×

zbMATH — the first resource for mathematics

Lacunary Taylor and Fourier series. (English) Zbl 0121.30102

PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Georges Alexits, Sur la convergence et la sommabilité des séries orthogonales lacunaires, Acta Sci. Math. Szeged 18 (1957), 179 – 188 (French). · Zbl 0095.27305
[2] László Alpár, Sur la divergence de certaines séries de Taylor lacunaires, Ann. Univ. Sci. Budapest. EöTvös Sect. Math. 3 – 4 (1960/1961), 19 – 26 (French).
[3] A. Beurling and P. Malliavin, Acta Math. (to appear).
[4] Ludwig Bieberbach, Analytische Fortsetzung, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 3, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955 (German). · Zbl 0004.15901
[5] P. Erdös, Note on the converse of Fabry’s gap theorem, Trans. Amer. Math. Soc. 57 (1945), 102 – 104. · Zbl 0060.20303
[6] P. Erdös, On the uniform but not absolute convergence of power series with gaps, Ann. Soc. Polon. Math. 25 (1952), 162 – 168 (1953). · Zbl 0048.31002
[7] P. Erdős, On trigonometric sums with gaps, Magyar Tud. Akad. Mat. Kutató Int. Közl 7 (1962), 37 – 42 (English, with Russian summary). · Zbl 0116.04703
[8] P. Erdös and A. J. Macintyre, Integral functions with gap power series, Proc. Edinburgh Math. Soc. (2) 10 (1954), 62 – 70. · Zbl 0058.06301
[9] E. Fabry, Sur les séries de Taylor qui ont une infinité de points singuliers, Acta Math. 22 (1899), no. 1, 65 – 87 (French). · JFM 29.0209.04
[10] Géza Freud, Über trigonometrische Approximation und Fouriersche Reihen, Math. Z. 78 (1962), 252 – 262 (German). · Zbl 0101.04802
[11] J. Hadamard, Essai sur l’étude des fonctions données par leur développement de Taylor, J. Math. 8 (1892), 101-186. · JFM 24.0359.01
[12] G. H. Hardy, Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), no. 3, 301 – 325. · JFM 46.0401.03
[13] SigurÄ’ur Helgason, Lacunary Fourier series on noncommutative groups, Proc. Amer. Math. Soc. 9 (1958), 782 – 790. · Zbl 0091.10905
[14] Edwin Hewitt and H. S. Zuckerman, Some theorems on lacunary Fourier series, with extensions to compact groups, Trans. Amer. Math. Soc. 93 (1959), 1 – 19. · Zbl 0141.12601
[15] M. Izumi, S. I. Izumi and J.-P. Kahane, Théorèmes élémentaires sur les séries de Fourier lacunaires, J. Analyse Math. (to appear). · Zbl 0133.02303
[16] M. Kac, Probability methods in some problems of analysis and number theory, Bull. Amer. Math. Soc. 55 (1949), 641 – 665. · Zbl 0036.30502
[17] S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, Monogr. Mat., Varsovie, 1935. · JFM 61.1119.05
[18] Jean-Pierre Kahane, Sur quelques problèmes d’unicité et de prolongement, relatifs aux fonctions approchables par des sommes d’exponentielles, Ann. Inst. Fourier, Grenoble 5 (1953 – 1954), 39 – 130 (1955) (French). · Zbl 0064.35903
[19] Jean-Pierre Kahane, Sur les fonctions moyenne-périodiques bornées, Ann. Inst. Fourier, Grenoble 7 (1957), 293 – 314 (French). · Zbl 0083.34401
[20] Jean-Pierre Kahane, Pseudo-périodicité et séries de Fourier lacunaires, Ann. Sci. École Norm. Sup. (3) 79 (1962), 93 – 150 (French). · Zbl 0105.28601
[21] J.-P. Kahane, Sur les coefficients de Fourier-Bohr, Studia Math. 21 (1961/1962), 103 – 106 (French). · Zbl 0101.05903
[22] Jean-Pierre Kahane and Raphaël Salem, Ensembles parfaits et séries trigonométriques, Actualités Sci. Indust., No. 1301, Hermann, Paris, 1963 (French). · Zbl 0112.29304
[23] J.-P. Kahane, Mary Weiss, and Guido Weiss, On lacunary power series, Ark. Mat. 5 (1963), 1 – 26 (1963). · Zbl 0134.05701
[24] P. B. Kennedy, Fourier series with gaps, Quart. J. Math. Oxford Ser. (2) 7 (1956), 224 – 230. · Zbl 0096.04602
[25] P. B. Kennedy, Remark on a theorem of Zygmund, J. London Math. Soc. 33 (1958), 71 – 72. · Zbl 0083.05302
[26] A. Kolmogoroff, Une contribution à l’étude de la convergence des séries de Fourier, Fund. Math. 5 (1924), 96-97. · JFM 50.0206.04
