Celebratio Mathematica — Zygmund

[1] W. Sierpiński and A. Zygmund: “Sur une fonction qui est discontinue sur tout ensemble de puissance du continu” [On a function which is dicontinuous on all sets of continuum power], Fundam. Math. 4 (1923), pp. 316–318. JFM 49.0179.02 article

[2] A. Zygmund: “Sur la théorie riemannienne des séries trigonométriques” [On the riemannian theory of trigonometric series], C. R. Acad. Sci., Paris 177 (1923), pp. 521–523. Errata for this article were published in C. R. Acad. Sci., Paris 177 (1923). JFM 49.0198.07 article

[3] A. Zygmund: “Sur les séries trigonométriques” [On trigonometric series], C. R. Acad. Sci., Paris 177 (1923), pp. 576–579. JFM 49.0198.08 article

[4] A. Zygmund: [Dissertation on the Riemannian theory of trigonometric series]. Ph.D. thesis, University of Warsaw, 1923. Advised by A. Rajchman and S. Mazurkiewicz. phdthesis

[5] A. Zygmund: “Errata: ‘Sur la théorie riemannienne des séries trigonométriques’” [Errata: ‘On the riemannian theory of trigonometric series’], C. R. Acad. Sci., Paris 177 (1923), pp. 804. Errata for an article published in C. R. Acad. Sci., Paris 177 (1923). article

[6] A. Zygmund: “Sur une généralisation de la methode de Cesàro” [On a generalization of the method of Cesàro], C. R. Acad. Sci., Paris 179 (1924), pp. 870–872. JFM 50.0157.03 article

[7] S. Saks and A. Zygmund: “Sur les faisceaux des tangentes à une courbe” [On the sheaves of tangents to a curve], Fund. Math. 6 (1924), pp. 117–121. JFM 50.0187.03 article

[8] A. Zygmund: “Sur les séries de Fourier restreintes” [On restricted Fourier series], C. R. Acad. Sci., Paris 178 (1924), pp. 181–182. JFM 50.0202.01 article

[9] A. Zygmund: “Sur la dérivation des séries de Fourier” [On the derivation of Fourier series], Bull. Acad. Polon. A 1924 (1924), pp. 243–249. JFM 51.0218.01 article

[10] A. Zygmund: “Sur la sommation des séries trigonométriques conjuguées aux séries de Fourier” [On the summation of trigonometric series conjugate to Fourier series], Bull. Acad. Polon. A 1924 (1924), pp. 251–258. JFM 51.0224.01 article

[11] A. Zygmund: “O module ciagłości sumy szeregu sprzcedillaoneżonego z szeregiem Fouriera” [On the continuity module of the sum of series conjugate to a Fourier series], Prace Mat.-Fiz. 33 (1924), pp. 125–132. With French summary. JFM 52.0274.03 article

[12] A. Zygmund: “Sur un théorème de M. Marcel Riesz” [On a theorem of M. Marcel Riesz], Ann. Soc. Polon. Math. 3 (1925), pp. 153. JFM 51.0186.11 article

[13] A. Rajchman and A. Zygmund: “Sur les principes et les problèmes de la théorie riemannienne des séries trigonométriques” [On the principles and problems from the riemannian theory of trigonometric series], Ann. Soc. Polon. Math. 3 (1925), pp. 147–148. JFM 51.0236.10 article

[14] A. Zygmund: “Sur la sommation des séries trigonométriques et celles de puissances par les moyennes typiques” [On the summation of trigonometric series and power series for typical means], C. R. Acad. Sci., Paris 181 (1925), pp. 1122–1123. JFM 51.0244.02 article

[15] S. Saks and A. Zygmund: “Un teorema sulle curve continue” [A theorem on continuous curves], Boll. Un. Mat. Ital. 4 (1925), pp. 7–10. JFM 51.0537.03 article

[16] A. Rajchmann and A. Zygmund: “Sur la relation du procédé de sommation de Cesàro et celui de Riemann” [On the relation of the summation procedure of Cesàro and that of Riemann], Bulletin Acad. Polonaise 1925 (1925), pp. 69–80. Addenda to this were published in Acta Math. 52:1 (1929). JFM 52.0217.03 article

[17] A. Zygmund: “Sur la sommation des séries par les procédé des moyennes typiques” [On the summation of series for the procedure of typical averages], Bull. Acad. Polon. A 1925 (1925), pp. 265–287. JFM 52.0218.01 article

[18] A. Zygmund: “Sur la sommabilité des séries de Fourier des fonctions vérifiant la condition de Lipschitz” [On the summability of Fourier series of functions satisfying the Lipschitz condition], Bull. Acad. Polon. A 1925 (1925), pp. 1–9. JFM 52.0271.02 article

[19] A. Zygmund: “Sur un théorème de M. Gronwall” [On a theorem of M. Gronwall], Bull. Acad. Polon. A 1925 (1925), pp. 207–217. JFM 52.0271.04 article

[20] A. Zygmund: “Remarque sur la sommabilité des séries de fonctions orthogonales” [Remark on the summability of series of orthogonal functions], Bull. Acad. Polon. A 1926 (1926), pp. 185–191. JFM 52.0277.03 article

[21] A. Zygmund: “Contribution à l’unicité du développement trigonométrique” [Contribution to the uniqueness of trigonometric development], Math. Z. 24 : 1 (1926), pp. 40–46. MR 1544747 JFM 51.0219.03 article

[22] A. Zygmund: “Sur la théorie riemannienne des séries trigonométriques” [On the riemannian theory of trigonometric series], Math. Z. 24 : 1 (1926), pp. 47–104. MR 1544748 JFM 51.0219.04 article

[23] A. Rajchman and A. Zygmund: “Sur la possibilité d’appliquer la méthode de Riemann aux séries trigonométriques sommables par le procédé de Poisson” [On the possibility of applying Riemann’s method to trigonometric series summable by the Poisson procedure], Math. Z. 25 : 1 (1926), pp. 261–273. MR 1544808 JFM 52.0272.01 article

[24] A. Zygmund: “Sur les séries trigonométriques sommables par le procédé de Poisson” [On trigonometric series summable by the Poisson procedure], Math. Z. 25 : 1 (1926), pp. 274–290. MR 1544809 JFM 52.0272.02 article

[25] A. Zygmund: “Sur un théorème de la théorie de la sommabilité” [On a theorem from the theory of summability], Math. Z. 25 : 1 (1926), pp. 291–296. MR 1544810 JFM 52.0217.02 article

[26] A. Zygmund: “Une remarque sur un théorème de M. Kaczmarz” [A remark on a theorem of M. Kaczmarz], Math. Z. 25 : 1 (1926), pp. 297–298. MR 1544811 JFM 52.0278.01 article

[27] A. Zygmund: “Über einige Sätze aus der Theorie der divergenten Reihen” [On a few theorems from the theory of divergent series], Bull. Acad. Polon. A 1927 (1927), pp. 309–331. JFM 53.0190.02 article

[28] A. Zygmund: “Remarque sur les sommes positives des séries trigonométriques” [Remark on some positive sums of trigonometric series], Bull. Acad. Polon. A 1927 (1927), pp. 331–341. JFM 53.0256.01 article

[29] A. Zygmund: “Remarque sur un théorème de M. Fekete” [Remark on a theorem of M. Fekete], Bull. Acad. Polon. A 1927 (1927), pp. 343–347. JFM 53.0256.02 article

[30] A. Zygmund: “Sur la sommation des sèries de fonctions orthogonales” [On the summation of series of orthogonal functions], Bull. Acad. Polon. A 1927 (1927), pp. 295–308. JFM 53.0266.01 article

[31] A. Zygmund: “Sur l’application de la première moyenne arithmétique dans la théorie des séries de fonctions orthogonales” [On the application of the first arithmetic mean to the theory of series of orthogonal functions], Fundam. Math. 10 (1927), pp. 356–362. JFM 53.0267.04 article

[32] A. Zygmund: “Sur les fonctions conjuguées” [On conjugate functions], C. R. Acad. Sci., Paris 187 (1928), pp. 1025–1026. Research announcement for an article published in Fundam. Math. 13 (1929). JFM 54.0307.02 article

[33] A. Zygmund: “Über die Beziehungen der Eindeutigkeitsfragen in den Theorien der trigonometrischen Reihen und Integrale” [On the question of uniqueness relations in the theory of trigonometric series and integrals], Math. Ann. 99 : 1 (1928), pp. 562–589. MR 1512466 JFM 54.0320.03 article

