Applications of the theory of reproducing kernels to the Tikhonov regularization

• Project/Area Number: 21540111
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Gunma University
• Principal Investigator: SAITOH Saburou 群馬大学, 名誉教授 (10110397)
• Co-Investigator(Kenkyū-buntansha): MATSUURA Tsutomu 群馬大学, 大学院・工学研究科, 准教授 (80181692) ; WATANABE Syuuji 群馬大学, 大学院・工学研究科, 教授 (90222405)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
• Keywords: 再生核 / チホノフ正則化法 / 逆問題 / アルゴリズム / 数値解析 / 近似解積分方程式 / 微分方程式 / 積分方程式 / 行 / 近似解 / チコノフ正則化法 / 一般化逆 / 有界線形作用素 / 近似法 / 離散化 / 離散微分方程式 / 特異積分方程式 / 境界値問題

Research Abstract

We obtained the concept of discrete differential equations and derived various concrete results. For a finite number of input data, we obtain a new concept that we can give natural input data as other input data. We introduced the new definition of general fractional functions and their representations, and methods how to calculate them practically, by computers. We gave the approximate solutions of integral equations with convolution type kernels and with Toeplitz and Hankel kernels. New convolutions and related convolution norm inequalities were derived. We wrote a new research book manuscript with 314 pages.

