Skip to main content

Distributions and Function Spaces

  • Chapter

Part of the Springer Series in Computational Mathematics book series (SSCM,volume 46)

Abstract

Chapter 1 provides a brief survey of basic results from linear functional analysis, particularly Banach and Hilbert space theory, and an overview of standard results from the theory of distributions and function spaces, including isotropic and anisotropic Sobolev spaces, Besov spaces, Fourier multipliers and mollifiers in function spaces, and function space interpolation.

Keywords

  • Fourier Multiplier
  • Mollifiers
  • Linear Functionals
  • Besov Spaces
  • Normed Linear Space

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-1-4471-5460-0_1
  • Chapter length: 90 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   99.00
Price excludes VAT (USA)
  • ISBN: 978-1-4471-5460-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   129.00
Price excludes VAT (USA)
Hardcover Book
USD   139.99
Price excludes VAT (USA)

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Pure and Applied Mathematics Series, vol. 140. Elsevier/Academic Press, Amsterdam (2003)

    MATH  Google Scholar 

  3. Bartle, R.G.: The Elements of Integration and Lebesgue Measure. Wiley, New York (1995)

    CrossRef  MATH  Google Scholar 

  4. Bergh, J., Löfström, J.: Interpolation Spaces, an Introduction. Grundlehren der mathematischen Wissenschaften, vol. 228. Springer, Berlin (1976)

    CrossRef  MATH  Google Scholar 

  5. Besov, O.V., Il’in, V.P., Nikol’skiĭ, S.M.: Integral Representations of Functions and Imbedding Theorems. Nauka, Moscow (1975). (Russian)

    Google Scholar 

  6. Dražić, M.: Convergence rates of difference approximations to weak solutions of the heat transfer equation. Technical Report 86/22, Oxford University Computing Laboratory, Numerical Analysis Group, Oxford (1986)

    Google Scholar 

  7. Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Interscience, New York (1957)

    Google Scholar 

  8. Dyda, B.: A fractional order Hardy inequality. Ill. J. Math. 48(2), 575–588 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Edwards, R.E.: Fourier Series: A Modern Introduction vol. 2, 2nd edn. Springer, New York (1982)

    CrossRef  MATH  Google Scholar 

  10. Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, Berlin (1969)

    MATH  Google Scholar 

  11. Gagliardo, E.: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Semin. Mat. Univ. Padova 27, 284–305 (1957)

    MathSciNet  MATH  Google Scholar 

  12. Gel’fand, I.M., Shilov, G.E.: Generalized Functions, Vol. I: Properties and Operations. Academic Press, New York (1964)

    MATH  Google Scholar 

  13. Grafakos, L.: Classical Fourier Analysis, 2nd edn. Graduate Texts in Mathematics, vol. 249. Springer, New York (2008)

    MATH  Google Scholar 

  14. Grisvard, P.: Équations différentielles abstraites. Ann. Sci. Éc. Norm. Super. 4(2), 311–395 (1969). (French)

    MathSciNet  Google Scholar 

  15. Grisvard, P.: Elliptic Problems in Non-smooth Domains. Pitman, London (1985)

    Google Scholar 

  16. Hörmander, L.: Lectures on linear partial differential operators. Mimeographed Notes, Stanford University (1960)

    Google Scholar 

  17. Hörmander, L.: Linear Partial Differential Operators 4th edn. Die Grundlehren der mathematischen Wissenschaften, vol. 116. Springer, Berlin (1969)

    MATH  Google Scholar 

  18. Jacobsen, N.: Basic Algebra, vol. II. Freeman, San Francisco (1980)

    Google Scholar 

  19. Kreiss, H.-O., Thomée, V., Widlund, O.: Smoothing of initial data and rates of convergence for parabolic difference equations. Commun. Pure Appl. Math. 23(2), 241–259 (1970)

    CrossRef  MATH  Google Scholar 

  20. Kufner, A., John, O., Fučik, S.: Function Spaces. Nordhoff International Publ., Leyden (1977)

    MATH  Google Scholar 

  21. Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications. Dunod, Paris (1968)

