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Moment Spaces and Resonance Theorems

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Inequalities

Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete ((MATHE2,volume 30))

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Abstract

A central idea of analysis, which can be used to connect vast fields of study that at first glance may seem quite unrelated, can be expressed in the following simple form:

“An element of a linear space S can often be characterized most readily and revealingly in terms of its interaction with a suitably chosen set of elements in a dual space S′.”

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Bibliographical Notes

  1. Banach, S.: Théorie des operations linéaires. Monografje Matematyczne I. Warsaw: Z Subwencji Funduszu Kultury Narodowej 1932.

    Google Scholar 

  2. Zygmund, A.: Trigonometrical series. Monografje Matematyczne V. Warsaw: Z Subwencju Funduszu Kultury Narodowej 1935.

    Google Scholar 

  3. Rosenbloom, P.: Quelque classes de problémes extrémaux. Bull Soc. Math. France 79, 1–58 (1951);

    MATH  MathSciNet  Google Scholar 

  4. Rosenbloom, P.: Quelque classes de problèmes extrémaux. Bull Soc. Math. France 80, 183–215 (1952).

    MATH  Google Scholar 

  5. Mathias, M.: Ober positive Fourier-Integrale. Math. Z. 16, 103–125 (1923).

    Article  MathSciNet  Google Scholar 

  6. Bochner, S.: Vorlesungen über Fouriersche Integrale. Leipzig: Akademische Verlagsgesellschaft 1932.

    Google Scholar 

  7. Cooper, J. L. B.: Positive definite functions of a real variable. Proc. London Math. Soc. 10, 53–66 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  8. Fan, K.: Les fonctions définies-positives et les fonctions completement monotones. Mémorial Sci. Math. 64. Paris: Gauthier-Villars 1950.

    Google Scholar 

  9. Widder, D. V.: The Laplace transform. Princeton, N. J.: Princeton University Press 1941.

    Google Scholar 

  10. Godemont, R.: Les fonctions de type positif et la théorie des groupes. Trans. Am. Math. Soc. 63, 1–84 (1948).

    Google Scholar 

  11. Cartan, H., and R. Godemont: Théorie de la dualité et analyse harmonique dans les groupes abéliens localement compacts. Ann. Sci. École Norm. Sup. (3) 64, 79–99 (1947).

    MATH  Google Scholar 

  12. Shohat, J., and J. Tamarkin: The problem of moments. Mathematical Surveys, vol. 2. New York: American Mathematical Society 1943.

    Google Scholar 

  13. Karlin, S., and L. Shapley: Geometry of moment spaces, Memoirs of the American Mathematical Society, 12. New York: American Mathematical Society 1953.

    Google Scholar 

  14. Mallows, C. L.: Generalizations of Tcebycheff’s inequalities. J. Roy. Stat. Soc. (B) 18, 139–176 (1956).

    MathSciNet  Google Scholar 

  15. Fan, K., I. Glicksberg and A. Hoffman: Systems of inequalities involving convex functions. Proc. Am. Math. Soc. 8, 617–622 (1957).

    Article  MATH  MathSciNet  Google Scholar 

  16. Fan, K.: On systems of linear inequalities. Ann. Math. Stud. 38, 99–156. Princeton, N. J.: Princeton University Press 1956.

    Google Scholar 

  17. Fan, K.: Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations. Math. Z. 68, 205–216 (1957).

    Article  MATH  MathSciNet  Google Scholar 

  18. Neyman, J., and E. Pearson: On the problem of most efficient tests of statistical hypotheses. Phil. Trans. Roy. Soc. (A) 231, 289–337 (1933).

    Google Scholar 

  19. Weyl, H.: The method of orthogonal projection in potential theory. Duke Math. J. 7, 411–444 (1940).

    Article  MathSciNet  Google Scholar 

  20. Lax, P. D.: A remark on the method of orthogonal projection. Comm. Pure Appl. Math. 4, 457–464 (1951).

    Article  MATH  MathSciNet  Google Scholar 

  21. Graves, L. M.: Extensions of the lemma of HAAR in the calculus of variations. Bull. Am. Math. Soc. 65, 319–321 (1959).

    Article  MATH  MathSciNet  Google Scholar 

  22. Berwald, L.: Über Haars Verallgemeinerung das Lemmas von Du Bois-Reymond und verwandte Sätze. Acta Math. 79, 39–49 (1947).