[34] A. Zygmund: “Remarque sur la convergence absolue des series de Fourier” [Remark on the absolute convergence of Fourier series], J. London Math. Soc. 3 : 3 (1928), pp. 194–196. MR 1574161 JFM 54.0303.01 article

[35] A. Zygmund: “Sur la convergence absolue des séries de Fourier” [On the absolute convergence of Fourier series], Ann. Soc. Polon. Math. 7 (1929), pp. 266–267. JFM 55.0170.16 article

[36] A. Zygmund: “Sur le problème d’unicité pour certaines systèmes orthogonaux” [On the problem of uniqueness for certain orthogonal systems], Ann. Soc. Math. 7 (1929), pp. 267. JFM 55.0170.17 article

[37] A. Zygmund: “Sur les fonctions conjuguées” [On conjugate functions], Fundam. Math. 13 (1929), pp. 284–303. Errata and remarks for this article were published in Fund. Math. 18 (1932). A research announcement was published in C. R. Acad. Sci., Paris 187 (1928). JFM 55.0751.02 article

[38] A. Zygmund: “Uwaga o funkcjach ciagłych nieróżniczkowalnych” [A remark on nondifferentiable continuous functions], Mathesis Polska 4 : 1–2 (1929), pp. 1–7. JFM 57.1397.04 article

[39] A. Rajchman and A. Zygmund: “Addenda: ‘Sur la relation du procédé de sommation de Cesàro et celui de Riemann’” [Addenda: ‘On the relation of the summation procedure of Cesàro and that of Riemann’], Acta Math. 52 : 1 (1929), pp. A80. Addenda to an article published in Bulletin Acad. Polonaise 1925 (1925). MR 1555288 article

[40] A. Zygmund: “Über einseitige Lokalisation” [On one-sided localisations], Jahresber. Dtsch. Math.-Ver. 39 (1930), pp. 47–52. JFM 56.0245.01 article

[41] A. Zygmund: “A note on series of sines,” Q. J. Math., Oxf. Ser. 1 : 1 (1930), pp. 102–107. JFM 56.0245.02 article

[42] A. Zygmund: “Remarques sur les ensembles d’unicité dans quelques systèmes orthogonaux” [Remarks on unique sets in some orthogonal systems], pp. 309–311 in Atti del Congresso Internazionale dei Matematici [Proceedings of the International Congress of Mathematicians] (Bologna, 3–10 September 1928), vol. 2. Nicola Zanichelli (Bologna), 1930. JFM 56.0245.03 incollection

[43] A. Zygmund: “On the convergence of lacunary trigonometric series,” Fundam. Math. 16 (1930), pp. 90–107. JFM 56.0252.01 article

[44] R. E. A. C. Paley and A. Zygmund: “On some series of functions, I,” Proc. Camb. Philos. Soc. 26 : 3 (July 1930), pp. 337–357. JFM 56.0254.01 article

[45] R. E. A. C. Paley and A. Zygmund: “On the partial sums of Fourier series,” Stud. Math. 2 (1930), pp. 221–227. JFM 56.0940.03 article

[46] A. Zygmund: “Un théorème sur les séries orthogonales” [A theorem on orthogonal series], Stud. Math. 2 (1930), pp. 181–182. JFM 56.0944.03 article

[47] A. Zygmund: “Sur la théorie riemannienne de certains systèmes othogonaux, I” [On the riemannian theory of certain orthogonal systems, I], Stud. Math. 2 (1930), pp. 97–170. Part II was published in Prace Mat.-Fiz. 39 (1932). JFM 56.0945.01 article

[48] A. Zygmund: “O pewnych nierównościach” [On some inequalities], Mathesis Polska 5 : 5–6 (1930), pp. 83–100. JFM 57.1374.03 article

[49] A. Zygmund: “Sur les séries trigonométriques lacunaires” [On lacunary trigonometric series], J. London Math. Soc. 5 : 2 (1930), pp. 138–145. Corrigenda to this article were published in J. London Math. Soc 5:4 (1930). MR 1574219 JFM 56.0251.03 article

[50] A. Zygmund: “Corrigenda: ‘Sur les séries trigonométriques lacunaires’” [Corrigenda: ‘On lacunary trigonometric series’], J. London Math. Soc. 5 : 4 (1930), pp. 272. Corrigenda to an article published in J. London Math. Soc 5:2 (1930). MR 1574569 article

[51] A. Zygmund: “On a theorem of Ostrowski,” J. London Math. Soc. 6 : 3 (1931), pp. 162–163. MR 1574733 JFM 57.0374.02 Zbl 0002.18902 article

If a power series \[ \sum_0^{\infty}a_{\nu}z^{\nu} = f(z), \] whose radius of convergence is equal to 1, possesses an infinite number of “large gaps”, i.e. if \( a_{\nu} = 0 \) for \( n_{\kappa} \leq \nu \leq n^{\prime}_{\kappa} \), \( n^{\prime}_{\kappa}/n_{\kappa} > 1 + \epsilon \) (\( \kappa = 1, 2, \dots \)), then the partial sums \( s_{n_{\kappa}} \) of the series converge uniformly in a full neighbourhood of every regular point of \( f(z) \) lying on the circle \( |z| = 1 \).

This theorem (a generalization of a well-known theorem of Hadamard) has been proved by Ostrowski [1926]. The purpose of this note is to show that Ostrowski’s theorem is a consequence of a lemma (of a Tauberian character) on numerical series.

[52] A. Zygmund: “On a theorem of Privaloff,” Stud. Math. 3 (1931), pp. 239–247. JFM 57.0321.01 Zbl 0003.20401 article

[53] A. Zygmund: “Quelques théoremes sur les séries trigonométriques et celles de puissances” [Some theorems on trigonometric series and power series], Stud. Math. 3 (1931), pp. 77–91. JFM 57.0317.02 Zbl 0003.25301 article

[54] A. Zygmund: “O pewnych zagadnieniach teorji szeregów trygonometrycznych” [On some questions in the theory of trigonometric series], Mathesis Polska 7 (1932), pp. 49–58. JFM 58.0268.04 article

[55] A. Zygmund: “Errata and remarks for my article ‘On conjugate functions’,” Fundam. Math. 18 (1932), pp. 312. Errata and remarks for an article published in Fund. Math. 13 (1929). JFM 58.0295.03 article

[56] A. Zygmund: “On lacunary trigonometric series,” Bull. Am. Math. Soc. 38 : 5 (1932), pp. 344. Abstract only. Abstract for an article published in Trans. Am. Math. Soc. 34:3 (1932). JFM 58.0296.09 article

[57] A. Zygmund: “Sur un théorème de M. Pólya” [On a theorem of M. Pólya], pp. 38 in Verhandlungen des Internationalen Mathematiker-Kongresses [Proceedings of the International Congress of Mathematicians] (Z/”urich, 4–12 September 1932), vol. 2. Edited by W. Saxner. Orell Füssli (Zürich and Leipzig), 1932. Abstract only. JFM 58.0353.02 incollection

[58] A. Zygmund: “On lacunary trigonometric series,” Trans. Am. Math. Soc. 34 : 3 (1932), pp. 435–446. An abstract for this article was published in Bull. Am. Math. Soc. 38:5 (1932). MR 1501647 JFM 58.0280.01 Zbl 0005.06303 article

[59] A. Zygmund: “A remark on conjugate series,” Proc. London Math. Soc. (2) 34 : 1 (1932), pp. 392–400. MR 1576159 JFM 58.0276.01 Zbl 0005.35301 article

[60] R. E. A. C. Paley and A. Zygmund: “A note on analytic functions in the unit circle,” Proc. Camb. Philos. Soc. 28 : 3 (July 1932), pp. 266–272. JFM 58.1076.03 Zbl 0005.06602 article

[61] A. Zygmund: “Sur la théorie riemannienne de certains systèmes orthogonaux, II” [On the riemannian theory of certain orthogonal systems, II], Prace Mat.-Fiz. 39 (1932), pp. 73–117. Part I was published in Stud. Math. 2 (1930). JFM 58.0289.01 Zbl 0005.29004 article

[62] R. E. A. C. Paley and A. Zygmund: “On some series of functions, III,” Proc. Camb. Philos. Soc. 28 : 2 (1932), pp. 190–205. JFM 58.1070.01 Zbl 0006.19802 article

