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doi: 10.3934/dcdss.2021105
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Harmonic analysis of network systems via kernels and their boundary realizations

1. 

Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA

2. 

Mathematical Reviews, 416 4th Street, Ann Arbor, MI 48103-4816, USA

* Corresponding author: James Tian

Received  March 2021 Revised  June 2021 Early access September 2021

With view to applications to harmonic and stochastic analysis of infinite network/graph models, we introduce new tools for realizations and transforms of positive definite kernels (p.d.) $ K $ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of factorizations: (i) Probabilistic: Starting with a positive definite kernel $ K $ we analyze associated Gaussian processes $ V $. Properties of the Gaussian processes will be derived from certain factorizations of $ K $, arising as a covariance kernel of $ V $. (ii) Geometric analysis: We discuss families of measure spaces arising as boundaries for $ K $. Our results entail an analysis of a partial order on families of p.d. kernels, a duality for operators and frames, optimization, Karhunen–Loève expansions, and factorizations. Applications include a new boundary analysis for the Drury-Arveson kernel, and for certain fractals arising as iterated function systems; and an identification of optimal feature spaces in machine learning models.

Citation: Palle Jorgensen, James Tian. Harmonic analysis of network systems via kernels and their boundary realizations. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021105
References:
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D. Alpay and H. Dym, On reproducing kernel spaces, the Schur algorithm, and interpolation in a general class of domains, in Operator Theory and Complex Analysis (Sapporo, 1991), vol. 59 of Oper. Theory Adv. Appl., Birkhäuser, Basel, 1992, 30–77.

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show all references

References:
[1]

D. Alpay and V. Bolotnikov, On tangential interpolation in reproducing kernel Hilbert modules and applications, in Topics in Interpolation Theory (Leipzig, 1994), vol. 95 of Oper. Theory Adv. Appl., Birkhäuser, Basel, 1997, 37–68.

[2]

D. AlpayV. Bolotnikov and H. T. Kaptanoğlu, The Schur algorithm and reproducing kernel Hilbert spaces in the ball, Linear Algebra Appl., 342 (2002), 163-186.  doi: 10.1016/S0024-3795(01)00448-7.

[3]

D. AlpayP. Cerejeiras and U. Kähler, Gleason's problem associated to the fractional Cauchy-Riemann operator, Fueter series, Drury-Arveson space and related topics, Proc. Amer. Math. Soc., 145 (2017), 4821-4835.  doi: 10.1090/proc/13613.

[4]

D. Alpay, P. Dewilde and H. Dym, Lossless inverse scattering and reproducing kernels for upper triangular operators, in Extension and Interpolation of Linear Operators and Matrix Functions, vol. 47 of Oper. Theory Adv. Appl., Birkhäuser, Basel, (1990), 61–135.

[5]

D. Alpay, A. Dijksma, J. Rovnyak and H. S. V. de Snoo, Realization and factorization in reproducing kernel Pontryagin spaces, in Operator Theory, System Theory and Related Topics (Beer-Sheva/Rehovot, 1997), vol. 123 of Oper. Theory Adv. Appl., Birkhäuser, Basel, (2001), 43–65.

[6]

D. Alpay and C. Dubi, Some remarks on the smoothing problem in a reproducing kernel Hilbert space, J. Anal. Appl., 4 (2006), 119-132. 

[7]

D. Alpay and H. Dym, On reproducing kernel spaces, the Schur algorithm, and interpolation in a general class of domains, in Operator Theory and Complex Analysis (Sapporo, 1991), vol. 59 of Oper. Theory Adv. Appl., Birkhäuser, Basel, 1992, 30–77.

[8]

D. Alpay and H. Dym, On a new class of structured reproducing kernel spaces, J. Funct. Anal., 111 (1993), 1-28.  doi: 10.1006/jfan.1993.1001.

[9]

D. Alpay and P. Jorgensen, Spectral theory for Gaussian processes: Reproducing kernels, boundaries, and $L^2$-wavelet generators with fractional scales, Numer. Funct. Anal. Optim., 36 (2015), 1239-1285.  doi: 10.1080/01630563.2015.1062777.

[10]

D. AlpayP. Jorgensen and D. Levanony, On the equivalence of probability spaces, J. Theoret. Probab., 30 (2017), 813-841.  doi: 10.1007/s10959-016-0667-7.

[11]

D. Alpay and P. E. T. Jorgensen, Stochastic processes induced by singular operators, Numer. Funct. Anal. Optim., 33 (2012), 708-735.  doi: 10.1080/01630563.2012.682132.

