A reproducing kernel theory with some general applications. (English) Zbl 1394.46020
Qian, Tao (ed.) et al., Mathematical analysis, probability and applications – plenary lectures. ISAAC congress, August 3–8, 2015, Macau, China. Cham: Springer (ISBN 978-3-319-41943-5/hbk; 978-3-319-41945-9/ebook). Springer Proceedings in Mathematics & Statistics 177, 151-182 (2016).
Summary: In this paper, some essences of the general theory of reproducing kernels from the viewpoint of general applications and general interest will be introduced by our recent results.
For the entire collection see [Zbl 1352.35004].
For the entire collection see [Zbl 1352.35004].
MSC:
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
| 46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |
Keywords:
reproducing kernel; generalized delta function; Aveiro discretization; approximation; Tikhonov regularization; general fraction; division by zero; integral transform; initial value problem; convolutionReferences:
| [1] | Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337-404 (1950) · Zbl 0037.20701 · doi:10.1090/S0002-9947-1950-0051437-7 |
| [2] | Bergman, S.: 1970 The Kernel Function and Conformal Mapping. American Mathematical Society, Providence, R.I. (1950) · Zbl 0040.19001 |
| [3] | Bergstra, J.A., Hirshfeld, Y., Tucker, J.V.: Meadows and the equational specification of division. arXiv:0901.0823v1 [math.RA] 7 Jan 2009 (2009) · Zbl 1172.68039 |
| [4] | Berlinet, A., Thomas-Agnan, C.T.: Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer Akademic Publishers, Boston (2004) · Zbl 1145.62002 · doi:10.1007/978-1-4419-9096-9 |
| [5] | Berlinet, A.: Reproducing kernels in probability and statistics. In: More Progresses In Analysis, Proceedings of the 5th International ISAAC Congress, pp. 153-162 (2010) · Zbl 1187.46021 |
| [6] | Castro, L.P., Saitoh, S.: Fractional functions and their representations. Complex Anal. Oper. Theory 7(4), 1049-1063 (2013) · Zbl 1285.46024 · doi:10.1007/s11785-011-0154-1 |
| [7] | Castro, L.P., Saitoh, S., Tuan, N.M.: Convolutions, integral transforms and integral equations by means of the theory of reproducing kernels. Opuscula Math. 32(4), 633-646 (2013) · Zbl 1279.30004 · doi:10.7494/OpMath.2012.32.4.633 |
| [8] | Castro, L.P., Fujiwara, H., Saitoh, S., Sawano, Y., Yamada, A., Yamada, M.: Fundamental error estimates inequalities for the Tikhonov regularization using reproducing kernels. In: International Series of Numerical Mathematics, Inequalities and Applications 2010, vol. 161, pp. 87-101. Springer, Basel (2010) · Zbl 1253.46037 |
| [9] | Castro, L.P., Fujiwara, H., Rodrigues, M.M., Saitoh, S., Tuan, V.K.: Aveiro discretization method in mathematics: a new discretization principle. In: Pardalos, P., Rassias, T.M. (eds.) Mathematics Without Boundaries: Surveys in Pure Mathematics, pp. 37-92. Springer (2014) · Zbl 1319.65103 |
| [10] | Castro, L.P., Fujiwara, H., Qian, T., Saitoh, S.: How to catch smoothing properties and analyticity of functions by computers? In: Pardalos, P., Rassias, T.M. (eds.) Mathematics Without Boundaries: Surveys in Interdisciplinary Research, pp. 101-116. Springer (2014) · Zbl 1328.65126 |
| [11] | Castro, L.P., Rodorigues, M.M., Saitoh, S.: A fundamental theorem on initial value problems by using the theory of reproducing kernels. Complex Anal. Oper. Theory 9, 87-98 (2015) · Zbl 1342.47088 · doi:10.1007/s11785-014-0375-1 |
| [12] | Fujiwara, H.: Applications of reproducing kernel spaces to real inversions of the Laplace transform. RIMS Koukyuuroku 1618, 188-209 (2008) |
| [13] | Fujiwara, H., Matsuura, T., Saitoh, S., Sawano, Y.: Numerical real inversion of the Laplace transform by using a high-accuracy numerical method. In: Further Progress in Analysis, pp. 574-583. World Science Publisher, Hackensack, NJ (2009) · Zbl 1180.65169 |
| [14] | Fujiwara, H.: Numerical real inversion of the Laplace transform by reproducing kernel and multiple-precision arithmetric. In: Proceedings of the 7th International ISAAC Congress Progress in Analysis and Its Applications, pp. 289-295. World Scientific (2010) · Zbl 1266.65204 |