[63] A. Zygmund: “On continuability of power series,” Acta Litt. Sci. Szeged 6 : 2–3 (1932–1934), pp. 80–84. JFM 59.0322.02 Zbl 0007.31202 article

[64] A. Zygmund: “O zachowaniu sie pewnych szeregów funkcyjnych” [On the behavior of some function series], Mathesis Polska 8 (1933), pp. 76–87. JFM 59.1015.02 article

[65] R. E. A. C. Paley, N. Wiener, and A. Zygmund: “Notes on random functions,” Math. Z. 37 : 1 (1933), pp. 647–668. MR 1545426 JFM 59.1015.01 Zbl 0007.35402 article

Norbert Wiener	Related
Raymond Edward Alan Christopher Paley	Related

[66] A. Zygmund: “On an integral inequality,” J. Lond. Math. Soc. 8 : 3 (1933), pp. 175–178. MR 1574115 JFM 59.0992.02 Zbl 0007.30205 article

[67] A. Zygmund: “Some points in the theory of trigonometric and power series,” Bull. Am. Math. Soc. 40 : 3 (1934), pp. 210. Abstract only. Research announcement for an article published in Trans. Am. Math. Soc. 36:3 (1934). JFM 60.0242.01 article

[68] A. Zygmund: “Some points in the theory of trigonometric and power series,” Trans. Am. Math. Soc. 36 : 3 (1934), pp. 586–617. A research announcement for this article was published in Bull. Am. Math. Soc. 40:3 (1934). MR 1501757 JFM 60.0988.01 Zbl 0010.06201 article

[69] S. Saks and A. Zygmund: “On functions of rectangles and their application to analytic functions,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 3 : 1 (1934), pp. 27–32. MR 1556719 JFM 60.0243.01 article

[70] A. Zygmund: “On the differentiability of multiple integrals,” Fundam. Math. 23 (1934), pp. 143–149. JFM 60.0219.02 Zbl 0010.01404 article

[71] A. Zygmund: Trigonometrical series. Monografie Matematyczne 5. Polska Academia Nauk (Warsaw), 1935. Reprinted in 1952 and 1955, then expanded into volumes I and II in 1959. These were later translated into Russian (1965), then combined back into a single English volume, being republished in 1969, 1977, 1988 and 2002. JFM 61.0263.03 Zbl 0011.01703 book

[72] B. Jessen, J. Marcinkiewicz, and A. Zygmund: “Note on the differentiability of multiple integrals,” Fundam. Math. 25 (1935), pp. 217–234. JFM 61.0255.01 Zbl 0012.05901 article

Józef Marcinkiewicz	Related
Børge Christian Jessen	Related

[73] A. Zygmund: “Sur le caractère de divergence des séries orthogonales” [On the character of divergence of orthogonal series], Mathematica, Cluj 9 (1935), pp. 86–88. JFM 61.1121.03 Zbl 0014.01402 article

[74] J. Marcinkiewicz and A. Zygmund: “On the differentiability of functions and summability of trigonometrical series,” Fundam. Math. 26 (1936), pp. 1–43. JFM 62.0287.01 Zbl 0014.11102 article

[75] A. Zygmund: “A remark on Fourier transforms,” Proc. Camb. Philos. Soc. 32 : 2 (1936), pp. 321–327. JFM 62.0468.02 Zbl 0014.25803 article

[76] A. Zygmund: “Note on trigonometrical and Rademacher’s series,” Prace Mat.-Fiz. 44 (1937), pp. 91–107. JFM 62.0281.04 Zbl 0016.10902 article

[77] J. Marcinkiewicz and A. Zygmund: “Some theorems on orthogonal systems,” Fundam. Math. 28 (1937), pp. 309–335. JFM 63.0202.01 Zbl 0016.20502 article

[78] J. Marcinkiewicz and A. Zygmund: “Sur les fonctions indépendantes” [On independent functions], Fundam. Math. 29 (1937), pp. 60–90. JFM 63.0946.02 Zbl 0016.40901 article

[79] J. Marcinkiewicz and A. Zygmund: “Mean values of trigonometrical polynomials,” Fundam. Math. 28 (1937), pp. 131–166. JFM 63.0221.03 Zbl 0017.25102 article

[80] J. Marcinkiewicz and A. Zygmund: “Two theorems on trigonometrical series,” Rec. Math. Moscou, n. Ser. 2(44) : 4 (1937), pp. 733–737. JFM 63.0208.01 Zbl 0018.01802 article

[81] J. Marcinkiewicz and A. Zygmund: “Remarque sur la loi du logarithme itéré” [Remark on the law of the iterated logarithm], Fundam. Math. 29 (1937), pp. 215–222. JFM 63.1076.03 Zbl 0018.03204 article

[82] J. Marcinkiewicz and A. Zygmund: “Proof of a gap theorem,” Duke Math. J. 4 : 3 (1938), pp. 469–472. MR 1546068 JFM 64.0223.01 Zbl 0019.27004 article

[83] J. Marcinkiewicz and A. Zygmund: “A theorem of Lusin,” Duke Math. J. 4 : 3 (1938), pp. 473–485. MR 1546069 JFM 64.0268.01 Zbl 0019.42001 article

[84] J. Marcinkiewicz and A. Zygmund: “Quelques théorèmes sur les fonctions indépendantes” [Some theorems on independent functions], Stud. Math. 7 (1938), pp. 104–120. JFM 64.0214.01 Zbl 0018.07504 article

[85] A. Zygmund: “On the convergence and summability of power series on the circle of convergence, I,” Fundam. Math. 30 (1938), pp. 170–196. Part II was published in Proc. London Math. Soc. 47:1. JFM 64.1054.01 Zbl 0019.01602 article

[86] A. Zygmund: “Proof of a theorem of Paley,” Proc. Camb. Philos. Soc. 34 : 2 (1938), pp. 125–133. JFM 64.0227.02 Zbl 0019.01701 article

[87] A. Zygmund: “A remark on conjugate series,” Bull. Sém. Math. Univ. Wilno 1 (1938), pp. 16–18. JFM 64.1042.02 Zbl 0019.30102 article

[88] J. Marcinkiewicz and A. Zygmund: “Sur les séries de puissances” [On power series], Mathematica, Cluj 14 (1938), pp. 21–30. JFM 64.1051.04 Zbl 0020.23103 article

[89] S. Saks and A. Zygmund: Funkcje analityczne: Wykłady uniwersyteckie [Analytic functions: University lectures]. Monografie Matematyczne 10. PWN Polish Scientific Publishers (Warsaw), 1938. Republished in 1948. An English edition was published in 1952. JFM 64.1048.01 book

[90] J. Marcinkiewicz and A. Zygmund: “Sur la dérivée seconde généralisée” [On a generalized second derivative], Bull. Sém. Math. Univ. Wilno 2 (1939), pp. 35–40. MR 0000045 JFM 65.0206.02 Zbl 0021.01603 article

[91] A. Zygmund: “Sur un théorèm de M. Fejér” [On a theorem of M. Fejér], Bull. Sém. Math. Univ. Wilno 2 (1939), pp. 3–12. MR 0000052 JFM 65.0308.01 Zbl 0022.15303 article

[92] A. Zygmund: “Note on the formal multiplication of trigonometrical series,” Bull. Sém. Math. Univ. Wilno 2 (1939), pp. 52–56. MR 0000065 JFM 65.0254.03 Zbl 0021.12002 article

[93] J. Marcinkiewicz and A. Zygmund: “Quelques inégalités pour les opérations linéaires” [Some inequalities for linear operations], Fundam. Math. 32 (1939), pp. 115–121. JFM 65.0506.02 Zbl 0021.40703 article

[94] J. Marcinkiewicz and A. Zygmund: “On the summability of double Fourier series,” Fundam. Math. 32 (1939), pp. 122–132. JFM 65.0266.01 Zbl 0022.01804 article

[95] J. Marcinkiewicz and A. Zygmund: “On the behavior of trigonometric series and power series,” Trans. Am. Math. Soc. 50 : 3 (1941), pp. 407–453. MR 0005130 JFM 67.0225.02 Zbl 0025.40002 article

[96] A. Zygmund: “On the convergence and summability of power series on the circle of convergence, II,” Proc. London Math. Soc. (2) 47 : 1 (1942), pp. 326–350. Part I was published in Fundam. Math. 30. MR 0007042 Zbl 0060.20208 article

[97] A. Zygmund: “Two notes on inequalities,” J. Math. Phys., Mass. Inst. Tech. 21 : 1–4 (April 1942), pp. 117–123. MR 0007425 Zbl 0060.14801 article

