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.
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References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Pure and Applied Mathematics Series, vol. 140. Elsevier/Academic Press, Amsterdam (2003)
Bartle, R.G.: The Elements of Integration and Lebesgue Measure. Wiley, New York (1995)
Bergh, J., Löfström, J.: Interpolation Spaces, an Introduction. Grundlehren der mathematischen Wissenschaften, vol. 228. Springer, Berlin (1976)
Besov, O.V., Il’in, V.P., Nikol’skiĭ, S.M.: Integral Representations of Functions and Imbedding Theorems. Nauka, Moscow (1975). (Russian)
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)
Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Interscience, New York (1957)
Dyda, B.: A fractional order Hardy inequality. Ill. J. Math. 48(2), 575–588 (2004)
Edwards, R.E.: Fourier Series: A Modern Introduction vol. 2, 2nd edn. Springer, New York (1982)
Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, Berlin (1969)
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)
Gel’fand, I.M., Shilov, G.E.: Generalized Functions, Vol. I: Properties and Operations. Academic Press, New York (1964)
Grafakos, L.: Classical Fourier Analysis, 2nd edn. Graduate Texts in Mathematics, vol. 249. Springer, New York (2008)
Grisvard, P.: Équations différentielles abstraites. Ann. Sci. Éc. Norm. Super. 4(2), 311–395 (1969). (French)
Grisvard, P.: Elliptic Problems in Non-smooth Domains. Pitman, London (1985)
Hörmander, L.: Lectures on linear partial differential operators. Mimeographed Notes, Stanford University (1960)
Hörmander, L.: Linear Partial Differential Operators 4th edn. Die Grundlehren der mathematischen Wissenschaften, vol. 116. Springer, Berlin (1969)
Jacobsen, N.: Basic Algebra, vol. II. Freeman, San Francisco (1980)
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)
Kufner, A., John, O., Fučik, S.: Function Spaces. Nordhoff International Publ., Leyden (1977)
Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications. Dunod, Paris (1968)
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)
Massey, W.S.: Singular Homology Theory. Graduate Texts in Mathematics, vol. 70. Springer, New York (1980)
Maz’ya, V.G.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 342. Springer, Heidelberg (2011)
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)
Maz’ya, V.G., Shaposhnikova, T.O.: Multiplikatory v prostranstvakh differentsiruemykh funktsii. Leningrad. Univ., Leningrad (1986)
Narasimhan, R.: Several Complex Variables. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1995). Reprint of the 1971 original
Nikol’skiĭ, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Nauka, Moscow (1977). (Russian)
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)
Oberguggenberger, M.: Generalized functions in nonlinear models—a survey. Nonlinear Anal. 47(8), 5029–5040 (2001)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis. Academic Press, San Diego (1980)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1986)
Rudin, W.: Functional Analysis, 2nd edn. International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1991)
Schmeisser, H.J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Wiley, Chichester (1987)
Schwartz, L.: Théorie des distributions I, II. Herman, Paris (1950/1951)
Stein, E.M.: Singular Integrals and Differentiability of Functions. Princeton Univ. Press, Princeton (1970)
Stein, E.M., Weiss, G.L.: Introduction to Harmonic Analysis on Euclidean Spaces. Princeton Mathematical Series. Princeton Univ. Press, Princeton (1971)
Thomée, V., Wahlbin, L.B.: Convergence rates of parabolic difference schemes for non-smooth data. Math. Comput. 28(125), 1–13 (1974)
Triebel, H.: Fourier Analysis and Function Spaces. Teubner, Leipzig (1977)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Deutscher Verlag der Wissenschaften, Berlin (1978)
Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1983)
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)
Vladimirov, V.S.: Generalized Functions in Mathematical Physics. Mir, Moscow (1979). (English edn.)
Zygmund, A.: Trigonometric Series, 2nd edn. Cambridge University Press, Cambridge (1988), vols. 1 and 2 combined
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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
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