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Published online by Cambridge University Press:  05 May 2016

Adrian Constantin
Adrian Constantin
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Universität Wien, Austria
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Fourier Analysis , pp. 343 - 348
Publisher: Cambridge University Press
Print publication year: 2016

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References

Ahlfors, L. V. 1979. Complex Analysis.McGraw-Hill, New York.
Albiac, F., and Kalton, N. J. 2006. Topics in Banach Space Theory.Springer, New York.
Alinhac, S., and Gérard, P. 2007. Pseudo-differential Operators and the Nash-Moser Theorem.American Mathematical Society, Providence, RI.
Aliprantis, C. D., and Burkinshaw, O. 1999. Problems in Real Analysis. A Workbook with Solutions.Academic Press, Inc., San Diego, CA.
Ambrosio, L., DaPrato, G., and Mennucci, A. 2011. Introduction to Measure Theory and Integration.Edizioni della Scuola Normale Superiore di Pisa, Pisa.
Androulakis, G., Beanland, K., Dilworth, S. J., and Sanacory, F. 2006. Embedding l∞ into the space of bounded operators on certain Banach spaces. Bull. London Math. Soc., 38, 979–990.Google Scholar
Ash, J. M. 1976. Multiple trigonometric series. Pages 76–96 of: Ash, J. M. (ed), Studies in Harmonic Analysis.Mathematical Association of America, Washington, DC.
Ash, J. M., and Gluck, L. 1972. A divergent multiple Fourier series of power series type. Studia Math., 44, 477–491.Google Scholar
Ash, M. J. 2013. A survey of multidimensional generalizations of Cantor's uniqueness theorem for trigonometric series. Pages 49–61 of: Bilyk, D., Carli, L. De, Petukhov, A., Stokolos, A. M., and Wick, B. D. (eds), Recent Advances in Harmonic Analysis and Applications.Springer, New York.
Benedetto, J. J. 1997. Harmonic Analysis and Applications.CRC Press, Boca Raton, FL.
Boas, R. P. 1987. Invitation to Complex Analysis.Random House, New York.
Bôcher, M. 1906. Introduction to the theory of Fourier's series. Ann. of Math., 7, 81–152.Google Scholar
Borzellino, J. E., and Sherman, M. 2012. When is a trigonometric polynomial not a trigonometric polynomial?Amer. Math. Monthly, 119(5), 422–425.Google Scholar
Bourgain, J. 1996. Spherical summation and uniqueness of multiple trigonometric series. Internat. Math. Res. Notices, 3, 93–107.Google Scholar
Bramanti, M. 2014. An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields.Springer, Cham, Switzerland.
Brézis, H. 2011. Functional Analysis, Sobolev Spaces and Partial Differential Equations.Springer, New York.
Bruckner, A. M., and Leonard, J. T. 1966. Derivatives. Amer. Math. Monthly, 73, 24–56.Google Scholar
Bruckner, A. M., Bruckner, J. B., and Thomson, B. S. 1997. Real Analysis.Prentice-Hall, Upper Saddle River.
Burckel, R. B. 1979. An Introduction to Classical Complex Analysis.Birkhäuser Verlag, Basel.
Bustamante, J., and Jiménez, M. A. 2000. Chebyshev and Hölder approximation. Aportaciones Mat. Comun., 27, 23–31.Google Scholar
Carleson, L. 1966. On convergence and growth of partial sums of Fourier series. Acta Math., 116, 135–157.Google Scholar
Chernoff, P. R. 1980. Convergence of Fourier series. Amer. Math. Monthly, 87, 399–400.Google Scholar
Ciesielski, Z. 1960. On the isomorphisms of the spaces Hα and m. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astronom. Phys., 8, 217–222.Google Scholar
Constantin, A., and Strauss, W. A. 2000. Stability of a class of solitary waves in compressible elastic rods. Phys. Lett. A, 270, 140–148.Google Scholar
Conway, J. B. 1990. A Course in Functional Analysis.Springer-Verlag, New York.
Crone, L. 1971. A characterization of matrix operators on l2. Math. Z., 123, 315–317.Google Scholar
Day, M. M. 1940. The spaces Lp with 0 < p < 1. Bull. Amer. Math. Soc., 46, 816–823.Google Scholar
Day, M.M. 1941. Reflexive Banach spaces not isomorphic to uniformly convex spaces. Bull. Amer. Math. Soc., 47, 313–317.Google Scholar
Dirichlet, P. G. L. 1829. Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données. J. Reine Angew. Math., 4, 157–169.Google Scholar
du Bois-Reymond, P. 1874. Über die sprungweise Wertänderungen analytischer Funktionen. Math. Ann., 241–261.
du Bois-Reymond, P. 1876. Untersuchungen über die Convergenz und Divergenz der Fourierschen Darstellungsformeln. Abh. Math.-Phys. Cl. K. Bay. Akad. Wiss., 12, 1–13.Google Scholar
Dunham, W. 2005. Touring the Calculus gallery. Amer. Math. Monthly, 112, 1–19.Google Scholar
Duren, P. 2000. Theory of Hp Spaces.Dover, New York.