[98] A. Zygmund: “Complex methods in the theory of Fourier series,” Bull. Am. Math. Soc. 49 : 11 (1943), pp. 805–822. MR 0009221 Zbl 0060.18901 article

[99] A. Zygmund: “A theorem on generalized derivatives,” Bull. Am. Math. Soc. 49 : 12 (1943), pp. 917–923. MR 0009624 Zbl 0061.11307 article

[100] A. Zygmund: “A property of the zeros of Legendre polynomials,” Trans. Am. Math. Soc. 54 : 1 (1943), pp. 39–56. MR 0009657 Zbl 0063.08415 article

[101] J. D. Tamarkin and A. Zygmund: “Proof of a theorem of Thorin,” Bull. Am. Math. Soc. 50 : 4 (1944), pp. 279–282. MR 0009964 Zbl 0060.24104 article

[102] A. Zygmund: “On certain integrals,” Trans. Am. Math. Soc. 55 (1944), pp. 170–204. MR 0009966 Zbl 0061.13902 article

[103] A. Zygmund: “On the degree of approximation of functions by Fejér means,” Bull. Am. Math. Soc. 51 : 4 (1945), pp. 274–278. MR 0012146 Zbl 0061.13701 article

[104] A. Zygmund: “Proof of a theorem of Littlewood and Paley,” Bull. Am. Math. Soc. 51 : 6 (1945), pp. 439–446. MR 0012306 Zbl 0060.14703 article

[105] A. Zygmund: “Smooth functions,” Duke Math. J. 12 : 1 (1945), pp. 47–76. MR 0012691 Zbl 0060.13806 article

[106] A. Zygmund: “A theorem on fractional derivatives,” Duke Math. J. 12 : 3 (1945), pp. 455–464. MR 0013403 Zbl 0061.11308 article

[107] R. Salem and A. Zygmund: “Lacunary power series and Peano curves,” Duke Math. J. 12 : 4 (1945), pp. 569–578. MR 0015154 Zbl 0060.20402 article

[108] A. Zygmund: “The approximation of functions by typical means of their Fourier series,” Duke Math. J. 12 : 4 (1945), pp. 695–704. MR 0015539 Zbl 0060.18804 article

[109] A. Zygmund: “On the theorem of Fejér–Riesz,” Bull. Am. Math. Soc. 52 : 4 (1946), pp. 310–318. MR 0015536 Zbl 0060.18803 article

[110] R. Salem and A. Zygmund: “Capacity of sets and Fourier series,” Trans. Am. Math. Soc. 59 : 1 (1946), pp. 23–41. MR 0015537 Zbl 0060.18511 article

In [1939] A. Beurling proved that if the series \[ \sum n(a_n^2 + b_n^2) \] converges then the points of divergence of the Fourier series \[ a_0/2 + \sum_1^{\infty}a_n\cos nx + b_n\sin nx \] form a set of logarithmic capacity zero. In the same paper, he stated the following result, the proof of which was due to appear, but to our knowledge has not appeared, in the Arkiv für Matematik: if \( 0 < \alpha \leq 1 \) and if the series \[ \sum n^{\alpha-\epsilon}(a_n^2 + b_n^2) \] converges for every \( \epsilon > 0 \), the above Fourier series is summable by Abel’s method except in a set whose \( \beta \)-capacity is zero for every \( \beta > 1 - \alpha \).

The present paper has its origin in the desire of the authors to find a more direct and simpler proof of Beurling’s results and to extend them. In the course of the work, however, some other theorems of independent interest have been established, and are presented here.

[111] R. Salem and A. Zygmund: “The approximation by partial sums of Fourier series,” Trans. Am. Math. Soc. 59 : 1 (1946), pp. 14–22. MR 0015538 Zbl 0060.18510 article

[112] R. Salem and A. Zygmund: “On a theorem of Banach,” Proc. Natl. Acad. Sci. U. S. A. 33 : 10 (October 1947), pp. 293–295. MR 0021611 Zbl 0029.20801 article

[113] A. Zygmund: “On trigonometric integrals,” Ann. Math. (2) 48 : 2 (April 1947), pp. 393–440. MR 0021612 Zbl 0029.11903 article

[114] A. Zygmund: “A remark on characteristic functions,” Ann. Math. Stat. 18 : 2 (1947), pp. 272–276. MR 0021613 Zbl 0032.29202 article

[115] R. Salem and A. Zygmund: “On lacunary trigonometric series,” Proc. Natl. Acad. Sci. U. S. A. 33 : 11 (November 1947), pp. 333–338. MR 0022263 Zbl 0029.11902 article

Let us consider a lacunary trigonometric series \begin{equation*}\tag{1} \sum_{k=1}^{\infty}(a_k\cos n_kx + b_k\sin n_kx), \end{equation*} with \( n_{k+1}/n_k > q > 1 \), and let us confine our attention to the interval \( 0 \leq x \leq 2\pi \). Let \( S_N(X) \) denote the \( N \)-th partial sum of (1), that is to say the sum of the terms with \( k = 1, 2, \dots \), \( N \). Let \[ C_N = \bigl\{\tfrac{1}{2}(a_1^2 + b_1^2 + \cdots + a_N^2 + b_N^2)\bigr\}^{1/2} .\] In a recent note [1947], J. Ferrand and R. Fortet considered the behavior of the ratio \( S_N(x)/C_N \). They stated, without proof, that if \( C_N \to +\infty \) as \( N \to\infty \), the distribution function of \( S_N(x)/C_N \) tends to the Gaussian distribution with mean value 0 and dispersion 1. Clearly, the result as stated cannot be correct since, if \[ c_k = (a_k^2 + b_k^2)^{1/2} \] increases very rapidly, the distribution function tends to that of cosine. Thus some kind of restriction on the \( c_k \) is necessary. (Mr. Erdős informs us that a proof of the result, under the condition \( C_n\to\infty \), \( c_n = O(1) \), will be published in a paper written jointly by him, Mlle. Ferrand, Fortet and Kac.) In this note we propose to give a complete solution of the problem.

[116] A. Zygmund: “On the summability of multiple Fourier series,” Am. J. Math. 69 : 4 (October 1947), pp. 836–850. MR 0022618 Zbl 0034.04601 article

[117] R. Salem and A. Zygmund: “On lacunary trigonometric series, II,” Proc. Natl. Acad. Sci. U. S. A. 34 : 2 (February 1948), pp. 54–62. MR 002936 Zbl 0029.35601 article

[118] M. Kac, R. Salem, and A. Zygmund: “A gap theorem,” Trans. Am. Math. Soc. 63 : 2 (1948), pp. 235–243. MR 0023937 Zbl 0032.27402 article

Raphaël Salem	Related
Marek Kac	Related

[119] A. Zygmund: “On certain methods of summability associated with conjugate trigonometric series,” Studia Math. 10 (1948), pp. 97–103. MR 0025592 Zbl 0038.21901 article

[120] A. Zygmund: “On a theorem of Hadamard,” Ann. Soc. Polon. Math. 21 (1948), pp. 52–69. Errata were published in Ann. Soc. Polon. Math. 21 (1948). MR 0026685 Zbl 0032.06301 article

[121] R. Salem and A. Zygmund: “A convexity theorem,” Proc. Natl. Acad. Sci. U. S. A. 34 : 9 (September 1948), pp. 443–447. MR 0027083 Zbl 0034.33501 article

[122] A. Zygmund: “Two notes on the summability of infinite series,” Colloq. Math. 1 : 3 (1948), pp. 225–229. MR 0028436 Zbl 0038.21401 article

[123] A. Zygmund: “An example in Fourier series,” Studia Math. 10 (1948), pp. 113–119. MR 0029439 Zbl 0038.21802 article

[124] A. Zygmund: “Errata to the paper ‘On a theorem of Hadamard.’,” Ann. Soc. Polon. Math. 21 (1948), pp. 357–358. Errata to an article published in \text{Ann. Soc. Polon. Math.} 21 (1948). MR 0030595 article

[125] S. Saks and A. Zygmund: Funkcje analityczne: Wykłady uniwersyteckie [Analytic functions: University lectures], 2nd edition. Monografie Matematyczne 10. Sp\.ołdzielnia Wydawnicza (Warsaw), 1948. Republication of 1938 original. Zbl 0031.29804 book

