Concrete Hilbert Spaces

  • Kenneth Lange
Part of the Statistics and Computing book series (SCO)


In this chapter we consider an infinite-dimensional generalization of Euclidean space introduced by the mathematician David Hilbert. This generalization preserves two fundamental geometric notions of Euclidean space— namely, distance and perpendicularity. Both of these geometric properties depend on the existence of an inner product. In the infinite-dimensional case, however, we take the inner product of functions rather than of vectors. Our emphasis here will be on concrete examples of Hilbert spaces relevant to statistics. To keep our discussion within bounds, some theoretical facts are stated without proof. Relevant proofs can be found in almost any book on real or functional analysis [6, 12]. Applications of our examples to numerical integration, wavelets, and other topics appear in later chapters.


Hilbert Space Orthogonal Polynomial Cauchy Sequence Reproduce Kernel Hilbert Space Laguerre Polynomial 
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  1. 1.
    Aronszajn N (1950) Theory of reproducing kernels. Amer Math Soc Trans 63:337-404CrossRefMathSciNetGoogle Scholar
  2. 2.
    Berlinet A, Thomas-Agnan C (2004) Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer, Boston, MAMATHGoogle Scholar
  3. 3.
    Burges CJC (1998) A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery 2:121-167CrossRefGoogle Scholar
  4. 4.
    Dym H, McKean HP (1972) Fourier Series and Integrals. Academic Press, New YorkMATHGoogle Scholar
  5. 5.
    Hastie T, Tibshirani R, Friedman J (2001) The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New YorkMATHGoogle Scholar
  6. 6.
    Hewitt E, Stromberg K (1965) Real and Abstract Analysis. Springer, New YorkMATHGoogle Scholar
  7. 7.
    Hochstadt H (1986) The Functions of Mathematical Physics. Dover, New YorkMATHGoogle Scholar
  8. 8.
    Ismail MEH (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, CambridgeMATHGoogle Scholar
  9. 9.
    Newton HJ (2002) A Conversation with Emanuel Parzen. Stat Science 17:357-378MATHCrossRefGoogle Scholar
  10. 10.
    Parthasarathy KR (1977) Introduction to Probability and Measure. Springer, New YorkMATHGoogle Scholar
  11. 11.
    Pearce ND, Wand MP (2006) Penalised splines and reproducing kernel methods. Amer Statistician 60:233-240CrossRefMathSciNetGoogle Scholar
  12. 12.
    Rudin W (1973) Functional Analysis. McGraw-Hill, New YorkMATHGoogle Scholar
  13. 13.
    Schölkopf B, Smola AJ (2002) Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge, MAGoogle Scholar
  14. 14.
    Vapnik V (1995) The Nature of Statistical Learning Theory. Springer, New YorkMATHGoogle Scholar
  15. 15.
    Wahba G (1990) Spline Models for Observational Data. SIAM, PhiladelphiaMATHGoogle Scholar

Copyright information

© Springer New York 2010

Authors and Affiliations

  1. 1.Departments of Biomathematics, Human Genetics, and Statistics David Geffen School of MedicineUniversity of California, Los AngelesLos AngelesUSA

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