[27] Norman Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, v. 26, American Mathematical Society, New York, 1940. · Zbl 0145.08003
[28] S. Mandelbrojt, Séries lacunaires, Hermann, Paris, 1936. · JFM 62.0328.02
[29] S. Mandelbrojt, Séries de Fourier et classes quasi-analytiques de fonctions, Gauthier-Villars, Paris, 1935. · Zbl 0013.11006
[30] M. E. Noble, Coefficient properties of Fourier series with a gap condition, Math. Ann. 128 (1954), 55 – 62; correction, 256. · Zbl 0055.29502
[31] R. E. A. C. Paley, On the lacunary coefficients of power series, Ann. of Math. (2) 34 (1933), no. 3, 615 – 616. · Zbl 0007.24401
[32] R. E. A. C. Paley, On lacunary power series, Proc, Nat. Acad. Sci. U.S.A. 19 (1933), 271-272. · Zbl 0006.19705
[33] Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. · Zbl 0123.30104
[34] George Pólya, On converse gap theorems, Trans. Amer. Math. Soc. 52 (1942), 65 – 71. · Zbl 0060.20302
[35] Walter Rudin, Remarks on a theorem of Paley, J. London Math. Soc. 32 (1957), 307 – 311. · Zbl 0078.05904
[36] Walter Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203 – 227. · Zbl 0091.05802
[37] R. Salem and A. Zygmund, Lacunary power series and Peano curves, Duke Math. J. 12 (1945), 569 – 578. · Zbl 0060.20402
[38] R. Salem and A. Zygmund, On lacunary trigonometric series, Proc. Nat. Acad. Sci. U. S. A. 33 (1947), 333 – 338. · Zbl 0029.11902
[39] R. Salem and A. Zygmund, La loi du logarithme itéré pour les séries trigonométriques lacunaires, Bull. Sci. Math. (2) 74 (1950), 209 – 224 (French). · Zbl 0039.07001
[40] Laurent Schwartz, Étude des sommes d’exponentielles. 2ième éd, Publications de l’Institut de Mathématique de l’Université de Strasbourg, V. Actualités Sci. Ind., Hermann, Paris, 1959 (French). · Zbl 0092.06302
[41] S. Sidon, Verallgemeinerung eines Satzes über die absolute Konvergenz von Fourierreihen mit Lücken, Math. Ann. 97 (1927), no. 1, 675 – 676 (German). · JFM 53.0252.02
[42] S. Sidon, Ein Satz über trigonometrische Polynome mit Lücken und seine Anwendung in der Theorie der Fourier-Reihen, J. Reine Agnew. Math. 163 (1930), 251-252. · JFM 56.0253.01
[43] S. Sidon, Ein Satz über trigonometrische Polynome und seine Anwendung in der Theorie der Fourier-Reihen, Math. Ann. 106 (1932), no. 1, 536 – 539 (German). · JFM 58.0268.06
[44] S. B. Stečkin, On absolute convergence of Fourier series, Izv. Akad. Nauk SSSR. Ser. Mat. 20 (1956), 385 – 412 (Russian).
[45] F. Sunyer Balaguer, Properties of entire functions (of finite order) represented by lacunary Taylor series, Collectanea Math. 2 (1949), 129 – 174 (Spanish). · Zbl 0041.40402
[46] M. Tomić, Sur la convergence de certaines séries de Fourier lacunaires, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 3 – 4 (1960/1961), 363 – 367 (French).
[47] Paul Turán, Über lakunäre Potenzreihen, Rev. Math. Pures Appl. 1 (1956), no. 3, 27 – 32 (German). · Zbl 0063.09085
[48] P. Turán, On a certain problem in the theory of power series with gaps, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif, 1962, pp. 404 – 409.
[49] K. Weierstrass, Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen, Königl. Akad. Wiss. (1872), Mathematische Werke II, 71-74.
[50] Mary Weiss, Concerning a theorem of Paley on lacunary power series, Acta Math. 102 (1959), 225 – 238. · Zbl 0091.06702
[51] Mary Weiss, The law of the iterated logarithm for lacunary trigonometric series., Trans. Amer. Math. Soc. 91 (1959), 444 – 469. · Zbl 0143.28801
[52] M. Weiss and G. Weiss, On the Picard property of lacunary power series, Studia Math. 22 (1962/1963), 221 – 245. · Zbl 0111.27202
[53] A. Zygmund, On the convergence of lacunary trigonometric series, Fund. Math. 16 (1930), 90-107. · JFM 56.0252.01
[54] Antoni Zygmund, On lacunary trigonometric series, Trans. Amer. Math. Soc. 34 (1932), no. 3, 435 – 446. · Zbl 0005.06303
[55] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. · Zbl 0085.05601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.