Dvoretzky, A., and Rogers, C. 1950. Absolute and unconditional convergence in normed linear spaces. Proc. Nat. Acad. Sci. USA, 36(192–197).Google Scholar
Dym, H., and McKean, H. P. 1972. Fourier Series and Integrals.Academic Press, New York–London.
Edwards, R. E. 1967. Fourier Series: a Modern Introduction. Vol. I. & II. Holt, Rinehart and Winston, Inc., New York.
Ehrenpreis, L. 1954. Solution of some problems of division. I. Amer. J. Math., 76, 883–903.Google Scholar
Elekes, M., and Keleti, T. 2006. Is Lebesgue measure the only s-finite invariant Borel measure?J. Math. Anal. Appl., 321, 445–451.Google Scholar
Evans, L. C. 1990. Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS Series, Amer. Math. Soc., Providence, RI.
Evans, L. C., and Gariepy, R. F. 1992. Measure Theory and Fine Properties of Functions.CRC Press, Boca Raton, FL.
Federer, H. 1969. Geometric Measure Theory.Springer, Heidelberg.
Fefferman, C. 1971a. On the convergence of Fourier series. Bull. Amer. Math. Soc., 77, 744–745.Google Scholar
Fefferman, C. 1971b. On the divergence of Fourier series. Bull. Amer. Math. Soc., 77, 191–195.Google Scholar
Fejér, L. 1904. Untersuchungen über Fouriersche Reihen. Math. Ann., 58, 51–69.Google Scholar
Feldman, M. B. 1981. A proof of Lusin's theorem. Amer. Math. Monthly, 88, 191–192.Google Scholar
Folland, G. B. 1999. Real Analysis: Modern Techniques and their Applications.Wiley-Interscience, New York.
Friedlander, F. G. 1998. Introduction to the Theory of Distributions.Cambridge University Press, Cambridge.
Friedman, A. 1982. Foundations of Modern Analysis.Dover Publications, Inc., New York.
Gamelin, T. W. 2001. Complex Analysis.Springer, New York.
Gårding, L. 1997. Some Points of Analysis and their History.American Mathematical Society, Providence, RI.
Gelbaum, B. R., and Olmsted, J. M. H. 2003. Counterexamples in Analysis.Dover Publications, Inc., Mineola, NY.
Gibbs, J. W. 1898. Letter to the Editor. Nature, 59, 606.Google Scholar
Giblin, P. J. 1981. Graphs, Surfaces and Homology. An Introduction to Algebraic Topology.Chapman and Hall, London and New York.
Goffman, C. 1977. A bounded derivative which is not Riemann integrable. Amer. Math. Monthly, 84, 205–206.Google Scholar
Gohberg, I., Goldberg, S., and Kaashoek, M. A. 2003. Basic Classes of Linear Operators.Birkhäuser Verlag, Basel.
Grafakos, L. 2008. Classical Fourier Analysis.Springer, New York.
Gray, J. D., and Morris, S. A. 1978. When is a function that satisfies the Cauchy– Riemann equations analytic?Amer. Math. Monthly, 85, 246–256.Google Scholar
Gröchenig, K. 1996. An uncertainty principle related to the Poisson summation formula. Studia Math., 121, 87–104.Google Scholar
Guillemin, V., and Pollack, A. 1974. Differential Topology.Prentice Hall, Inc., Englewood Cliffs, NJ.
Hamadouche, D. 2000. Invariance principles in Hölder spaces. Portugaliae Math., 57, 127–151.Google Scholar
Hamilton, R. S. 1982. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc., 7, 65–222.Google Scholar
Haroske, D. D., and Triebel, H. 2008. Distributions, Sobolev Spaces, Elliptic Equations. Europ. Math. Soc., Züric.
Henstock, R. 1991. The General Theory of Integration.Oxford University Press, Oxford.
Hewitt, E., and Hewitt, R. E. 1979. The Gibbs-Wilbraham phenomenon. An episode in Fourier Analysis. Arch. Hist. Exact Sci., 21, 129–160.Google Scholar
Hewitt, E., and Stromberg, K. 1965. Real and Abstract Analysis.Springer-Verlag, New York.
Hogan, J. A., and Lakey, J. D. 2004. Time-frequency and Time-scale Methods.Springer, Berlin.
Hörmander, L. 1983. The Analysis of Linear Partial Differential Operators. Vol. I. Springer-Verlag, Berlin.