    Article  MATH  MathSciNet  Google Scholar 

  23. Schwartz, L.: Théorie des distributions, I, II. Actual Sci. Ind. 1091, 1122. Paris: Hermann et Cie. 1950, 1951

    Google Scholar 

  24. Ursell, H. D.: Inequalities between sums of powers. Proc. London Math. Soc. 9, 432–450 (1959).

    Article  MATH  MathSciNet  Google Scholar 

  25. Eggleston, H. G.: Convexity. Cambridge Tracts, no. 47. London: Cambridge University Press 1958.

    Google Scholar 

  26. Bellman, R., and D. Blackwell: On moment spaces. Ann. of Math. 54, 272–274 (1951).

    Article  MATH  MathSciNet  Google Scholar 

  27. Royden, H.: Bounds for distribution functions when the first n moment are given. Ann. Math. Stat. 24, 361–376 (1953).

    Article  MATH  MathSciNet  Google Scholar 

  28. Riesz, F.: Untersuchungen Über Systems integrierbarer Funktionen. Math. Ann. 69, 449–497 (1910)

    Article  MATH  MathSciNet  Google Scholar 

  29. Helly, E.: Über lineare Funktionaloperationen. Wiener Ber. 121, 265–297 (1912)

    MATH  Google Scholar 

  30. Riesz, F.: Les systemes d’equations linéaires a une infinité d’inconnues. Paris: Gauthier-Villars 1913.

    MATH  Google Scholar 

  31. Boas, R. P.: A general moment problem. Am. J. Math. 63, 361–370 (1941).

    Article  MathSciNet  Google Scholar 

  32. Bernstein, S.: Leçons sur les propriétés extrémales et la meileure approximation des fonctions analytiques d’une variable réelle. Paris: Gauthier-Villars 1926.

    Google Scholar 

  33. Walsh, J. L.: Interpolation and approximation by rational functions in the complex domain. American Mathematical Society Colloquium Publications, vol. 20. New York: American Mathematical Society 1935.

    Google Scholar 

  34. Dresher, M., and S. Karlin: Solutions of convex games as fixed points. Ann. of Math. Studies, no. 28, 75–86. Princeton, N. J.: Princeton University Press 1953.

    Google Scholar 

  35. Weston, J. D.: A note on the extension of linear functionals. Am. Math. Monthly 67, 444–445 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  36. Szegö, G.: Orthogonal polynomials. American Mathematical Society Colloquium Publications, vol. 23. New York: American Mathematical Society 1939.

    Google Scholar 

  37. Fischer, E.: Über das Carathéodorysche Problem, Potenzreihen mit positivem reellen Teil betreffend. Rend. Circ. Mat. Palermo 32, 240–256 (1911).

    Article  MATH  Google Scholar 

  38. Picard, E.: Mémoire sur les fonctions entières. Ann. Sci. École Norm. Supér. (2) 9, 147–166 (1880).

    Google Scholar 

  39. Borel, É.: Démonstration élémentaire d’un théorème de M. Picard sur les fontions entrères. Compt. Rend. (Paris) 122, 1045–1048 (1896).

    MATH  Google Scholar 

  40. Landau, E.: Über eine Verallgemeinerung des Picardschen Satzes. Sitz.ber. preuß. Akad. Wiss. 1904, 1118–1133.

    Google Scholar 

  41. Fejér, L.: Über trigonometrische Polynome. J. Reine Angew. Math. 146, 53–82 (1915).

    Google Scholar 

  42. Carathéodory, C.: Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911).

    Article  MATH  Google Scholar 

  43. Carathéodory, C., and L. Fejér: Über den Zusammenhang der Extremum von harmonischen Funktionen mit ihren Koeffizienten und über den PicardLandauschen Satz. Rend. Circ. Mat. Palermo 32, 218–239 (1911).

    Article  MATH  Google Scholar 

  44. Herglotz, G.: Über Potenzreihen mit positivem, reellem Teil im Einheitskreis. Sitz.ber. Sachs. Akad. Wiss. 63, 501–511 (1911).

    Google Scholar 

  45. Bieberbach, L.: Lehrbuch der Funktionentheorie, bd. 2, p. 217 ff. Berlin: B. G. Teubner 1931.

    MATH  Google Scholar 

  46. Landau, G.: Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie. 2nd ed., chapter 7. Ergebnisse der Math. Berlin: J. Springer-Verlag 1929.

    MATH  Google Scholar 

  47. Kac, M., L. Murdock, and G. Szegö: On the eigenvalues of certain hermitian forms. J. Rat. Mech. Analysis 9, 767–800 (1953).

    Google Scholar 

  48. Grenander, U., and G. Szegö: Toeplitz forms and their applications. Berkeley and Los Angeles: University of California Press 1958.

    MATH  Google Scholar 

  49. Maradudin, A., and G. H. Weiss: The disordered lattice problem, a review. J. Soc. Ind. Appl. Math. 6, 302–329 (1958).

