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Theory of Reproducing Kernels

  • Saburou Saitoh
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 10)

Abstract

In this survey article, we would like to show that the theory of reproducing kernels is fundamental, is beautiful and is applicable widely in mathematics. At the same time, we shall present some operator versions of our fundamental theory in the general theory of reproducing kernels, as original results.

Keywords

Hilbert Space Interpolation Problem Inversion Formula Reproduce Kernel Hilbert Space Norm Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Aikawa, H., Hayashi, N., Saitoh, S.: The Bergman space on a sector and the heat equation. Complex Variables, Theory Appl. 15 (1990), 27–36.MathSciNetzbMATHGoogle Scholar
  2. [2]
    Aikawa, H., Hayashi, N., Saitoh, S.: Isometrical identities for the Bergman and the Szegö spaces on a sector. J. Math. Soc. Japan 43 (1991), 196–201.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Amano, K., Saitoh, S., Syarif, A.: A real inversion formula for the Laplace transform in a Sobolev space. Zeitschrift für Analysis und ihre Anwendungen 18 (1999), 1031–1038.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Amano, K., Saitoh, S., Yamamoto, M.: Error estimates of the real inversion formulas of the Laplace transform. Graduate School of Math. Sci. The University of Tokyo, Preprint Series 1998, 98–29. Integral Transforms and Special Functions 10 (2000), 1–14.MathSciNetGoogle Scholar
  5. [5]
    Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 (1950), 337–404.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Bergman, S.: The kernel function and conformal mapping. Amer. Math. Soc., Providence, R. I. 1970.Google Scholar
  7. [7]
    Byun, D.-W., Saitoh, S.: A real inversion formula for the Laplace transform. Zeitschrift für Analysis and ihre Anwendungen 12 (1993), 597–603.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Byun, D.-W., Saitoh, S.: Approximation by the solutions of the heat equation. J. Approximation Theory 78 (1994), 226–238.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Byun, D.-W., Saitoh, S.: Best approximation in reproducing kernel Hilbert spaces. Proc. of the 2th International Colloquium on Numerical Analysis, VSPHolland, 1994, 55–61.Google Scholar
  10. [10]
    Hayashi, N.: Analytic function spaces and their applications to nonlinear evolution equations. Analytic Extension Formulas and their Applications, Kluwer Academic Publishers, 2001, 59–86.Google Scholar
  11. [11]
    Hayashi, N., Saitoh, S.: Analyticity and smoothing effect for the Schrödinger equation. Ann. Inst. Henri Poincaré 52 (1990), 163–173.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Higgins, J.R.: A sampling principle associated with Saitoh’s fundamental theory of linear transformations. Analytic Extension Formulas and their Applications, Kluwer Academic Publishers, Dordrecht 2001, 73–86.Google Scholar
  13. [13]
    Kajiwara, J., Tsuji, M.: Program for the numerical analysis of inverse formula for the Laplace transform. Proceedings of the Second Korean-Japanese Colloquium on Finite or Infinite Dimensional Complex Analysis, 1994, 93–107.Google Scholar
  14. [14]
    Kajiwara, J., Tsuji, M.: Inverse formula for Laplace transform. Proceedings of the 5th International Colloquium on Differential Equations, VHP-Holland 1995, 163–172.Google Scholar
  15. [15]
    Körezlioglu, H.: Reproducing kernels in separable Hilbert spaces. Pacific J. Math. 25 (1968), 305–314.zbMATHGoogle Scholar
  16. [16]
    Nakamura, G., Saitoh, S., Syarif, A.: Representations of initial heat distributions by means of their heat distributions as functions of time. Inverse Problems 15 (1999), 1255–1261.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Rassias, Th.M., Saitoh, S.: The Pythagorean theorem and linear mappings. PanAmerican Math. J. 12 (2002), 1–10.MathSciNetzbMATHGoogle Scholar
  18. [18]
    Saitoh, S.: Hilbert spaces induced by Hilbert space valued functions. Proc. Amer. Math. Soc. 89 (1983), 74–78.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Saitoh, S.: The Weierstrass transform and an isometry in the heat equation. Applicable Analysis 16 (1983), 1–6.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Saitoh, S.: Theory of Reproducing Kernels and its Applications. Pitman Research Notes in Mathematics Series 189. Longman Scientific & Technical, Harlow 1998.Google Scholar