[12]

D. Alpay and D. Levanony, On the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions, Potential Anal., 28 (2008), 163-184.  doi: 10.1007/s11118-007-9070-4.

[13]

D. Alpay and T. M. Mills, A family of Hilbert spaces which are not reproducing kernel Hilbert spaces, J. Anal. Appl., 1 (2003), 107-111. 

[14]

N. Arcozzi and M. Levi, On a class of shift-invariant subspaces of the Drury-Arveson space, Concr. Oper., 5 (2018), 1-8.  doi: 10.1515/conop-2018-0001.

[15]

N. Aronszajn, La théorie des noyaux reproduisants et ses applications. I, Proc. Cambridge Philos. Soc., 39 (1943), 133-153.  doi: 10.1017/S0305004100017813.

[16]

N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404.  doi: 10.1090/S0002-9947-1950-0051437-7.

[17]

N. Aronszajn and K. T. Smith, Characterization of positive reproducing kernels. Applications to Green's functions, Amer. J. Math., 79 (1957), 611-622.  doi: 10.2307/2372565.

[18]

W. Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math., 181 (1998), 159-228.  doi: 10.1007/BF02392585.

[19]

G. ChengX. Hou and C. Liu, The singular integral operator induced by Drury-Arveson kernel, Complex Anal. Oper. Theory, 12 (2018), 917-929.  doi: 10.1007/s11785-016-0537-4.

[20]

F. Cucker and S. Smale, On the mathematical foundations of learning, Bull. Amer. Math. Soc. (N.S.), 39 (2002), 1–49 (electronic). doi: 10.1090/S0273-0979-01-00923-5.

[21]

S. W. Drury, A generalization of von Neumann's inequality to the complex ball, Proc. Amer. Math. Soc., 68 (1978), 300-304.  doi: 10.2307/2043109.

[22]

D. E. Dutkay and P. E. T. Jorgensen, Affine fractals as boundaries and their harmonic analysis, Proc. Amer. Math. Soc., 139 (2011), 3291-3305.  doi: 10.1090/S0002-9939-2011-10752-4.

[23]

D. E. Dutkay and P. E. T. Jorgensen, Unitary groups and spectral sets, J. Funct. Anal., 268 (2015), 2102-2141.  doi: 10.1016/j.jfa.2015.01.018.

[24]

W. E and S. Wojtowytsch, Kolmogorov width decay and poor approximators in machine learning: Shallow neural networks, random feature models and neural tangent kernels, Res. Math. Sci., 8 (2021), Paper No. 5, 28 pp. doi: 10.1007/s40687-020-00233-4.

[25]

M. Geiger, A. Jacot, S. Spigler, F. Gabriel, L. Sagun, S. d'Ascoli, G. Biroli, C. Hongler and M. Wyart, Scaling description of generalization with number of parameters in deep learning, J. Stat. Mech. Theory Exp., (2020), 023401, 23 pp. doi: 10.1088/1742-5468/ab633c.

[26]

M. Geiger, S. Spigler, A. Jacot and M. Wyart, Disentangling feature and lazy training in deep neural networks, J. Stat. Mech. Theory Exp., (2020), 113301, 27 pp. doi: 10.1088/1742-5468/abc4de.

[27]

Y. HaoP. Li and K. Zhao, Regularities of semigroups, Carleson measures and the characterizations of BMO-type spaces associated with generalized Schrödinger operators, Banach J. Math. Anal., 13 (2019), 1-25.  doi: 10.1215/17358787-2018-0013.

[28]

J. E. Herr, P. E. T. Jorgensen and E. S. Weber, A matrix characterization of boundary representations of positive matrices in the Hardy space, in Frames and Harmonic Analysis, vol. 706 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2018), 255–270. doi: 10.1090/conm/706/14211.

[29]

J. E. HerrP. E. T. Jorgensen and E. S. Weber, A characterization of boundary representations of positive matrices in the Hardy space via the Abel product, Linear Algebra Appl., 576 (2019), 51-66.  doi: 10.1016/j.laa.2018.02.023.

[30]

T. Hida, Quadratic functionals of Brownian motion, J. Multivariate Anal., 1 (1971), 58-69.  doi: 10.1016/0047-259X(71)90029-7.

[31]

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Figure 1.  Current flows in a connected resistance network
Figure 2.  Transition probabilities $ p_{xy} $ at a vertex $ x $ $ \left(\mbox{in }V\right) $
Figure 3.  $ v_{x}\left(\cdot\right) = \cdot\wedge x $
Figure 4.  ${}^{1}\!\!\diagup\!\!{}_{4}\; $-Cantor set
Figure 5.  A Swiss role
Figure 6.  SVM using Gaussian kernel
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