| [15] | Fujiwara, H., Higashimori, N.: Numerical inversion of the Laplace transform by using multiple-precision arithmetic. Libertas Mathematica (New Series) 34(2), 5-21 (2014) · Zbl 1330.65195 |
| [16] | Fujiwara, H., Saitoh, S.: The general sampling theory by using reproducing kernels. In: Pardalos, P., Rassias, Th. M. (eds.) Contributions in Mathematics and Engineering in Honor of Constantin Caratheodory Springer (in press) · Zbl 1381.94048 |
| [17] | Fukumizu, K., Bach, F.R., Jordan, M.I.: Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces. J. Mach. Learn. Res. 5, 73-99 (2004) · Zbl 1222.62069 |
| [18] | Fukumizu, K., Bach, F.R., Gretton, A.: Statistical consistency of kernel canonical correlation analysis. J. Mach. Learn. Res. 8, 361-383 (2007) · Zbl 1222.62063 |
| [19] | Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34, 561-580 (1992) · Zbl 0770.65026 · doi:10.1137/1034115 |
| [20] | Lawson, C.L., Hanson, R.J.: Solving Least Squares Problems. Prentice-Hall, Englewood Cliffs (1974) · Zbl 0860.65028 |
| [21] | Kuroda, M., Michiwaki, H., Saitoh, S., Yamane, M.: New meanings of the division by zero and interpretations on \(100/0 = 0\) and on \(0/0 = 0\). Int. J. Appl. Math. 27(2), 191-198 (2014) · Zbl 1291.30049 · doi:10.12732/ijam.v27i2.9 |
| [22] | Kolmogorov, A.N.: Stationary sequences in Hilbert’s space. Bolletin Moskovskogo Gosudarstvenogo Universiteta, Matematika 2, 40 (1941) (in Russian) · Zbl 0063.03291 |
| [23] | Matsuura, T., Saitoh, S.: Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley-Wiener spaces. Appl. Anal. 85, 901-915 (2006) · Zbl 1101.65116 · doi:10.1080/00036810600643662 |
| [24] | Matsuura, T., Saitoh, S.: Matrices and division by zero z/0 = 0. Adv. Linear Algebra Matrix Theory 6, 51-58 (2016) |
| [25] | Michiwaki H, Saitoh S, Yamada M.: Reality of the division by zero \(z/0=0\). IJAPM Int. J. Appl. Phys. Math. 6, 1-8 (2016) |
| [26] | Michiwaki, H., Okumura, H., Saitoh, S.: Division by zero z/0 = 0 in Euclidean spaces. Int. J. Math. Comput. 28 (2017) (in press) |
| [27] | Mo, Y., Qian, T.: Support vector machine adapted Tikhonov regularization method to solve Dirichlet problem. Appl. Math. Comput. 245, 509-519 (2014) · Zbl 1337.65172 |
| [28] | Rocha, E.M.: A reproducing kernel Hilbert discretization method for linear PDEs with nonlinear right-hand side. Libertas Mathematica (New Series) 34(2), 91-104 (2014) · Zbl 1330.65169 |
| [29] | Saitoh, S.: Integral Transforms, Reproducing Kernels and their Applications. Pitman Research Notes in Mathematical Series 369. Addison Wesley Longman, Harlow, CRC Press, Taylor & Francis Group, Boca Raton London, New York (in hard cover) (1997) |
| [30] | Saitoh, S.: Various operators in Hilbert space induced by transforms. Int. J. Appl. Math. 1, 111-126 (1999) · Zbl 1171.46308 |
| [31] | Saitoh, S.: Theory of reproducing kernels: applications to approximate solutions of bounded linear operator equations on Hilbert spaces. Am. Math. Soc. Transl. Ser. 2(230), 107-137 (2010) · doi:10.1090/trans2/230/06 |
| [32] | Saitoh, S.: Generalized inversions of Hadamard and tensor products for matrices. Adv. Linear Algebra Matrix Theory 4(2), 87-95 (2014) · doi:10.4236/alamt.2014.42006 |
| [33] | Saitoh, S., Sawano, Y.: Generalized delta functions as generalized reproducing kernels (manuscript) · Zbl 1383.30004 |
| [34] | Saitoh, S., Sawano, Y.: General initial value problems using eigenfunctions and reproducing kernels (manuscript) · Zbl 1383.30004 |
| [35] | Takahasi, S.E., Tsukada, M.: Kobayashi Y Classification of continuous fractional binary operators on the real and complex fields. Tokyo J. Math. 38, 369-380 (2015) · Zbl 1364.30005 |
| [36] | Vapnik, V.N.: Statistical Learning Theory. Wiley, New York (1998) · Zbl 0935.62007 |
| [37] | Vourdas, A.: Analytic representations in quantum mechanics. J. Phys. A: Math. Gen. 39, 65-141 (2006) · Zbl 1093.81038 |
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