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On the degree of polynomial approximation to harmonic and analytic functions. (English) Zbl 0035.17102


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[1] Torsten Carleman, Über die Fourierkoeffizienten einer stetigen Funktion, Acta Math. 41 (1916), no. 1, 377 – 384 (German). Aus einem Brief an Herrn A. Wiman. · JFM 46.0444.01 · doi:10.1007/BF02422951
[2] M. L. Cartwright On analytic functions regular in the unit circle, Quart. J. Math. Oxford Ser. vol. 4 (1933) pp. 246-257. · JFM 59.0325.01
[3] J. H. Curtiss, A note on the degree of polynomial approximation, Bull. Amer. Math. Soc. 42 (1936), no. 12, 873 – 878. · Zbl 0016.01903
[4] G. Faber Über Tschebyscheffsche Polynome, J. Reine Angew. Math. vol. 150 (1920) pp. 79-106. · JFM 47.0315.01
[5] M. Fekete Über einen Satz des Herrn Serge Bernstein, J. Reine Angew. Math. vol. 146 (1916) pp. 88-94. · JFM 45.0408.02
[6] Dunham Jackson, The theory of approximation, American Mathematical Society Colloquium Publications, vol. 11, American Mathematical Society, Providence, RI, 1994. Reprint of the 1930 original. · Zbl 0008.05803
[7] C. de la Vallée Poussin Leçons sur l’approximation des fonctions d’une variable réelle, Paris, 1919. · JFM 47.0908.02
[8] E. Lindelöf Sur un principe général de l’analyse, Acta Societatis Scientiarum Fennicae vol. 46 (1915) No. 4. · JFM 45.0665.02
[9] N. Lusin and J. Privaloff Sur l’unicité et la multiplicité de fonctions analytiques, Ann. École Norm. (3) vol. 42 (1925) pp. 143-191. · JFM 51.0245.01
[10] P. Montel Sur les polynômes d’approximation, Bull. Soc. Math. France vol. 46 (1919) pp. 151-192. · JFM 46.0417.02
[11] G. Pólya and G. Szegö Aufgaben und Lehrsätze aus der Analysis, Berlin, 1925. · JFM 51.0173.01
[12] J. Priwaloff, Sur les fonctions conjuguées, Bull. Soc. Math. France 44 (1916), 100 – 103 (French). · JFM 46.0540.03
[13] -Sur certaines propriétés métriques des fonctions analytiques, J. École Polytech. (2) vol. 24 (1924) pp. 77-112.
[14] Marcel Riesz, Sur les fonctions conjuguées, Math. Z. 27 (1928), no. 1, 218 – 244 (French). · JFM 53.0259.02 · doi:10.1007/BF01171098
[15] A. C. Schaeffer and G. Szegö, Inequalities for harmonic polynomials in two and three dimensions, Trans. Amer. Math. Soc. 50 (1941), 187 – 225. · JFM 67.1001.03
[16] W. Seidel, Über die Ränderzuordnung bei konformen Abbildungen, Math. Ann. 104 (1931), no. 1, 182 – 243 (German). · JFM 57.0398.01 · doi:10.1007/BF01457932
[17] W. E. Sewell, Degree of Approximation by Polynomials in the Complex Domain, Annals of Mathematical Studies, no. 9, Princeton University Press, Princeton, N. J., 1942. · Zbl 0063.08342
[18] V. Smirnoff Sur les formules de Cauchy et de Green et quelques problèmes qui s’y rattachent, Bull. Acad. Sci. URSS. Sér. Math. vol. 7 (1932) pp. 337-371. · Zbl 0005.10702
[19] Gabriel Szegő, Über trigonometrische und harmonische Polynome, Math. Ann. 79 (1919), no. 4, 323 – 339 (German). · JFM 47.0263.02 · doi:10.1007/BF01498414
[20] -Über einen Satz des Herrn Serge Bernstein, Schriften der Königsberger Gelehrten Gesellschaft, Naturwissenschaftliche Klasse vol. 5 (1928) pp. 59-70. · JFM 54.0311.03
[21] Masatsugu Tsuji, On the Green’s function, Jap. J. Math. 18 (1942), 379 – 383. · Zbl 0061.23405
[22] J. L. Walsh, On the degree of approximation to a harmonic function, Bull. Amer. Math. Soc. 33 (1927), no. 5, 591 – 598. · JFM 53.0466.03
[23] -Über die Entwicklung einer harmonischen Funktion nach harmonischen Polynomen, J. Reine Angew. Math. vol. 159 (1928) pp. 197-209. · JFM 54.0510.02
[24] J. L. Walsh, The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions, Bull. Amer. Math. Soc. 35 (1929), no. 4, 499 – 544. · JFM 55.0889.05
[25] -Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Colloquium Publications, vol. 20, New York, 1935.
[26] J. L. Walsh, Maximal convergence of sequences of harmonic polynomials, Ann. of Math. (2) 38 (1937), no. 2, 321 – 354. · Zbl 0017.10801 · doi:10.2307/1968557
[27] J. L. Walsh and W. E. Sewell, Sufficient conditions for various degrees of approximation by polynomials, Duke Math. J. 6 (1940), 658 – 705. · Zbl 0023.40303
[28] J. L. Walsh and W. E. Sewell, Note on the relation between continuity and degree of polynomial approximation in the complex domain, Bull. Amer. Math. Soc. 43 (1937), no. 8, 557 – 563. · Zbl 0017.39503
[29] J. L. Walsh and W. E. Sewell, Note on degree of trigonometric and polynomial approximation to an analytic function, Bull. Amer. Math. Soc. 44 (1938), no. 12, 865 – 873. · Zbl 0020.21303
[30] J. L. Walsh and W. E. Sewell, On the degree of polynomial approximation to analytic functions: problem \?, Trans. Amer. Math. Soc. 49 (1941), 229 – 257. · Zbl 0025.32004
[31] Stefan Warschawski, Bemerkung zu meiner Arbeit: Über das Randverhalten der Ableitung der Abbildungsfunktion bei konformer Abbildung, Math. Z. 38 (1934), no. 1, 669 – 683 (German). · Zbl 0009.26101 · doi:10.1007/BF01170662
[32] A. Zygmund Trigonometrical series, Monografje Matematyczne, vol. V, Warsaw-Lwów, 1935. · JFM 61.0263.03
[33] A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47 – 76. · Zbl 0060.13806
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