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    Fourier Analysis
    • Online ISBN: 9781107358508
    • Book DOI: https://doi.org/10.1017/CBO9781107358508
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Book description

Fourier analysis aims to decompose functions into a superposition of simple trigonometric functions, whose special features can be exploited to isolate specific components into manageable clusters before reassembling the pieces. This two-volume text presents a largely self-contained treatment, comprising not just the major theoretical aspects (Part I) but also exploring links to other areas of mathematics and applications to science and technology (Part II). Following the historical and conceptual genesis, this book (Part I) provides overviews of basic measure theory and functional analysis, with added insight into complex analysis and the theory of distributions. The material is intended for both beginning and advanced graduate students with a thorough knowledge of advanced calculus and linear algebra. Historical notes are provided and topics are illustrated at every stage by examples and exercises, with separate hints and solutions, thus making the exposition useful both as a course textbook and for individual study.

Reviews

'[Fourier Analysis: Volume l - Theory is] fabulous … Constantin structures his exercise sets beautifully, I think: they are abundant and long, covering a spectrum of levels of difficulty; each set is followed immediately by a section of hints (in one-one correspondence); finally the hints sections are followed by very detailed and well-written solutions (also bijectively). Can there be any clearer homage to the maxim that to learn mathematics one has to get one’s hands really dirty? To boot, attention to detail is ubiquitous: it’s everywhere in Constantin’s presentation of proofs and arguments, as well as examples, all throughout the narrative itself. The entire presentation is very much to the point and the student who works through this book will come out knowing some real mathematics very well.'

Michael Berg Source: MAA Reviews

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Contents

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.
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.
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.
Borzellino, J. E., and Sherman, M. 2012. When is a trigonometric polynomial not a trigonometric polynomial?Amer. Math. Monthly, 119(5), 422–425.
Bourgain, J. 1996. Spherical summation and uniqueness of multiple trigonometric series. Internat. Math. Res. Notices, 3, 93–107.
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.
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.
Carleson, L. 1966. On convergence and growth of partial sums of Fourier series. Acta Math., 116, 135–157.
Chernoff, P. R. 1980. Convergence of Fourier series. Amer. Math. Monthly, 87, 399–400.
Ciesielski, Z. 1960. On the isomorphisms of the spaces Hα and m. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astronom. Phys., 8, 217–222.
Constantin, A., and Strauss, W. A. 2000. Stability of a class of solitary waves in compressible elastic rods. Phys. Lett. A, 270, 140–148.
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.
Day, M. M. 1940. The spaces Lp with 0 < p < 1. Bull. Amer. Math. Soc., 46, 816–823.
Day, M.M. 1941. Reflexive Banach spaces not isomorphic to uniformly convex spaces. Bull. Amer. Math. Soc., 47, 313–317.
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.
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.
Dunham, W. 2005. Touring the Calculus gallery. Amer. Math. Monthly, 112, 1–19.
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).
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.
Elekes, M., and Keleti, T. 2006. Is Lebesgue measure the only s-finite invariant Borel measure?J. Math. Anal. Appl., 321, 445–451.
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.
Fefferman, C. 1971b. On the divergence of Fourier series. Bull. Amer. Math. Soc., 77, 191–195.
Fejér, L. 1904. Untersuchungen über Fouriersche Reihen. Math. Ann., 58, 51–69.
Feldman, M. B. 1981. A proof of Lusin's theorem. Amer. Math. Monthly, 88, 191–192.
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.
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.
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.
Gröchenig, K. 1996. An uncertainty principle related to the Poisson summation formula. Studia Math., 121, 87–104.
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.
Hamilton, R. S. 1982. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc., 7, 65–222.
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.
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.
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.
James, R. C. 1982. Bases in Banach spaces. Amer. Math. Monthly, 89, 625–640.
Jordan, C. 1881. Sur la série de Fourier. C. R. Acad. Sci. Paris, 92, 228–230.
Kahane, J.-P. 2000. Baire's category theorem and trigonometric series. J. d'Analyse Math., 80, 143–181.
Kahane, J. P., and Katznelson, Y. 1966. Sur les ensembles de divergence des séries trigonométriques. Studia Math., 26, 305–306.
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.
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.
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.
Lieb, E. H. 1990. Gaussian kernels have only Gaussian maximizers. Invent. Math., 102, 179–208.
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.
Mazur, S., and Ulam, S. 1932. Sur les transformations isométriques d'espaces vectoriels normés. C. R. Acad. Sci. Paris, 194, 946–948.
Michelson, A. A. 1898. Letter to the Editor. Nature, 59, 544–545.
Miller, T. L., and Olin, R. F. 1984. Analytic curves. Amer. Math. Monthly, 91, 127–130.
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.
Peano, G. 1890. Sur une courbe, qui remplit toute une aire plane. Math. Ann., 36, 157–160.
Pinsky, M. A. 1993. Fourier inversion for piecewise smooth functions in several variables. Proc. Amer. Math. Soc., 118, 903–910.
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.
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.
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.
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.
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.
van Douwen, E. K. 1989. Fubini's theorem for null sets. Amer. Math. Monthly, 96, 718–721.
van Neerven, J. M. A. M. 1997. The norm of a complex Banach lattice. Positivity, 1, 381–390.
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.
Wheeler, G. F., and Crummett, W. P. 1987. The vibrating string controversy. Amer. J. Phys., 55, 33–37.
Wilbraham, H. 1848. On a certain periodic function. Cambridge & Dublin Math. J., 3, 198–201.
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.
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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