  21. [21]
    Saitoh, S.: Interpolation problems of Pick-Nevanlinna type. Pitman Research Notes in Mathematics Series 212 (1989), 253–262.MathSciNetGoogle Scholar
  22. [22]
    Saitoh, S.: Representations of the norms in Bergman-Selberg spaces on strips and half planes. Complex Variables, Theory Appl. 19 (1992); 231–241.MathSciNetzbMATHGoogle Scholar
  23. [23]
    Saitoh, S.: One approach to some general integral transforms and its applications. Integral Transforms and Special Functions 3 (1995), 49–84.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Saitoh, S.: Natural norm inequalities in nonlinear transforms. General Inequalities 7 (1997), 39–52.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Saitoh, S.: Representations of inverse functions. Proc. Amer. Math. Soc. 125 (1997), 3633–3639.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Saitoh, S.: Integral Transforms, Reproducing Kernels and their Applications. Pitman Research Notes in Mathematics Series 369. Addison Wesley Longman, Harlow 1997.Google Scholar
  27. [27]
    Saitoh, S., Yamamoto, M.: Stability of Lipschitz type in determination of initial heat distribution. J. of Inequa. & Appl. 1 (1997), 73–83.MathSciNetzbMATHGoogle Scholar
  28. [28]
    Saitoh, S.: Nonlinear transforms and analyticity of functions. Nonlinear Mathematical Analysis and Applications. Hadronic Press, Palm Harbor, 1998, 223–234.Google Scholar
  29. [29]
    Saitoh, S.: Various operators in Hilbert space induced by transforms. International J. of Applied Math. 1 (1999), 111–126.MathSciNetzbMATHGoogle Scholar
  30. [30]
    Saitoh, S.: Applications of the general theory of reproducing kernels. Reproducing Kernels and their Applications. Kluwer Academic Publishers, Dordrecht 1999, 165–188.Google Scholar
  31. [31]
    Saitoh, S., Yamamoto, M.: Integral transforms involving smooth functions. Reproducing Kernels and their Applications. Kluwer Academic Publishers, Dordrecht 1999, 149–164.Google Scholar
  32. [32]
    Saitoh, S.: Linear integro-differential equations and the theory of reproducing kernels. Volterra Equations and Applications. C. Corduneanu and I.W. Sandberg (eds), Gordon and Breach Science Publishers, Amsterdam 2000.Google Scholar
  33. [33]
    Saitoh, S.: Representations of the solutions of partial differential equations of parabolic and hyperbolic types by means of time observations. Applicable Analysis 76 (2000), 283–289.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    Saitoh, S.: Analytic extension formulas, integral transforms and reproducing kernels. Analytic Extension Formulas and their Applications. Kluwer Academic Publishers, Dordrecht, 2001, 207–232.Google Scholar
  35. [35]
    Saitoh, S., Than, V.K., Yamamoto, M.: Conditional Stability of a Real Inverse Formula for the Laplace Transform. Zeitschrift für Analysis und ihre Anwendungen 20 (2001), 193–202.zbMATHGoogle Scholar
  36. [36]
    Saitoh, S.: Applications of the reproducing kernel theory to inverse problems. Comm. Korean Math. Soc. 16 (2001), 371–383.MathSciNetzbMATHGoogle Scholar
  37. [37]
    Saitoh, S., Mori, M.: Representations of analytic functions in terms of local values by means of the Riemann mapping function. Complex Variables, Theory Appl. 45 (2002), 387–393.MathSciNetGoogle Scholar
  38. [38]
    Saitoh, S.: Principle of telethoscope. Proceedings of the Graz Workshop on “Functional-analytic and complex methods, their interaction and applications to Partial Differential Equations”, World Scientific, Singapore, 2001, 101–117.CrossRefGoogle Scholar
  39. [39]
    Schwartz, L.: Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (noyaux reproduisants). J. Analyse Math. 13 (1963), 115–256.CrossRefGoogle Scholar
  40. [40]
    Tuan, V.K., Saitoh, S., Saigo, M.: Size of support of initial heat distribution in the 1D heat equation. Applicable Analysis 74 (2000), 439–446.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Saburou Saitoh
    • 1
  1. 1.Department of Mathematics Faculty of EngineeringGunma UniversityKiryuJapan

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