Hörmander, L. 1985. The Analysis of Linear Partial Differential Operators. III. Pseudodifferential Operators.Springer Verlag, Berlin.
Hörmander, L. 1995. Lectures on Harmonic Analysis.Lund University, Lund, Sweden.
Hunt, R. A. 1968. On the convergence of Fourier series. Pages 235–255 of: Haimo, D. T. (ed), Orthogonal Expansions and their Continuous Analogues.Southern Illinois University Press, Carbondale.
Hunt, R. A. 1976. Developments related to the a.e. convergence of Fourier series. Pages 20–37 of: Ash, J. M. (ed), Studies in Harmonic Analysis.Mathematical Association of America, Washington, DC.
Ingelstam, L. 1963. Hilbert algebras with identity. Bull. Amer. Math. Soc., 69, 794–796.Google Scholar
Iorio, R. J., and Iorio, V. M. 2001. Fourier Analysis and Partial Differential Equations.Cambridge University Press, Cambridge.
James, R. C. 1951. A non-reflexive Banach space isometric with its second conjugate space. Proc. Nat. Acad. Sci. USA, 37, 174–177.Google Scholar
James, R. C. 1982. Bases in Banach spaces. Amer. Math. Monthly, 89, 625–640.Google Scholar
Jordan, C. 1881. Sur la série de Fourier. C. R. Acad. Sci. Paris, 92, 228–230.Google Scholar
Kahane, J.-P. 2000. Baire's category theorem and trigonometric series. J. d'Analyse Math., 80, 143–181.Google Scholar
Kahane, J. P., and Katznelson, Y. 1966. Sur les ensembles de divergence des séries trigonométriques. Studia Math., 26, 305–306.Google Scholar
Katznelson, Y. 1968. An Introduction to Harmonic Analysis.J. Wiley & Sons, Inc., New York.
Kolmogorov, A. 1926. Une série de Fourier–Lebesgue divergente partout. C. R. Acad. Sci. Paris, 183, 1327–1328.Google Scholar
Koosis, P. 1998. Introduction to Hp Spaces.Cambridge University Press, Cambridge.
Köthe, G. 1969. Topological Vector Spaces. Vol. I.Springer-Verlag, Berlin.
Köthe, G. 1979. Topological Vector Spaces. Vol. II.Springer-Verlag, Berlin.
Krantz, S. G. 1999. A Panorama of Harmonic Analysis.Mathematical Association of America, Washington, DC.
Kupka, J. 1986. Measure theory: the heart of the matter. Math. Intell., 8, 47–56.Google Scholar
Lax, P. D. 2002. Functional Analysis.Wiley-Interscience, New York.
Leoni, G. 2009. A First Course in Sobolev Spaces.American Mathematical Society, Providence, RI.
Lévy-Leblond, J.-M. 1997. If Fourier had known Argand … A geometrical point of view on Fourier transforms. Math. Intell., 19, 63–71.Google Scholar
Lieb, E. H. 1990. Gaussian kernels have only Gaussian maximizers. Invent. Math., 102, 179–208.Google Scholar
Lindenstrauss, J., and Tzafriri, L. 1977. Classical Banach Spaces.Springer, Berlin-Heidelberg-New York.
Loomis, L. H. 1953. An Introduction to Abstract Harmonic Analysis.D. Van Nostrand Company, Inc., Toronto-New York-London.
Lunardi, A. 2013. Analytic Semigroups and Optimal Regularity in Parabolic Problems.Birkhäuser Verlag, Basel.
Malgrange, B. 1955. Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution. Ann. Inst. Fourier, 6, 271–355.Google Scholar
Mazur, S., and Ulam, S. 1932. Sur les transformations isométriques d'espaces vectoriels normés. C. R. Acad. Sci. Paris, 194, 946–948.Google Scholar
Michelson, A. A. 1898. Letter to the Editor. Nature, 59, 544–545.Google Scholar
Miller, T. L., and Olin, R. F. 1984. Analytic curves. Amer. Math. Monthly, 91, 127–130.Google Scholar
Mukherjea, A., and Pothoven, K. 1986. Real and Functional Analysis.Plenum Press, New York & London.
Munkres, J. R. 2000. Topology.Prentice Hall Inc., NJ.
Natanson, I. P. 1955. Theory of Functions of a Real Variable.F. Ungar Publishing Co., New York.
Needham, T. 2000. Visual Complex Analysis.Oxford University Press, Oxford.
Niven, I. 1947. A simple proof that Π is irrational. Bull. Amer. Math. Soc., 53, 509.Google Scholar
Peano, G. 1890. Sur une courbe, qui remplit toute une aire plane. Math. Ann., 36, 157–160.Google Scholar
Pinsky, M. A. 1993. Fourier inversion for piecewise smooth functions in several variables. Proc. Amer. Math. Soc., 118, 903–910.Google Scholar
Pinsky, M. A. 2009. Introduction to Fourier Analysis and Wavelets.American Mathematical Society, Providence, RI.