Research Products

• [Journal Article] Natural outputs and global inputs of linear systems with a finite number of input data (2012)
  • Author(s): L.P. Castro, Saitoh, S.
  • Journal Title: Applicable Analysis Volume: 91 Pages: 225-236
  • Peer Reviewed
• [Journal Article] General Inhomogeneous Discrete Linear Partial Differential Equations with Constant Coefficients on the Whole Spaces (2012)
  • Author(s): L.P. Castro, Saitoh, S., Y. Sawano, A.M. Simoes
  • Journal Title: Complex. Anal. Oper. Theory Volume: 6 Pages: 307-324
  • Peer Reviewed
• [Journal Article] General Inhomogeneous Discrete Linear Partial Differential Equations with Constant Coefficients on the Whole Spaces (2012)
  • Author(s): L.P.Castro
  • Journal Title: Complex.Anal.Oper.Theory Volume: 6 Pages: 307-324
  • Peer Reviewed
• [Journal Article] Interpolation and sampling (2011)
  • Author(s): Butzer, P. L., Ferreira, P. J. S. G., Higgins, J. R., Saitoh, S., Schmeisser, G., Stens, R. L. E. T. Whittaker, K. Ogura and their followers
  • Journal Title: Fourier Anal. Appl. Volume: 17no.2 Pages: 320-354
  • Peer Reviewed
• [Journal Article] Interpolation and sampling : E.T.Whittaker, K.Ogura and their followers (2011)
  • Author(s): P.L.Butzer
  • Journal Title: J.Fourier Anal.Appl. Volume: 17 Issue: 2 Pages: 320-354
  • DOI: 10.1007/s00041-010-9131-8
  • Peer Reviewed
• [Journal Article] Refined generalizations of the triangle inequality on Banach spaces (2010)
  • Author(s): Takahasi, Sin-Ei, Rassias, John M., Saitoh, S., Takahashi, Yasuji
  • Journal Title: Math. Inequal. Appl. Volume: 13no.4 Pages: 733-741
  • Peer Reviewed
• [Journal Article] Source inversion of heat conduction from a finite number of observation data (2010)
  • Author(s): Castro, L. P., Chen, Q., Saitoh, S.
  • Journal Title: Appl.Anal Volume: 89no.6 Pages: 801-813
  • Peer Reviewed
• [Journal Article] Singular integral inequalities and natural regularizations (2010)
  • Author(s): Sawano, Yoshihiro, Yamada, Masato, Saitoh, S.
  • Journal Title: Math. Inequal. Appl. Volume: 13no.2 Pages: 289-303
  • Peer Reviewed
• [Journal Article] Refined generalizations of the triangle inequality on Banach spaces (2010)
  • Author(s): Takahasi, Sin-Ei
  • Journal Title: Math. Inequal. Appl. Volume: 13 Pages: 733-741
  • Peer Reviewed
• [Journal Article] Source inversion of heat conduction from a finite number of observation data (2010)
  • Author(s): Castro, L.P.
  • Journal Title: Applicable Analysis Volume: 89 Pages: 801-813
  • Peer Reviewed
• [Journal Article] Singular integral inequalities and natural regularizations (2010)
  • Author(s): Sawano, Yoshihiro
  • Journal Title: Math. Inequal. Appl. Volume: 13 Pages: 289-303
  • Peer Reviewed
• [Journal Article] Inversion from different kinds of information and real inversion formulas of the Laplace transform from a finite number of data (2010)
  • Author(s): L.P.Castro
  • Journal Title: MATHEMATICS IN ENGINEERING, SCIENCE AND AEROSPACE Volume: 1 Pages: 105-116
  • Peer Reviewed
• [Journal Article] Theory of Reproducing Kernels : Applications to Approximate Solutions of Bounded Linear Operator Functions on Hilbert Spaces (2010)
  • Author(s): S.Saitoh
  • Journal Title: AMS Translations, Series 2 Volume: 230 Pages: 107-134
  • Peer Reviewed
• [Journal Article] A numerical construction of a natural inverse of any matrix by using the theory of reproducing kernels with the Tikhonov regularization (2010)
  • Author(s): K.Iwamura
  • Journal Title: Far East J. Math. Edu. Volume: 4 Pages: 141-149
  • Peer Reviewed
• [Journal Article] Integral and Direct representations of nonlinear inverse mapping (2010)
  • Author(s): Tsutomu Matsuura
  • Journal Title: Applied Mechanics and Materials Volume: 36 Pages: 476-484
  • Peer Reviewed
• [Journal Article] Normal solutions of linear ordinary differential equations of the second order. (2009)
  • Author(s): Uchida Keitaroh, Kumahara Keisaku, Saitoh S.
  • Journal Title: Int. J. Appl Volume: 22no.6 Pages: 981-996
  • Peer Reviewed
• [Journal Article] Numerical solutions of two non-linear simultaneous equations (2009)
  • Author(s): Yamada, Masato, Saitoh, S.
  • Journal Title: Appl. Anal Volume: 88no.2 Pages: 151-160
  • Peer Reviewed
• [Journal Article] Normal solutions of linear ordinary differential equations of the second order (2009)
  • Author(s): Keitaroh Uchida
  • Journal Title: Int.J.Appl.Math 22 Pages: 981-996
  • Peer Reviewed
• [Journal Article] Numerical solutions of two non-linear simultaneous equations (2009)
  • Author(s): M.Yamada
  • Journal Title: Appl.Anal. 88 Pages: 151-160
  • Peer Reviewed
• [Presentation] Applications of the theory of reproducing kernels to convolutions and integral equations (2011)
  • Author(s): Saitoh, S.
  • Organizer: IWOTA、 Sevilla 2011
  • Place of Presentation: Universidad de Sevilla (スペイン)
  • Year and Date: 2011-07-08