    Article  MATH  MathSciNet  Google Scholar 

  50. Fan, K.: On positive definite sequences. Ann. of Math. 47, 593–607 (1946).

    Article  MATH  MathSciNet  Google Scholar 

  51. Levitan, B.: A generalization of positive definiteness and a generalization of almost-periodic functions. Doklady Akad. Nauk SSSR 1947, 1593–1596.

    Google Scholar 

  52. Loomis, L.: An introduction to abstract harmonic analysis. New York: D. Van Nostrand Co. 1953.

    MATH  Google Scholar 

  53. Weil, A.: L’integration dans les groupes topologiques et ses applications. Actual. Sci. Industr. 869. Paris: Hermann et Cie. 1940.

    Google Scholar 

  54. Devinatz, A.: Some contributions to the extension of positive definite functions. Storrs, Conn.: University of Connecticut. OSR-TN-55–421, 1955.

    Google Scholar 

  55. Bochner, S.: Hilbert distances and positive functions, Ann. of Math. 42, 647–656 (1941).

    Article  MathSciNet  Google Scholar 

  56. Schoenberg, I. J.: Metric spaces and positive definite functions. Trans. Am. Math. Soc. 44, 522–536 (1938).

    Article  MathSciNet  Google Scholar 

  57. Schoenberg, I. J.: and J. von Neuman: Fourier integrals and metric geometry. Trans. Am. Math. Soc. 50, 226–251 (1941).

    Google Scholar 

  58. Sznagy, B., and A. Korinyi: Relation d’une problème de Nevanlinna et Pick avec la théorie des operateurs de l’espace Hilbertian. Acta Math. Hung. 7, 295–305 (1956).

    Article  Google Scholar 

  59. Weyl, H.: Über das Pick-Nevanlinnasche Interpolationsproblem und sein infinitesimales Analogen. Ann. of Math. 36, 230–254 (1935).

    Article  MathSciNet  Google Scholar 

  60. Aronszajn, N.: The theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950).

    Article  MATH  MathSciNet  Google Scholar 

  61. Aronszajn, N.: and K. T. Smith: Characterization of positive reproducing kernels. Applications to Green’s functions. Am. J. Math. 79, 611–622 (1957).

    Article  MATH  MathSciNet  Google Scholar 

  62. Liapounoff, A.: Sur les fonctions-vecteurs complètement additives. Bull. Acad. Sci. URSS Ser. Math. 4, 465–478 (1940).

    MathSciNet  Google Scholar 

  63. Landau, E.: Über einen Konvergenzsatz. Gott. Nachr. 1907, 25–27

    Google Scholar 

  64. Bellman, R.: On an application of the Banach-Steinhaus theorem to the study of the boundedness of solutions of nonlinear differential and difference equations. Ann. of Math. 49, 515–522 (1948).

    Article  MATH  MathSciNet  Google Scholar 

  65. Massera, J. L., and J. J. Schäffer: Linear differential equations and functional analysis, I. Ann. of Math. 67, 517–573 (1958).

    Article  MATH  MathSciNet  Google Scholar 

  66. Massera, J. L., and J. J. Schäffer: Linear differential equations and functional analysis, II. Ann. of Math. 69, 88–104 (1959).

    Article  MATH  MathSciNet  Google Scholar 

  67. Cesari, L.: Asymptotic behavior and stability problems in ordinary differential equations, p. 102. Ergebnisse der Math. Berlin: J. Springer-Verlag 1959.

    MATH  Google Scholar 

  68. Corduneanu, C.: Sur un théorème de Perron. Analele Stiintifice ale Univ. Jasi. 5, 33–36 (1959).

    MathSciNet  Google Scholar 

  69. Gale, D.: Theory of linear economic models. New York: McGraw-Hill Book Co., Inc. 1960.

    Google Scholar 

  70. Weyl, H.: The elementary theory of convex polyhedra. Ann. Math. Studies 24, 3–18. Princeton, N. J.: Princeton University Press 1950.

    Google Scholar 

  71. Minkowset, H.: Zur Theorie der Einheiten in den algebraischen Zahlkörpern. Gesammelte Abhandlungen, I. Leipzig: B. G. Teubner 1911.