Piranian, G., Titus, C. J., and Young, G. S. 1952. Conformal mappings and Peano curves. Michigan Math. J., 1, 69–72.Google Scholar
Pommerenke, C. 2002. Conformal maps at the boundary. Pages 39–74 of: Kuhnau, R. (ed), Handbook of Complex Analysis: Geometric Function Theory.North-Holland, Amsterdam.
Reed, M., and Simon, B. 1980a. Methods of Mathematical Physics. I: Functional Analysis.Academic Press, Inc., San Diego, CA.
Reed, M., and Simon, B. 1980b. Methods of Mathematical Physics. II: Fourier Analysis, Self-adjointness.Academic Press, Inc., San Diego, CA.
Riesz, F., and Sz.-Nagy, B. 1955. Functional Analysis.F. Ungar Publ. Co., New York.
Rouse, J. 2012. Explicit bounds for sums of squares. Math. Res. Lett., 19, 359–376.Google Scholar
Rudin, W. 1974. Real and Complex Analysis.McGraw-Hill Book Co., New York.
Salem, R., and Zygmund, A. 1945. Lacunary power series and Peano curves. Duke Math. J., 12, 569–578.Google Scholar
Schlag, W. 2014. A Course in Complex Analysis and Riemann Surfaces.American Mathematical Society, Providence RI.
Segal, S. L. 2008. Nine Introductions in Complex Analysis.Elsevier, Amsterdam.
Siegmund-Schultze, R. 2008. Henri Lebesgue. Page 796 of: Gowers, T., Barrow-Green, J., and Leader, I. (eds), Princeton Companion to Mathematics.Princeton University Press.
Srivastava, S. M. 1998. A Course on Borel Sets.Springer-Verlag, New York.
Stein, E. M. 1976. Harmonic analysis on ℝn. Pages 97–135 of: Ash, J. M. (ed), Studies in Harmonic Analysis.Mathematical Association of America, Washington, DC.
Stein, E. M., and Shakarchi, R. 2003. Fourier Analysis. An Introduction.Princeton University Press, Princeton, NJ.
Stein, E. M., and Weiss, G. 1971. Introduction to Fourier Analysis on Euclidean Spaces.Princeton University Press, Princeton, NJ.
Strauss, W. A. 2008. Partial Differential Equations. An Introduction.John Wiley & Sons, Ltd., Chichester.
Suslin, M. Y. 1917. Sur une définition des ensembles measurables B sans nombres transfinis. C. R. Acad. Sci. Paris, 164, 88–91.Google Scholar
Taylor, M. 1981. Pseudo-differential Operators.Princeton University Press, Princeton, NJ.
Urysohn, P. 1923. Sur une fonction analytique partout continue. Fund. Math., 4, 144–150.Google Scholar
van Douwen, E. K. 1989. Fubini's theorem for null sets. Amer. Math. Monthly, 96, 718–721.Google Scholar
van Neerven, J. M. A. M. 1997. The norm of a complex Banach lattice. Positivity, 1, 381–390.Google Scholar
Wagon, S. 1985. The Banach-Tarski Paradox.Cambridge University Press, Cambridge.
Weiner, J. L., and Wilkens, G. R. 2005. Quaternions and rotations in E4. Amer. Math. Monthly, 112, 69–76.Google Scholar
Wheeler, G. F., and Crummett, W. P. 1987. The vibrating string controversy. Amer. J. Phys., 55, 33–37.Google Scholar
Wilbraham, H. 1848. On a certain periodic function. Cambridge & Dublin Math. J., 3, 198–201.Google Scholar
Yosida, K. 1995. Functional Analysis.Springer-Verlag, Berlin.
Zalcman, L. 1974. Real proofs of complex theorems (and vice versa). Amer. Math. Monthly, 81, 115–137.Google Scholar
Zhu, K. 2007. Operator Theory in Function Spaces.American Mathematical Society, Providence, RI.
Zygmund, A. 1959. Trigonometrical Series.Cambridge University Press, Cambridge.

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  • References
  • Adrian Constantin, Universität Wien, Austria
  • Book: Fourier Analysis
  • Online publication: 05 May 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781107358508.011
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  • References
  • Adrian Constantin, Universität Wien, Austria
  • Book: Fourier Analysis
  • Online publication: 05 May 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781107358508.011
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  • References
  • Adrian Constantin, Universität Wien, Austria
  • Book: Fourier Analysis
  • Online publication: 05 May 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781107358508.011
Available formats
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