[126] A. P. Caldéron, A. González Domínguez, and A. Zygmund: “Nota sobre los valores límites de funciones analíticas” [Note on the limit values of analytic functions], Revista Unión Mat. Argentina 14 (1949), pp. 16–19. MR 0031550 Zbl 0033.05901 article

Alberto González Domínguez	Related
Alberto Pedro Calderón	Related	Volume

[127] A. Zygmund: “On the boundary values of functions of several complex variables, I,” Fund. Math. 36 (1949), pp. 207–235. MR 0035832 Zbl 0040.19201 article

[128] A. Zygmund: “On a theorem of Littlewood,” Summa Brasil. Math. 2 : 5 (1949), pp. 51–57. MR 0036306 Zbl 0038.26202 article

[129] A. Zygmund: “A remark on functions of several complex variables,” Acta Sci. Math. 12 : B (1950), pp. 66–68. Dedicated to Leopoldo Fejér and Frederico Riesz on their 70th birthdays. MR 0034849 Zbl 0041.20601 article

Frigyes Riesz	Related
Lipót Fejér	Related

[130] A. Zygmund: Trigonometric interpolation. University of Chicago, 1950. Mimeograph of typed lecture notes. MR 0034865 Zbl 0039.06501 book

[131] A. P. Calderón and A. Zygmund: “Note on the boundary values of functions of several complex variables,” pp. 145–165 in A. Zygmund, W. Transue, M. Morse, A. P. Calderón, and S. Bochner: Contributions to Fourier analysis. Annals of Mathematics Studies 25. Princeton University Press (Princeton, NJ), 1950. MR 0035833 Zbl 0040.04001 incollection

Alberto Pedro Calderón	Related	Volume
Salomon Bochner	Related
Harold Calvin Marston Morse	Related
William Raoul Reagle Transue	Related

[132] A. P. Calderón and A. Zygmund: “On the theorem of Hausdorff–Young and its extensions,” pp. 166–188 in A. Zygmund, W. Transue, M. Morse, A. P. Calderón, and S. Bochner: Contributions to Fourier analysis. Annals of Mathematics Studies 25. Princeton University Press, 1950. MR 0037395 Zbl 0040.02902 incollection

Alberto Pedro Calderón	Related	Volume
Salomon Bochner	Related
Harold Calvin Marston Morse	Related
William Raoul Reagle Transue	Related

[133] R. Salem and A. Zygmund: “La loi du logarithme itéré pour les séries trigonométriques lacunaires” [The law of the iterated logarithm for lacunary trigonometric series], Bull. Sci. Math. (2) 74 (1950), pp. 209–224. MR 0039828 Zbl 0039.07001 article

[134] A. Zygmund: “A remark on the integral modulus of continuity,” Rev., Ser. A, Univ. Nac. Tucumán 7 (1950), pp. 259–269. MR 0042479 Zbl 0040.18101 article

[135] Contributions to Fourier analysis. Edited by A. Zygmund, W. Transue, M. Morse, A. P. Calderon, and S. Bochner. Annals of Mathematics Studies 25. Princeton University Press, 1950. Zbl 1341.42001 book

Alberto Pedro Calderón	Related	Volume
Salomon Bochner	Related
Harold Calvin Marston Morse	Related
William Raoul Reagle Transue	Related

[136] A. Zygmund: “Polish mathematics between the two wars (1919–39),” pp. 3–9 in Proceedings of the second Canadian Mathematics Congress (Vancouver, BC, 1949). University of Toronto Press, 1951. MR 0043016 Zbl 0042.24302 incollection

[137] A. Zygmund: “A remark on characteristic functions,” pp. 369–372 in Proceedings of the second Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 31 July–12 August 1950). Edited by J. Neyman. University of California Press (Berkeley and Los Angeles), 1951. MR 0044056 Zbl 0045.07303 incollection

[138] A. Zygmund: “An individual ergodic theorem for non-commutative transformations,” Acta Sci. Math. 14 : 2 (1951), pp. 103–110. MR 0045948 Zbl 0045.06403 article

[139] A. P. Calderón and A. Zygmund: “A note on the interpolation of linear operations,” Studia Math. 12 (1951), pp. 194–204. MR 0046567 Zbl 0044.11902 article

[140] A. P. Calderon and A. Zygmund: “On the existence of certain singular integrals,” Acta Math. 88 (December 1952), pp. 85–139. Dedicated to Professor Marcel Riesz, on the occasion of his 65th birthday. MR 0052553 Zbl 0047.10201 article

Alberto Pedro Calderón	Related	Volume
Marcel Riesz	Related

[141] S. Saks and A. Zygmund: Analytic functions. Monografie Matematyczne 28. Polskie Towarzystwo Matematyczne (Warsaw), 1952. English translation of 1938 Polish original. MR 0055432 Zbl 0048.30803 book

[142] A. Zygmund: Trigonometrical series, 2nd edition. Chelsea Publishing (New York), 1952. Republication of 1935 original. MR 0076084 book

[143] R. Salem and A. Zygmund: “Sur les séries trigonométriques dont les coefficients ont des signes aléatoires” [On trigonometric series whose coefficients have random signs], C. R. Acad. Sci. Paris 236 (1953), pp. 571–573. MR 0055478 Zbl 0050.07303 article

[144] A. Zygmund: “Book review: H. G. Hardy, J. E. Littlewood and G. Pólya , ‘Inequalities’,” Bull. Am. Math. Soc. 59 : 4 (1953), pp. 411–412. MR 1565505 article

J. E. Littlewood	Related
Godfrey Harold Hardy	Related
George Pólya	Related

[145] A. P. Calderón and A. Zygmund: “Singular integrals and periodic functions,” Studia Math. 14 (1953–1954), pp. 249–271. MR 0069310 Zbl 0064.10401 article

[146] G. Szegő and A. Zygmund: “On certain mean values of polynomials,” J. Analyse Math. 3 : 1 (1954), pp. 225–244. Dedicated to the memory of Otto Szász. MR 0064910 Zbl 0055.30303 article

Otto Szász	Related
Gábor Szegö	Related

[147] R. Salem and A. Zygmund: “Some properties of trigonometric series whose terms have random signs,” Acta Math. 91 : 1 (1954), pp. 245–301. Dedicated to Professor Hugo Steinhaus for his 65th birthday. MR 0065679 Zbl 0056.29001 article

Raphaël Salem	Related
Władysław Hugo Dionizy Steinhaus	Related

[148] A. P. Calderón and A. Zygmund: “On a problem of Mihlin,” Trans. Am. Math. Soc. 78 : 1 (1955), pp. 209–224. An addenda to this was paper published in Trans. Am. Math. Soc. 84:2 (1957). MR 0068028 Zbl 0065.04104 article

[149] R. Salem and A. Zygmund: “Sur un théorème de Piatetçki–Shapiro” [On a theorem of Piatetçki–Shapiro], C. R. Acad. Sci. Paris 240 : part 2 (1955), pp. 2040–2042. MR 0071580 Zbl 0065.29702 article

[150] R. Salem and A. Zygmund: “Sur les ensembles parfaits dissymétriques à rapport constant” [On the asymmetric perfect sets of constant ratio], C. R. Acad. Sci. Paris 240 : part 2 (1955), pp. 2281–2283. MR 0071581 Zbl 0066.31701 article

[151] A. Zygmund: Trigonometrical series, 2nd edition. Dover Publications (New York), 1955. Unabridged and unaltered republication of the edition published in 1935. MR 0072976 Zbl 0065.05604 book

[152] A. Zygmund: “On the Littlewood–Paley function \( g^*(\theta) \),” Proc. Natl. Acad. Sci. U. S. A. 42 : 4 (April 1956), pp. 208–212. MR 0077700 Zbl 0072.07201 article

[153] A. Zygmund: “On a theorem of Marcinkiewicz concerning interpolation of operations,” J. Math. Pures Appl. (9) 35 (1956), pp. 223–248. MR 0080887 Zbl 0070.33701 article

[154] A. P. Calderón and A. Zygmund: “A note on the interpolation of sublinear operations,” Am. J. Math. 78 : 2 (April 1956), pp. 282–288. MR 0082647 Zbl 0071.33601 article

[155] R. Salem and A. Zygmund: “A note on random trigonometric polynomials,” pp. 243–246 in Proceedings of the third Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 26–31 December 1954 and July–August 1955), vol. 2. Edited by J. Neyman. University of California Press (Berkeley and Los Angeles), 1956. MR 0084622 Zbl 0071.28401 incollection