    Google Scholar 

  72. Farkas, J.: Ober die Theorie der einfachen Ungleichungen. J. Reine Angew. Math. 124, 1–27 (1901).

    MATH  Google Scholar 

  73. Stiemke, E.: Ober positive Lösungen homogener linearer Gleichungen. Math. Ann. 76, 340–342 (1915).

    Article  MATH  MathSciNet  Google Scholar 

  74. Dines, L. L.: Systems of linear inequalities. Ann. of Math. 20, 191–199 (1919).

    Article  MATH  MathSciNet  Google Scholar 

  75. Carver, W. B.: Systems of linear inequalities. Ann. of Math. 23, 212–220 (1920–1922).

    Article  MathSciNet  Google Scholar 

  76. Motzkin, T.: Beiträge zur Theorie der linearen Ungleichungen. Diss., Basel; 1936.

    MATH  Google Scholar 

  77. Ernikov, S. N.: Systems of linear inequalities. Uspekhi Matem. Nauk, N. S. 8, 7–73 (1953).

    Google Scholar 

  78. Dantzig, G. B., A. Orden and P. Wolfe: The generalized simplex method for minimizing a linear form under linear inequality restraints. Pacific J. Math. 5, 183–195 (1955).

    MATH  MathSciNet  Google Scholar 

  79. Ford, L. R., J., and D. R. Fulkerson: A simple algorithm for finding maximal network flows and an application to the Hitchcock problem. Canad. J. Math. 9, 210–218 (1957).

    Article  MATH  MathSciNet  Google Scholar 

  80. Riley, V., and S. I. Gass: Linear programming and associated techniques. Baltimore, Md.: Operations Research Office, John Hopkins University 1958.

    Google Scholar 

  81. Lehman, S.: On confirmation and rational betting. J. Symbolic Logic 20, 251–262 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  82. Nikaidó, H.: Sur un théorème de Stiemke. Osaka, Japan: Inst. Social Economic Research, Osaka University 1961.

    Google Scholar 

  83. Gale, D., H. Kuhn and A. W. Tucker: Nonlinear programming. Second Berkeley Symposium on Mathematical Statistics and Probability. Berkeley and Los Angeles: University of California Press 1951.

    Google Scholar 

  84. Duffin, R. J.: Infinite programs. Pittsburgh: Department of Mathematics, University of Pittsburgh, Technical Report No. 20, 1955.

    Google Scholar 

  85. Lehman, S.: On the continuous simplex method. Santa Monica, Calif.: The RAND Corporation, Research Memorandum RM-1386, 1954.

    Google Scholar 

  86. Bellman, R., and S. Lehman: Studies on bottleneck problems in a production processes. Santa Monica, Calif.: The Rand Corporation, Paper P-492, 1954.

    Google Scholar 

  87. Neumann, J. Von, and O. Morgenstern: Theory of games and economic behavior. Princeton, N. J.: Princeton University Press 1953.

    MATH  Google Scholar 

  88. Borel, É.: Applications aux jeux de hasard. Traité du calcul des probabilities et de ses applications. Paris: Gauthier-Villars 1913.

    Google Scholar 

  89. Williams, J. D.: The compleat strategyst. New York: McGraw-Hill Book Co., Inc. 1954.

    MATH  Google Scholar 

  90. Looals, L. H.: On a theorem of Von Neumann. Proc. Nat. Acad. Sci. USA 32, 213–215 (1946).

    Article  Google Scholar 

  91. Robinson, J.: An iterative method for solving a game. Ann. of Math. 54, 296–301 (1951).

    Article  MATH  MathSciNet  Google Scholar 

  92. Gross, O.: A mechanical proof of the min-max theorem. Santa Monica, Calif.: The RAND Corporation, Research Memorandum RM-2320, 1960.

    Google Scholar 

  93. Nikaidô, H.: Coincidence and some systems of inequalities. J. Math. Soc. Japan 11, 354–373 (1959).

    Article  MathSciNet  Google Scholar 

  94. Nikaidô, H.: On a method of proof for the minimax theorem. Proc. Am. Math. Soc. 10, 205–212 (1959).

    Article  MATH  Google Scholar 

  95. Newman, J.: Another proof of the minimax theorem. Proc. Am. Math. Soc. 11, 692–693 (1960).

    Article  MATH  Google Scholar 

  96. Kaczmarz, S., and H. Steinhaus: Theorie der Orthogonalreihen. Monografje Matematyczne VI. Warsaw: Z Subwencji Funduszu Kultury Narodowej 1935.

    Google Scholar 

  97. Forsythe, G. E., and P. C. Rosenbloom: Numerical analysis and partial differential equations. Surveys in Applied Mathematics. New York: John Wiley and Sons 1958.

    Google Scholar 

  98. Lax, P. D.: Reciprocal extremal problems in function theory. Comm. Pure Appl. Math. 8, 437–453 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  99. Rogosinski, W. W., and H. S. Shapiro: On certain extremum problems for analytic functions. Acta Math. 90, 287–318 (1953).

    Article  MATH  MathSciNet  Google Scholar 

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Beckenbach, E.F., Bellman, R. (1965). Moment Spaces and Resonance Theorems. In: Inequalities. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-64971-4_3

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  • DOI: https://doi.org/10.1007/978-3-642-64971-4_3

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