certain linear extremal problems in Bergman spaces of analytic functions, Indiana Univ. Math. J. 46 (1997), no. 3, 933–974. MR 1488342, DOI 10.1512/iumj.1997.46.1375
- S. Ya. Khavinson, Two papers on extremal problems in complex analysis, Amer. Math. Soc. Transl. Ser. 2, No. 129 (1980).
- Paul Koosis, Introduction to $H_p$ spaces, 2nd ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998. With two appendices by V. P. Havin [Viktor Petrovich Khavin]. MR 1669574
- Rong Rong Lin, Hai Zhang Zhang, and Jun Zhang, On reproducing kernel Banach spaces: generic definitions and unified framework of constructions, Acta Math. Sin. (Engl. Ser.) 38 (2022), no. 8, 1459–1483. MR 4473864, DOI 10.1007/s10114-022-1397-7
- Qianru Liu, Rui Wang, Yuesheng Xu, and Mingsong Yan, Parameter choices for sparse regularization with the $\ell _1$ norm, Inverse Problems 39 (2023), no. 2, Paper No. 025004, 34. MR 4533333, DOI 10.1088/1361-6420/acad22
- G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29–43. MR 133024, DOI 10.1090/S0002-9947-1961-0133024-2
- X. Ma, and C. L. Nikias, Joint estimation of time delay and frequency delay in impulsive noise using fractional lower order statistics, IEEE Trans. Signal Process. 44 (1996), 2669–2687.
- A. J. Macintyre and W. W. Rogosinski, Extremum problems in the theory of analytic functions, Acta Math. 82 (1950), 275–325. MR 36314, DOI 10.1007/BF02398280
- Robert E. Megginson, An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, New York, 1998. MR 1650235, DOI 10.1007/978-1-4612-0603-3
- Vern I. Paulsen and Mrinal Raghupathi, An introduction to the theory of reproducing kernel Hilbert spaces, Cambridge Studies in Advanced Mathematics, vol. 152, Cambridge University Press, Cambridge, 2016. MR 3526117, DOI 10.1017/CBO9781316219232
- Wade C. Ramey and Heungsu Yi, Harmonic Bergman functions on half-spaces, Trans. Amer. Math. Soc. 348 (1996), no. 2, 633–660. MR 1303125, DOI 10.1090/S0002-9947-96-01383-9
- W. W. Rogosinski and H. S. Shapiro, On certain extremum problems for analytic functions, Acta Math. 90 (1953), 287–318. MR 59354, DOI 10.1007/BF02392438
- Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 255841
- Walter Rudin, Function theory in the unit ball of $\Bbb C^n$, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1980 edition. MR 2446682
- Bernhard Schölkopf, Ralf Herbrich, and Alex J. Smola, A generalized representer theorem, Computational learning theory (Amsterdam, 2001) Lecture Notes in Comput. Sci., vol. 2111, Springer, Berlin, 2001, pp. 416–426. MR 2042050, DOI 10.1007/3-540-44581-1_{2}7
- K. Slavakis, P. Bouboulis, and S. Theodoridis, Online learning in reproducing kernel Hilbert spaces, Academic press library in signal processing: Volume 1, signal processing theory and machine learning, Elsevier, 2014, pp. 883–987.
- Bharath K. Sriperumbudur, Kenji Fukumizu, and Gert R. G. Lanckriet, Universality, characteristic kernels and RKHS embedding of measures, J. Mach. Learn. Res. 12 (2011), 2389–2410. MR 2825431
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, NJ, 1971. MR 304972
- Béla Sz.-Nagy and Adam Korányi, Operatortheoretische Behandlung und Verallgemeinerung eines Problemkreises in der komplexen Funktionentheorie, Acta Math. 100 (1958), 171–202 (German). MR 130577, DOI 10.1007/BF02559538
- Michael Unser, A unifying representer theorem for inverse problems and machine learning, Found. Comput. Math. 21 (2021), no. 4, 941–960. MR 4298236, DOI 10.1007/s10208-020-09472-x
- Yuesheng Xu, Sparse machine learning in Banach spaces, Appl. Numer. Math. 187 (2023), 138–157. MR 4552266, DOI 10.1016/j.apnum.2023.02.011
- Yuesheng Xu and Qi Ye, Generalized Mercer kernels and reproducing kernel Banach spaces, Mem. Amer. Math. Soc. 258 (2019), no. 1243, vi+122. MR 3923255, DOI 10.1090/memo/1243
- Grace Wahba, Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 59, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR 1045442, DOI 10.1137/1.9781611970128
- Rui Wang and Yuesheng Xu, Representer theorems in Banach spaces: minimum norm interpolation, regularized learning and semi-discrete inverse problems, J. Mach. Learn. Res. 22 (2021), Paper No. 225, 65. MR 4329804
- Rui Wang, Yuesheng Xu, and Mingsong Yan, Sparse representer theorems for learning in reproducing kernel Banach spaces, J. Mach. Learn. Res. 25 (2024), Paper No. [93], 45. MR 4749129
- Wen-Jun Zeng, H. C. So, and Abdelhak M. Zoubir, An $\ell _p$-norm minimization approach to time delay estimation in impulsive noise, Digit. Signal Process. 23 (2013), no. 4, 1247–1254. MR 3061202, DOI 10.1016/j.dsp.2013.03.013
- Haizhang Zhang, Yuesheng Xu, and Jun Zhang, Reproducing kernel Banach spaces for machine learning, J. Mach. Learn. Res. 10 (2009), 2741–2775. MR 2579912
- Haizhang Zhang and Jun Zhang, Vector-valued reproducing kernel Banach spaces with applications to multi-task learning, J. Complexity 29 (2013), no. 2, 195–215. MR 3018139, DOI 10.1016/j.jco.2012.09.002
- Haizhang Zhang and Jun Zhang, Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products, Appl. Comput. Harmon. Anal. 31 (2011), no. 1, 1–25. MR 2795872, DOI 10.1016/j.acha.2010.09.007
Bibliographic Information
- Gilbert J. Groenewald
- Affiliation: School of Mathematical and Statistical Sciences, North-West University, Private Bag X6001, Potchefstroom 2520, South Africa
- MR Author ID: 349945
- ORCID: 0000-0002-1658-3648
- Email: Gilbert.Groenewald@nwu.ac.za
- Sanne ter Horst
- Affiliation: School of Mathematical and Statistical Sciences, North-West University, Private Bag X6001, Potchefstroom 2520, South Africa; and DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa
- MR Author ID: 776159
- ORCID: 0000-0003-2562-3370
- Email: Sanne.TerHorst@nwu.ac.za
- Hugo J. Woerdeman
- Affiliation: Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104
- MR Author ID: 183930
- ORCID: 0000-0002-8538-7173
- Email: hugo@math.drexel.edu
- Received by editor(s): February 22, 2024
- Received by editor(s) in revised form: December 3, 2024
- Published electronically: March 19, 2025
- Additional Notes: This work was based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers 118513 and 127364) and the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). In addition, the third author was partially supported by NSF grant DMS-2000037.
- © Copyright 2025 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 378 (2025), 4303-4333
- MSC (2020): Primary 46E15, 42B30, 30H10, 46B10, 46B25
- DOI: https://doi.org/10.1090/tran/9389