Raphaël Salem	Related
Jerzy Neyman	Related

[156] A. P. Calderón and A. Zygmund: “On singular integrals,” Am. J. Math. 78 : 2 (April 1956), pp. 289–309. MR 0084633 Zbl 0072.11501 article

[157] A. Zygmund: “Hilbert transforms in \( E^n \),” pp. 140–151 in Proceedings of the International Congress of Mathematicians (Amsterdam, 2–9 September 1954), vol. 3. Edited by J. C. H. Gerretsen and J. de Groot. Erven P. Noordhoff N.V. (Groningen), 1956. MR 0086187 Zbl 0073.32701 incollection

Johan Cornelis Hendrik Gerretsen	Related
Johannes de Groot	Related

[158] A. P. Calderón and A. Zygmund: “Algebras of certain singular operators,” Am. J. Math. 78 : 2 (April 1956), pp. 310–320. MR 0087810 Zbl 0072.11601 article

[159] A. P. Calderon and A. Zygmund: “Addenda to the paper ‘On a problem of Mihlin’,” Trans. Am. Math. Soc. 84 : 2 (March 1957), pp. 559–560. Addenda to a paper published in Trans. Am. Math. Soc. 78:1 (1955). MR 0084634 Zbl 0077.27204 article

[160] A. Zygmund: “On singular integrals,” Rend. Mat. e Appl. (5) 16 (1957), pp. 468–505. Based on C.I.M.E. Summer School lectures. MR 0096088 Zbl 0088.08302 article

[161] A.-P. Calderón and A. Zygmund: “Singular integral operators and differential equations,” Am. J. Math. 79 : 4 (October 1957), pp. 901–921. MR 0100768 Zbl 0081.33502 article

[162] A. Zygmund: “On the boundary values of functions of several complex variables,” pp. 236–247 in Seminars on Analytic Functions, vol. 1. Institute for Advanced Study (Princeton), 1958. Zbl 0116.06102 incollection

[163] M. Weiss and A. Zygmund: “A note on smooth functions,” Nederl. Akad. Wetensch. Proc. Ser. A 62 : 1 (1959), pp. 52–58. MR 0107122 Zbl 0085.05701 article

[164] A. Zygmund: Trigonometric series, 2nd edition, vol. I. Cambridge University Press (New York), 1959. First volume of an enlarged edition of 1935 original. MR 0107776 Zbl 0085.05601 book

[165] A. Zygmund: “On the preservation of classes of functions,” J. Math. Mech. 8 : 6 (1959), pp. 889–895. An erratum for this article was published in J. Math. Mech. 9:4 (1960). MR 0117498 Zbl 0092.28802 article

[166] A. Zygmund: Trigonometric series, 2nd edition, vol. II. Cambridge University Press (New York), 1959. Second volume of an enlarged edition of 1935 original. book

[167] M. Weiss and A. Zygmund: “On the existence of conjugate functions of higher order,” Fund. Math. 48 : 2 (1959–1960), pp. 175–187. MR 0113093 Zbl 0093.26604 article

[168] A. Zygmund: “Józef Marcinkiewicz,” Wiadom. Mat. (2) 4 (1960), pp. 11–41. MR 0115885 Zbl 0091.00617 article

[169] A.-P. Calderón and A. Zygmund: “A note on local properties of solutions of elliptic differential equations,” Proc. Natl. Acad. Sci. U.S.A. 46 : 10 (October 1960), pp. 1385–1389. An expanded version of this was published in Studia Math. 20:2 (1961). MR 0140829 Zbl 0196.40501 article

[170] A. Zygmund: “Erratum: ‘On the preservation of classes of functions’,” J. Math. Mech. 9 : 4 (1960), pp. 663. Erratum for an article published in J. Math. Mech. 8:6 (1959). article

[171] E. M. Stein and A. Zygmund: “Smoothness and differentiability of functions,” Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 3–4 (1960–1961), pp. 295–307. MR 0132135 Zbl 0104.04803 article

[172] A.-P. Calderón and A. Zygmund: “Local properties of solutions of elliptic partial differential equations,” Studia Math. 20 : 2 (1961), pp. 171–225. This expands on an article published in Proc. Natl. Acad. Sci. U.S.A. 46:10 (1960). MR 0136849 Zbl 0099.30103 article

[173] A. Zygmund: “On one-sided localization of trigonometric series,” pp. 435–447 in Studies in mathematical analysis and related topics: Essays in honor of G. Pólya. Edited by G. Szegö, C. Loewner, S. Bergman, M. M. Schiffer, J. Neyman, D. Gilbarg, and H. Solomon. Stanford Studies in Mathematics and Statistics 4. Stanford University Press (Stanford, CA), 1962. MR 0145261 Zbl 0119.05701 incollection

David Gilbarg	Related
George Pólya	Related
Charles Loewner	Related
Menahem Max Schiffer	Related
Herbert Solomon	Related
Jerzy Neyman	Related
Gábor Szegö	Related
Stefan Bergman	Related

[174] A.-P. Calderón and A. Zygmund: “On the differentiability of functions which are of bounded variation in Tonelli’s sense,” Rev. Un. Mat. Argentina 20 (1962), pp. 102–121. MR 0151557 Zbl 0116.31804 article

[175] A.-P. Calderón and A. Zygmund: “On higher gradients of harmonic functions,” pp. 57–60 in Outlines of the joint Soviet-American symposium on partial differential equations (Novosibirsk, August 1963). Academy of Sciences of the USSR, Siberian Branch (Moscow), 1963. A version of this was later published in Studia Math. 24:2 (1964). MR 0214798 incollection

[176] E. M. Stein and A. Zygmund: “On the differentiability of functions,” Studia Math. 23 : 3 (1963–1964), pp. 247–283. Dedicated to E. Hille on the occasion of his 70th birthday. MR 0158955 Zbl 0122.30203 article

Elias Menachem Stein	Related
Carl Einar Hille	Related

[177] A.-P. Calderón and A. Zygmund: “On higher gradients of harmonic functions,” Studia Math. 24 : 2 (1964), pp. 211–226. An earlier version of this was published in Outlines of the joint Soviet-American symposium on partial differential equations (1963). MR 0167631 Zbl 0168.37002 article

[178] E. M. Stein and A. Zygmund: “On the fractional differentiability of functions,” Proc. London Math. Soc. (3) 14A : 1 (1965), pp. 249–264. To J. E. Littlewood on his 80th birthday. MR 0177076 Zbl 0154.05501 article

J. E. Littlewood	Related
Elias Menachem Stein	Related

[179] A. Zigmund: Trigonometricheskie ryady [Trigonometric series], vol. I. Mir (Moscow), 1965. Russian translation of 1959 2nd edition. MR 0178296 Zbl 0121.05804 book

[180] S. Saks and A. Zygmund: Analytic functions. Monografie Matematyczne 28. Polskie Towarzystwo Matematyczne (Warsaw), 1965. Enlarged 2nd edition of 1952 English translation of 1938 Polish original. MR 0180658 Zbl 0136.37301 book

[181] A. Zigmund: Trigonometricheskie ryady [Trigonometric series], vol. II. Edited by N. K. Bari. Mir (Moscow), 1965. Russian translation of 1959 2nd edition. Zbl 0131.06703 book

[182] M. Weiss and A. Zygmund: “An example in the theory of singular integrals,” Studia Math. 26 : 1 (1965–1966), pp. 101–111. To Gabor Szegö on his seventieth birthday. MR 0190655 Zbl 0127.05702 article

Mary Bishop Weiss	Related
Gábor Szegö	Related

[183] A. Zygmund: “A note on the differentiability of integrals,” pp. 199–204 in Dedie a Franciszek Leja [Dedicated to Franciszek Leja], published as Colloq. Math. 16 (1967). MR 0210847 Zbl 0156.06301 incollection

[184] E. M. Stein and A. Zygmund: “Boundedness of translation invariant operators on Hölder spaces and \( L^p \)-spaces,” Ann. Math. (2) 85 : 2 (March 1967), pp. 337–349. MR 0215121 Zbl 0172.40102 article

[185] A. P. Calderón, M. Weiss, and A. Zygmund: “On the existence of singular integrals,” pp. 56–73 in Singular integrals (Chicago, 20–22 April 1966). Edited by A. P. Calderón. Proceedings of Symposia in Pure Mathematics 10. American Mathematical Society (Providence, RI), 1967. An addendum to this article was published in Studia Math. 46:3 (1973). MR 0338709 Zbl 0167.12202 incollection

