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On the degree of polynomial approximation to harmonic and analytic functions


Authors: J. L. Walsh, W. E. Sewell and H. M. Elliott
Journal: Trans. Amer. Math. Soc. 67 (1949), 381-420
MSC: Primary 31.0X
DOI: https://doi.org/10.1090/S0002-9947-1949-0033920-5
MathSciNet review: 0033920
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  • [1] T. Carleman Über die Fourierkoeffizienten einer stetigen Funktion, Acta Math. vol. 41 (1918) pp. 377-384. MR 1555157
  • [1] M. L. Cartwright On analytic functions regular in the unit circle, Quart. J. Math. Oxford Ser. vol. 4 (1933) pp. 246-257.
  • [1] John Curtiss A note on the degree of polynomial approximation, Bull. Amer. Math. Soc. vol. 42 (1936) pp. 873-878. MR 1563457
  • [1] G. Faber Über Tschebyscheffsche Polynome, J. Reine Angew. Math. vol. 150 (1920) pp. 79-106.
  • [1] M. Fekete Über einen Satz des Herrn Serge Bernstein, J. Reine Angew. Math. vol. 146 (1916) pp. 88-94.
  • [1] D. Jackson The theory of approximation, Amer. Math. Soc. Colloquium Publications, vol. 11, New York, 1930. MR 1451140 (98a:01022)
  • [1] C. de la Vallée Poussin Leçons sur l'approximation des fonctions d'une variable réelle, Paris, 1919.
  • [1] E. Lindelöf Sur un principe général de l'analyse, Acta Societatis Scientiarum Fennicae vol. 46 (1915) No. 4.
  • [1] N. Lusin and J. Privaloff Sur l'unicité et la multiplicité de fonctions analytiques, Ann. École Norm. (3) vol. 42 (1925) pp. 143-191.
  • [1] P. Montel Sur les polynômes d'approximation, Bull. Soc. Math. France vol. 46 (1919) pp. 151-192.
  • [1] G. Pólya and G. Szegö Aufgaben und Lehrsätze aus der Analysis, Berlin, 1925.
  • [1] J. Privaloff Sur les fonctions conjuguées, Bull. Soc. Math. France vol. 44 (1916) pp. 100-103. MR 1504751
  • [2] -Sur certaines propriétés métriques des fonctions analytiques, J. École Polytech. (2) vol. 24 (1924) pp. 77-112.
  • [1] M. Riesz Sur les fonctions conjuguées, Math. Zeit. vol. 27 (1928) pp. 218-244. MR 1544909
  • [1] A. C. Schaeffer and G. Szegö Inequalities for harmonic polynomials in two and three dimensions, Trans. Amer. Math. Soc. vol. 50 (1941) pp. 187-225. MR 0005164 (3:111b)
  • [1] W. Seidel Über die Ränderzuordnung bei konformen Abbildungen, Math. Ann. vol. 104 (1931) pp. 182-243. MR 1512661
  • [1] W. E. Sewell Degree of approximation by polynomials in the complex domain, Annals of Mathematics Studies, vol. 9, Princeton, 1942. MR 0007054 (4:78c)
  • [1] 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.
  • [1] G. Szegö Über trigonometrische und harmonische Polynome, Math. Ann. vol. 79 (1919) pp. 323-329. MR 1511934
  • [2] -Über einen Satz des Herrn Serge Bernstein, Schriften der Königsberger Gelehrten Gesellschaft, Naturwissenschaftliche Klasse vol. 5 (1928) pp. 59-70.
  • [1] M. Tsuji On the Green's function, Jap. J. Math. vol. 18 (1941-1943) pp. 379-383. MR 0017813 (8:203h)
  • [1] J. L. Walsh On the degree of approximation to a harmonic function, Bull. Amer. Math. Soc. vol. 33 (1927) pp. 591-598. MR 1561424
  • [2] -Über die Entwicklung einer harmonischen Funktion nach harmonischen Polynomen, J. Reine Angew. Math. vol. 159 (1928) pp. 197-209.
  • [3] -The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions, Bull. Amer. Math. Soc. vol. 35 (1929) pp. 499-544. MR 1561772
  • [4] -Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Colloquium Publications, vol. 20, New York, 1935.
  • [5] -Maximal convergence of sequences of harmonic polynomials, Ann. of Math. vol. 38 (1937) pp. 321-354. MR 1503339
  • [1] J. L. Walsh and W. E. Sewell Sufficient conditions for various degrees of approximation by polynomials, Duke Math. J. vol. 6 (1940) pp. 658-705. MR 0002592 (2:80c)
  • [2] -A note on the relation between continuity and degree of polynomial approximation in the complex domain, Bull. Amer. Math. Soc. vol. 43 (1937) pp. 557-563. MR 1563586
  • [3] -Note on degree of trigonometric and polynomial approximation to an analytic function, ibid. vol. 44 (1938) pp. 865-873. MR 1563892
  • [4] -On the degree of polynomial approximation to analytic functions: Problem $ \beta $, Trans. Amer. Math. Soc. vol. 49 (1941) pp. 229-257. MR 0003818 (2:276h)
  • [1] S. E. Warschawski Bemerkung zu meiner Arbeit: Über das Randverhalten der Ableitung der Abbildungsfunktion bei konformer Abbildung, Math. Zeit. vol. 38 (1934) pp. 669-683. MR 1545475
  • [1] A. Zygmund Trigonometrical series, Monografje Matematyczne, vol. V, Warsaw-Lwów, 1935.
  • [2] -Smooth functions, Duke Math. J. vol. 12 (1945) pp. 47-76. MR 0012691 (7:60b)

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DOI: https://doi.org/10.1090/S0002-9947-1949-0033920-5
Article copyright: © Copyright 1949 American Mathematical Society

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