• [Presentation] Applications of the theory of reproducing kernels to convolutions and in tegral equations (2011)
  • Author(s): S.Saitoh
  • Organizer: IWOTA Sevilla 2011
  • Place of Presentation: Universidad de Sevilla (Spain)
  • Year and Date: 2011-07-08
• [Presentation] Fundamental error estimate inequalitiea for the Tikhonov regularization using reproducing kernels (2010)
  • Author(s): S, Saitoh
  • Organizer: The 19^<th> seminar on function spaces 2010
  • Place of Presentation: Hokkaido University(Hokkaido)
  • Year and Date: 2010-12-24
• [Presentation] Fundamental error estimates inequalities for the Tikhonov regularization using reproducing kernels (2010)
  • Author(s): Saitoh, S.
  • Place of Presentation: Hajduszoboszlo ( ハンガリー)
  • Year and Date: 2010-09-22
• [Presentation] General inhomogeneous discrete linear partial differential equations with constant coefficients on the whole spaces. (2010)
  • Author(s): Y.Sawano
  • Organizer: Japaneae Mathematical Society
  • Place of Presentation: Nagoya University (Aichi)
  • Year and Date: 2010-09-22
• [Presentation] Fundamental error estimates inequalities for the Tikhonov regularization using reproducing kernels. (2010)
  • Author(s): S, Saitoh
  • Organizer: Conference on Inequalities and Applications 10
  • Place of Presentation: Hajduszoboszlo(Hungary)
  • Year and Date: 2010-09-22
• [Presentation] Constructions of the approximate solutions of singular integral equations by using the Tikhonov regularization and the theory of reproducing kernels (2010)
  • Author(s): Saitoh, S.
  • Organizer: ICNPAA 2010 World Congress : 8th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences
  • Place of Presentation: Sao Jose (ブラジル)
  • Year and Date: 2010-07-01
• [Presentation] Conatructions of the approximate solutions of singular integral equations by using the Tikhonov regularization and the theory of reproducing kernels. (2010)
  • Author(s): S.Saitoh
  • Organizer: ICNPAA 2010 World Congress : 8^<th> International Conference on Mathematical Problems in Engineering, Aerospace and Sciences
  • Place of Presentation: Sao Jose (Brazil)
  • Year and Date: 2010-07-01
• [Presentation] Heat conduction from a finite number of initial heat data (2010)
  • Author(s): S, Saitoh
  • Organizer: Firat Annual Workshop of Functional Analysis and Applicatians Group, CIDMA
  • Place of Presentation: University of Aveiro(Portugal)
  • Year and Date: 2010-05-05
• [Presentation] Analytical and numerical solutions of linear integral equations for bounded operators by using the theory of reproducing kernels. (2010)
  • Author(s): S, Saitoh
  • Organizer: Advanced School on Integral Equations
  • Place of Presentation: IST. (Lisbon)
  • Year and Date: 2010-04-07
• [Presentation] Explicit and direct representations of the solutions of non-linear simultaneous equations (2009)
  • Author(s): Saitoh, S.
  • Organizer: ISAAC
  • Place of Presentation: Imperial College London (イギリス )
  • Year and Date: 2009-07-16
• [Presentation] Explicit and direct representations of the solutions of non-linear simultaneous equations (2009)
  • Author(s): S.Saitoh
  • Organizer: ISAAC
  • Place of Presentation: Imperial College London, UK
  • Year and Date: 2009-07-16
• [Book] Theory of reproducing kernels : applications to approximate solutions of bounded linear operator equations on Hilbert spaces (2010)
  • Author(s): Saitoh, S.
• [Book] World Sci (2010)
  • Author(s): M. Yamada, Saitoh,S.
  • Publisher: Hackensack, NJ
• [Book] Progress in Analysis and its Applications, Proceedings of the 7^<th> International ISAAC Congress (2010)
  • Author(s): M.Yamada
  • Publisher: World Scientific
• [Book] World Sci (2009)
  • Author(s): Yamada, M., Saitoh,S.
  • Publisher: Hackensack, NJ
• [Book] World Sci (2009)
  • Author(s): Fjiwara, H., Matsuura, T., Saitoh, S., Sawano, Y.
  • Publisher: Hackensack
• [Book] More Progresses In Analysis (2009)
  • Author(s): H.Itoh
  • Total Pages: 8
  • Publisher: World Scientific
• [Book] More Progresses In Analysis (2009)
  • Author(s): T.Matsuura
  • Total Pages: 10
  • Publisher: World Scientific
• [Patent(Industrial Property Rights)] 逆ラプラス変換プログラム、逆ラプラス変換のためのテーブル作成プログラム、逆ラプラス変換の数値解算出プログラム、および逆ラプラス変換装置 (2010)
  • Inventor(s): 齋藤三郎, 松浦勉(群馬大学), 藤原宏志(京都大学)
  • Industrial Property Rights Holder: 国立大学法人群馬大学,国立大法人京都大学
  • Filing Date: 2010-08-12
  • Overseas
• [Patent(Industrial Property Rights)] 逆ラプラス変換プログラム、逆ラプラス変換のためのテーブル作成プログラム、逆ラプラス変換の数値解算出プログラム、および逆ラプラス変換装置 (2010)
  • Inventor(s): 齋藤三郎, 松浦勉(群馬大学), 藤原宏志(京都大学)
  • Industrial Property Rights Holder: 国立大学法人群馬大学,国立大法人京都大学
  • Filing Date: 2010-02-16
  • Overseas
• [Patent(Industrial Property Rights)] 逆ラプラス変換プログラム、逆ラプラス変換のためのテーブル作成プログラム、逆ラプラス変換の数値解算出プログラム、および逆ラプラス変換装置 (2010)
  • Inventor(s): 齋藤三郎, 松浦勉(群馬大学), 藤原宏志(京都大学)
  • Industrial Property Rights Holder: 国立大学法人群馬大学,国立大法人京都大学
  • Filing Date: 2010-02-12
  • Overseas