Alberto Pedro Calderón	Related	Volume
Mary Bishop Weiss	Related

[186] A. Zygmund: “Mary Weiss: December 11, 1930–October 8, 1966,” pp. xi–xviii in Orthogonal expansions and their continuous analogues (Edwardsville, IL, 27–29 April 1967). Edited by D. T. Haimo. Southern Illinois University Press (Carbondale, IL), 1968. MR 0230593 Zbl 0157.33210 incollection

Mary Bishop Weiss	Related
Deborah Tepper Haimo	Related

[187] E. M. Stein and A. Zygmund: “On the boundary behavior of harmonic functions,” pp. 127–141 in Orthogonal expansions and their continuous analogues (Edwardsville, IL, 27–29 April 1967). Edited by D. T. Haimo. Southern Illinois University Press (Carbondale, IL), 1968. MR 0234001 Zbl 0157.42501 incollection

Elias Menachem Stein	Related
Deborah Tepper Haimo	Related

[188] M. Weiss and A. Zygmund: “On multipliers preserving convergence of trigonometric series almost everywhere,” Studia Math. 30 : 1 (1968), pp. 111–120. MR 0234208 Zbl 0157.38204 article

[189] A. Zygmund: Trigonometric series, vol. I and II combined. 2nd. Cambridge University Press (London and New York), 1969. Two volumes bound as one, reprinted with corrections and some additions. 2nd edition volume I and II bound together. MR 0236587 book

[190] A. Zygmund: “On certain lemmas of Marcinkiewicz and Carleson,” J. Approx. Theory 2 : 3 (September 1969), pp. 249–257. Lecture delivered at Second Symposium on Inequalities at USAF Academy, Colorado, August 1967. MR 0261408 Zbl 0175.35303 article

[191] S. Saks and A. Zygmund: Fonctions analytiques [Analytic functions]. Masson et Cie (Paris), 1970. French translation of 1965 enlarged 2nd edition of 1952 English translation of 1938 Polish original. MR 0269813 Zbl 0199.12601 book

[192] A. Zygmund: “A theorem on the formal multiplication of trigonometric series,” pp. 224–227 in Functional analysis and related fields: Proceedings of a conference in honor of Professor Marshall Stone (Chicago, 20–24 May 1968). Edited by F. E. Browder. Springer (New York), 1970. MR 0275042 Zbl 0242.42007 incollection

Felix E. Browder	Related
Marshall Harvey Stone	Related	Volume

[193] S. Saks and A. Zygmund: Analytic functions, 3rd edition. Elsevier (Amsterdam), 1971. Republication of 1965 enlarged 2nd edition of 1952 English translation of 1938 Polish original. MR 0349963 book

[194] A. Zygmund: Intégrales singulières [Singular integrals]. Lecture Notes in Mathematics 204. Springer (Berlin), 1971. MR 0467191 Zbl 0216.15104 book

[195] A. Zygmund: “A Cantor–Lebesgue theorem for double trigonometric series,” Studia Math. 43 : 2 (1972), pp. 173–178. MR 0312149 Zbl 0214.32501 article

[196] Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I, published as Studia Math. 44 : 1. Państwowe Wydawnictwo Naukowe (Warsaw), 1972. MR 0327407 book

[197] Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, II, published as Studia Math. 44 : 2. Państwowe Wydawnictwo Naukowe (Warsaw), 1972. MR 0327408 book

[198] Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, III, published as Studia Math. 44 : 3. Państwowe Wydawnictwo Naukowe (Warsaw), 1972. MR 0327409 book

[199] Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, IV, published as Studia Math. 44 : 4. Państwowe Wydawnictwo Naukowe (Warsaw), 1972. MR 0327410 book

[200] Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, V, published as Studia Math. 44 : 5. Państwowe Wydawnictwo Naukowe (Warsaw), 1972. MR 0327411 book

[201] Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI, published as Studia Math. 44 : 6. Państwowe Wydawnictwo Naukowe (Warsaw), 1972. MR 0327412 book

[202] A. P. Calderón and A. Zygmund: “Addendum to the paper ‘On singular integrals’,” Studia Math. 46 : 3 (1973), pp. 297–299. Addendum to an article published in Singular integrals (1967). Errata for this addendum were published in Studia Math. 47:3 (1973). MR 0338710 Zbl 0263.44003 article

Let \( x\in E^n \), \[ \tilde{f}_{\varepsilon}(x) = (K_{\varepsilon}* f)(x) ,\] where \[ K_{\varepsilon}(x) = \Lambda(x)|x|^{-n} \] for \( |x| > \varepsilon \) and is 0 otherwise, and \( \Lambda(x) \) is a homogeneous function of degree 0 whose mean value over the unit sphere \( (\Sigma)|x| = 1 \) is 0. Let \( \omega_1(\delta) \) be the “rotational” modulus of continuity of \( \Lambda \) in the metric \( L^1 \): \[ \omega_1(\delta) = \sup_{|p| < \delta}\int_{\Sigma}|\Lambda(\rho x) - \Lambda(x)|\,d\sigma ,\] where \( \rho \) is an arbitrary rotation on \( \Sigma \), \( |\rho| \) its magnitude, and \( d\sigma \) the element of surface area. The note clarifies a proof of the following theorem: If \( \omega_1(\delta) \) satisfies the Dini condition, i.e. \[ \int_0^1\delta^{-1}\omega_1(\delta)\,d\delta < \infty ,\] then the operation \[ \sup_{\varepsilon}|\tilde{f}_{\varepsilon}(x)| = Tf \] is of weak type \( (1,1) \).

[203] A. P. Calderón and A. Zygmund: “Errata: ‘Addendum to the paper ‘On singular integrals’ ’,” Studia Math. 47 : 3 (1973), pp. 305. Errata for an addendum published in Studia Math. 46:3 (1973). MR 0338711 Zbl 0263.44004 article

[204] A. Zygmund: “On Fourier coefficients and transforms of functions of two variables,” Studia Math. 50 (1974), pp. 189–201. To Marceli Stark. MR 0387950 Zbl 0278.42005 article

Let \( f(x_1,x_2) \) be a function of two variables, of period 1 in each, and let \( c_{\mu} = c_{m,n} \) be the Fourier coefficients of \( f \). Then, if \[ 1 \leq p \leq \tfrac{4}{3} \quad\text{and}\quad q = \tfrac{1}{2}p^{\prime} = \tfrac{1}{2}p/(p-1) ,\] we have \[ \Bigl\{ \sum_{|\mu| = r}|c_{\mu}|^q \Bigr\}^{1/q} \leq A_p\|f\|_p \qquad(A_p = 5^{1/p^{\prime}}) \] for all \( r > 0 \). There is a corresponding result for Fourier transforms of functions \( f\in L^p(\mathbf{R}^2) \), \( 1 \leq p < \frac{4}{3} \), but the previous \( q = \frac{1}{2}p^{\prime} \) has to be replaced by \( q = \frac{1}{3}p^{\prime} \). Moreover, the result fails in the extreme case \( p = \frac{4}{3} \). The results are strictly two-dimensional.

[205] A. Zygmund: “The role of Fourier series in the development of analysis,” pp. 591–594 in Proceedings of the American Academy workshop on the evolution of modern mathematics (Boston, MA, 7–9 August 1974), published as Hist. Math. 2 : 4. Issue edited by G. Birkhoff. Elsevier (San Diego, CA), November 1975. MR 0479870 Zbl 0331.01009 incollection

[206] C. Fefferman, J.-P. Kahane, and E. M. Stein: “O dorobku naukowym Antoniego Zygmunda” [The scientific achievements of Antoni Zygmund], Wiadom. Mat. (2) 19 : 2 (1976), pp. 91–126. MR 0433115 article

Jean-Pierre Kahane	Related
Elias Menachem Stein	Related
Charles Louis Fefferman	Related

[207] A. Zygmund: “Notes on the history of Fourier series,” pp. 1–19 in Studies in harmonic analysis (Chicago, 29 June–2 July 1974). Edited by J. M. Ash. MAA Studies in Mathematics 13. Mathematical Association of America (Washington, DC), 1976. MR 0454489 Zbl 0358.42002 incollection

[208] R. L. Wheeden and A. Zygmund: Measure and integral: An introduction to real analysis. Monographs and Textbooks in Pure and Applied Mathematics 43. Marcel Dekker (New York), 1977. Republished in 2015. MR 0492146 Zbl 0362.26004 book