    MATH  Google Scholar 

  22. Lizorkin, P.I.: Generalized Liouville differentiation and the functional spaces \(L^{r}_{p}(E_{n})\). imbedding theorems. Mat. Sb. (N. S.) 60(102)(3), 325–353 (1963). (Russian)

    MathSciNet  Google Scholar 

  23. Massey, W.S.: Singular Homology Theory. Graduate Texts in Mathematics, vol. 70. Springer, New York (1980)

    CrossRef  MATH  Google Scholar 

  24. Maz’ya, V.G.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 342. Springer, Heidelberg (2011)

    MATH  Google Scholar 

  25. Maz’ya, V.G., Shaposhnikova, T.O.: Theory of Multipliers in Spaces of Differentiable Functions. Monographs and Studies in Mathematics, vol. 23. Pitman, Boston (1985)

    MATH  Google Scholar 

  26. Maz’ya, V.G., Shaposhnikova, T.O.: Multiplikatory v prostranstvakh differentsiruemykh funktsii. Leningrad. Univ., Leningrad (1986)

    Google Scholar 

  27. Narasimhan, R.: Several Complex Variables. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1995). Reprint of the 1971 original

    Google Scholar 

  28. Nikol’skiĭ, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Nauka, Moscow (1977). (Russian)

    Google Scholar 

  29. Oberguggenberger, M.: Multiplication of Distributions and Applications to Partial Differential Equations. Pitman Research Notes on Mathematics Series, vol. 259, pp. 269–3674. Longman Scientific and Technical, Harlow (1992)

    MATH  Google Scholar 

  30. Oberguggenberger, M.: Generalized functions in nonlinear models—a survey. Nonlinear Anal. 47(8), 5029–5040 (2001)

    MathSciNet  CrossRef  MATH  Google Scholar 

  31. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis. Academic Press, San Diego (1980)

    MATH  Google Scholar 

  32. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1986)

    Google Scholar 

  33. Rudin, W.: Functional Analysis, 2nd edn. International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1991)

    MATH  Google Scholar 

  34. Schmeisser, H.J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Wiley, Chichester (1987)

    Google Scholar 

  35. Schwartz, L.: Théorie des distributions I, II. Herman, Paris (1950/1951)

    Google Scholar 

  36. Stein, E.M.: Singular Integrals and Differentiability of Functions. Princeton Univ. Press, Princeton (1970)

    MATH  Google Scholar 

  37. Stein, E.M., Weiss, G.L.: Introduction to Harmonic Analysis on Euclidean Spaces. Princeton Mathematical Series. Princeton Univ. Press, Princeton (1971)

    Google Scholar 

  38. Thomée, V., Wahlbin, L.B.: Convergence rates of parabolic difference schemes for non-smooth data. Math. Comput. 28(125), 1–13 (1974)

    CrossRef  MATH  Google Scholar 

  39. Triebel, H.: Fourier Analysis and Function Spaces. Teubner, Leipzig (1977)

    MATH  Google Scholar 

  40. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Deutscher Verlag der Wissenschaften, Berlin (1978)

    Google Scholar 

  41. Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1983)

    CrossRef  Google Scholar 

  42. Vladimirov, V.S.: Equations of Mathematical Physics, 2nd English edn. Monographs and Textbooks in Pure and Applied Mathematics, vol. 3. Dekker, New York (1971). 2nd English edn.: Mir, Moscow (1983)

    Google Scholar 

  43. Vladimirov, V.S.: Generalized Functions in Mathematical Physics. Mir, Moscow (1979). (English edn.)

    Google Scholar 

  44. Zygmund, A.: Trigonometric Series, 2nd edn. Cambridge University Press, Cambridge (1988), vols. 1 and 2 combined

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer-Verlag London

About this chapter

Cite this chapter

Jovanović, B.S., Süli, E. (2014). Distributions and Function Spaces. In: Analysis of Finite Difference Schemes. Springer Series in Computational Mathematics, vol 46. Springer, London. https://doi.org/10.1007/978-1-4471-5460-0_1

Download citation