On the degree of polynomial approximation to harmonic and analytic functions

Walsh, J. L.; Sewell, W. E.; Elliott, H. M.

doi:10.1090/S0002-9947-1949-0033920-5

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
Full-text PDF Free Access

Torsten Carleman, Über die Fourierkoeffizienten einer stetigen Funktion, Acta Math. 41 (1916), no. 1, 377–384 (German). Aus einem Brief an Herrn A. Wiman. MR 1555157, DOI https://doi.org/10.1007/BF02422951

M. L. Cartwright On analytic functions regular in the unit circle, Quart. J. Math. Oxford Ser. vol. 4 (1933) pp. 246-257.

J. H. Curtiss, A note on the degree of polynomial approximation, Bull. Amer. Math. Soc. 42 (1936), no. 12, 873–878. MR 1563457, DOI https://doi.org/10.1090/S0002-9904-1936-06455-4

G. Faber Über Tschebyscheffsche Polynome, J. Reine Angew. Math. vol. 150 (1920) pp. 79-106.

M. Fekete Über einen Satz des Herrn Serge Bernstein, J. Reine Angew. Math. vol. 146 (1916) pp. 88-94.

Dunham Jackson, The theory of approximation, American Mathematical Society Colloquium Publications, vol. 11, American Mathematical Society, Providence, RI, 1994. Reprint of the 1930 original. MR 1451140

C. de la Vallée Poussin Leçons sur l’approximation des fonctions d’une variable réelle, Paris, 1919.

E. Lindelöf Sur un principe général de l’analyse, Acta Societatis Scientiarum Fennicae vol. 46 (1915) No. 4.

N. Lusin and J. Privaloff Sur l’unicité et la multiplicité de fonctions analytiques, Ann. École Norm. (3) vol. 42 (1925) pp. 143-191.

P. Montel Sur les polynômes d’approximation, Bull. Soc. Math. France vol. 46 (1919) pp. 151-192.

G. Pólya and G. Szegö Aufgaben und Lehrsätze aus der Analysis, Berlin, 1925.

J. Priwaloff, Sur les fonctions conjuguées, Bull. Soc. Math. France 44 (1916), 100–103 (French). MR 1504751

---Sur certaines propriétés métriques des fonctions analytiques, J. École Polytech. (2) vol. 24 (1924) pp. 77-112.

Marcel Riesz, Sur les fonctions conjuguées, Math. Z. 27 (1928), no. 1, 218–244 (French). MR 1544909, DOI https://doi.org/10.1007/BF01171098
A. C. Schaeffer and G. Szegö, Inequalities for harmonic polynomials in two and three dimensions, Trans. Amer. Math. Soc. 50 (1941), 187–225. MR 5164, DOI https://doi.org/10.1090/S0002-9947-1941-0005164-7
W. Seidel, Über die Ränderzuordnung bei konformen Abbildungen, Math. Ann. 104 (1931), no. 1, 182–243 (German). MR 1512661, DOI https://doi.org/10.1007/BF01457932
W. E. Sewell, Degree of Approximation by Polynomials in the Complex Domain, Annals of Mathematics Studies, No. 9, Princeton University Press, Princeton, N. J., 1942. MR 0007054

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.

Gabriel Szegő, Über trigonometrische und harmonische Polynome, Math. Ann. 79 (1919), no. 4, 323–339 (German). MR 1511934, DOI https://doi.org/10.1007/BF01498414

---Über einen Satz des Herrn Serge Bernstein, Schriften der Königsberger Gelehrten Gesellschaft, Naturwissenschaftliche Klasse vol. 5 (1928) pp. 59-70.

Masatsugu Tsuji, On the Green’s function, Jpn. J. Math. 18 (1942), 379–383. MR 17813, DOI https://doi.org/10.4099/jjm1924.18.0_379
J. L. Walsh, On the degree of approximation to a harmonic function, Bull. Amer. Math. Soc. 33 (1927), no. 5, 591–598. MR 1561424, DOI https://doi.org/10.1090/S0002-9904-1927-04416-0

---Über die Entwicklung einer harmonischen Funktion nach harmonischen Polynomen, J. Reine Angew. Math. vol. 159 (1928) pp. 197-209.

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. MR 1561772, DOI https://doi.org/10.1090/S0002-9904-1929-04753-0

---Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Colloquium Publications, vol. 20, New York, 1935.

J. L. Walsh, Maximal convergence of sequences of harmonic polynomials, Ann. of Math. (2) 38 (1937), no. 2, 321–354. MR 1503339, DOI https://doi.org/10.2307/1968557
J. L. Walsh and W. E. Sewell, Sufficient conditions for various degrees of approximation by polynomials, Duke Math. J. 6 (1940), 658–705. MR 2592
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. MR 1563586, DOI https://doi.org/10.1090/S0002-9904-1937-06603-1
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. MR 1563892, DOI https://doi.org/10.1090/S0002-9904-1938-06893-0
J. L. Walsh and W. E. Sewell, On the degree of polynomial approximation to analytic functions: problem $\beta $, Trans. Amer. Math. Soc. 49 (1941), 229–257. MR 3818, DOI https://doi.org/10.1090/S0002-9947-1941-0003818-X
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). MR 1545475, DOI https://doi.org/10.1007/BF01170662

A. Zygmund Trigonometrical series, Monografje Matematyczne, vol. V, Warsaw-Lwów, 1935.

A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47–76. MR 12691