[209] A. Zygmund: Trigonometric series, 2nd reprinted edition, vol. I and II combined. Cambridge University Press, 1977. Reprinting of the 1969 version of the 2nd edition volume I and II bound together. MR 0617944 Zbl 0367.42001 book

[210] A.-P. Calderón and A. Zygmund: “On singular integrals with variable kernels,” Appl. Anal. 7 : 3 (1977–1978), pp. 221–238. Dedicated to S. G. Mikhilin on the occasion of his 70th birthday. MR 0511145 Zbl 0451.42012 article

Alberto Pedro Calderón	Related	Volume
Solomon Grigorevich Mikhlin	Related

[211] A.-P. Calderón and A. Zygmund: “A note on singular integrals,” Studia Math. 65 : 1 (1979), pp. 77–87. MR 554542 Zbl 0431.42007 article

[212] A. Zygmund: “Stanisław Saks (1897–1942),” Wiadom. Mat. (2) 24 : 2 (1982), pp. 145–156. A slightly modified and abbreviated translation of this was published in Math. Intell. 9:1 (1987). MR 705775 Zbl 0529.01016 article

[213] Conference on harmonic analysis in honor of Antoni Zygmund (Chicago, 23–28 March 1981), vol. 1 and 2. Edited by W. Beckner, A. P. Calderón, R. Fefferman, and P. W. Jones. Wadsworth Mathematics Series. Wadsworth (Belmont, CA), 1983. MR 0730053 Zbl 0493.00009 book

Alberto Pedro Calderón	Related	Volume
William Beckner	Related
Robert Allen Fefferman	Related
Peter Wilcox Jones	Related

[214] A. P. Calderón: “Antoni Zygmund,” pp. xiii–xv in Conference on harmonic analysis in honor of Antoni Zygmund (Chicago, 23–28 March 1981), vol. 1. Edited by W. Beckner, A. P. Calderón, R. Fefferman, and P. W. Jones. Wadsworth Mathematics Series. Wadsworth (Belmont, CA), 1983. incollection

Alberto Pedro Calderón	Related	Volume
William Beckner	Related
Robert Allen Fefferman	Related
Peter Wilcox Jones	Related

[215] A. Zygmund: “Stanisław Saks, 1897–1942,” Math. Intell. 9 : 1 (1987), pp. 36–41. Slightly modified and abbreviated translation of an article in Wiadom. Mat. 24:2 (1982). MR 869538 Zbl 0602.01027 article

[216] A. Zygmund: “Aleksander Rajchman (1890–1940),” Wiadom. Mat. (2) 27 : 2 (1987), pp. 219–231. MR 908884 Zbl 0631.01027 article

[217] A. Zygmund: Trigonometric series, 2nd edition, vol. I and II combined. Cambridge Mathematical Library. Cambridge University Press, 1988. Republication of the 1977 reprinting of the 1969 2nd edition. MR 933759 Zbl 0628.42001 book

[218] R. R. Coifman and R. S. Strichartz: “The school of Antoni Zygmund,” pp. 343–368 in A century of mathematics in America, part III. Edited by P. Duren, R. Askey, U. C. Merzbach, and H. M. Edwards. History of Mathematics. American Mathematical Society (Providence, RI), 1989. With the collaboration of Gina Graziosi and Julia Hallquist. MR 1025352 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Ronald Raphael Coifman	Related
Harold Mortimer Edwards, Jr.	Related
Richard Allen Askey	Related	Volume

[219] A. Zygmund: Selected papers of Antoni Zygmund, vol. 1. Edited by A. Hulanicki, P. Wojtaszczyk, and W. Żelazko. Mathematics and its Applications: East European Series 41. Kluwer Academic (Dordrecht), 1989. MR 1112134 Zbl 0689.01020 book

Przemysław Wojtaszczyk	Related
Wieslaw Żelazko	Related
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[220] A. Zygmund: Selected papers of Antoni Zygmund, vol. 2. Edited by A. Hulanicki, P. Wojtaszczyk, and W. Żelazko. Mathematics and its Applications: East European Series 41. Kluwer Academic (Dordrecht), 1989. MR 1112135 book

Przemysław Wojtaszczyk	Related
Wieslaw Żelazko	Related
Andrzej Hulanicki	Related

[221] A. Zygmund: Selected papers of Antoni Zygmund, vol. 3. Edited by A. Hulanicki, P. Wojtaszczyk, and W. Żelazko. Mathematics and its Applications: East European Series 41. Kluwer Academic (Dordrecht), 1989. MR 1112136 book

Przemysław Wojtaszczyk	Related
Wieslaw Żelazko	Related
Andrzej Hulanicki	Related

[222] Volume dedicated to Antoni Zygmund, published as Colloq. Math. 60–61. Polska Akademia Nauk (Warsaw), 1990. book

[223] R. Askey: “In memoriam: Antoni Zygmund (December 26, 1900–May 30, 1992),” J. Approx. Theory 71 : 1 (October 1992), pp. iii, 1–2. MR 1180870 Zbl 0753.01016 article

[224] A. P. Calderón and E. M. Stein: “Antoni Zygmund: 1900–1992,” Notices Am. Math. Soc. 39 : 8 (1992), pp. 848–849. MR 1182969 Zbl 1194.01074 article

Alberto Pedro Calderón	Related	Volume
Elias Menachem Stein	Related

[225] “Antoni Szczepan Zygmund (December 26, 1900–May 30, 1992),” Stud. Math. 103 : 2 (1992), pp. 119–121. MR 1199320 Zbl 0809.01025 article

[226] G. G. Lorentz: “Antoni Zygmund and his work,” J. Approx. Theory 75 : 1 (October 1993), pp. 1–7. guest editorial, with partial list of publications. MR 1240252 Zbl 0782.01039 article

[227] G. Weiss: “Antoni Zygmund, 1900–1992,” J. Geom. Anal. 3 : 6 (November 1993), pp. 529–531. A version of this was reproduced in Fourier analysis and partial differential equations (1995). MR 1248083 Zbl 0788.01070 article

[228] D. Waterman: “The contributions of Antoni Zygmund to real analysis,” Real Anal. Exch. 19 : 1 (1993–1994), pp. 11–12. MR 1268827 article

[229] Harmonic analysis techniques for second order elliptic boundary value problems: Dedicated to the memory of Professor Antoni Zygmund. Edited by C. E. Kenig. CBMS Regional Conference Series in Mathematics 83. American Mathematical Society (Providence, RI), 1994. MR 1282720 Zbl 0812.35001 book

[230] G. Weiss: “Antoni Zygmund (1900–1992),” pp. 3–5 in Fourier analysis and partial differential equations (Madrid, 15–20 June 1992). Edited by J. García-Cuerva, E. Hernández, F. Soria, and J.-L. Torrea. Studies in Advanced Mathematics. CRC Press (Boca Raton, FL), 1995. Based on an article published in J. Geom. Anal. 3:6 (1993). MR 1330229 Zbl 0865.01037 incollection

José García-Cuerva	Related
Eugenio Hernández	Related
Fernando Soria	Related
José-Luis Torrea	Related
Guido Leopold Weiss	Related

[231] A. Zygmund: Trigonometric series, 3rd edition, vol. I and II combined. Cambridge Mathematical Library. Cambridge University Press, 2002. With a foreword by Robert A. Fefferman. Republication of the 1988 reprinting of the 1968 2nd edition. MR 1963498 Zbl 1084.42003 book

[232] J. Gray: “Zygmund, Antoni,” pp. 414–416 in New dictionary of scientific biography, vol. 7: Tammes–Zygmund. Edited by N. Koertge. Scribner (New York), 2008. incollection

Noretta Koertge	Related
Jeremy J. Gray	Related

[233] W. Piotrowski: “Przyczynek do biografii Antoniego Zygmunda” [A contribution to the biography of Antoni Zygmund], Wiadom. Mat. (2) 48 : 1 (2012), pp. 81–84. MR 2962205 article

[234] C. E. Kenig and W. Schlag: “Antoni Zygmund’s work on singular integrals,” Wiadom. Mat. (2) 48 : 2 (2012), pp. 119–125. MR 2986186 Zbl 1276.42001 article

Menahem Max Schiffer	Related
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[235] R. L. Wheeden and A. Zygmund: Measure and integral: An introduction to real analysis, 2nd edition. Monographs and Textbooks in Pure and Applied Mathematics 43. CRC Press (Boca Raton, FL), 2015. Republication of 1977 original. MR 3381284 Zbl 1326.26007 book

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