Bulletin of the American Mathematical Society

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Book Information:

Author: J. G. Llavona
Title: Approximation of continuously differentiable functions
Additional book information: North-Holland Mathematics Studies, vol. 130, North-Holland, Amsterdam, 1986, xiv + 241 pp., $48.00. ISBN 0-444-70128-1.

Teófilo Abuabara, A version of the Paley-Wiener-Schwartz theorem in infinite dimensions, Advances in holomorphy (Proc. Sem. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977) North-Holland Math. Stud., vol. 34, North-Holland, Amsterdam-New York, 1979, pp. 1–29. MR 520652

R. M. Aron, C. Hervés, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Functional Analysis 52 (1983), no. 2, 189–204. MR 707203, DOI 10.1016/0022-1236(83)90081-2

R. M. Aron and J. B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195–216. MR 552473, DOI 10.1515/crll.1980.313.195

Richard M. Aron and M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, J. Functional Analysis 21 (1976), no. 1, 7–30. MR 0402504, DOI 10.1016/0022-1236(76)90026-4

5.

S. Bernstein, Leçons sur les propriétés extremales de la meilleure approximation des fonctions analytiques d'une variable réele, Paris, 1926.

F. Bombal Gordón, The Arens product in locally convex algebras, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 70 (1976), no. 2, 337–346 (Spanish). MR 428036

Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849

Jaime Lesmes, On the approximation of continuously differentiable functions in Hilbert spaces, Rev. Colombiana Mat. 8 (1974), 217–223. MR 0358346

José G. Llavona, Approximations of differentiable functions, Studies in analysis, Adv. in Math. Suppl. Stud., vol. 4, Academic Press, New York-London, 1979, pp. 197–221. MR 546807

Leopoldo Nachbin, Sur les algèbres denses de fonctions différentiables sur une variété, C. R. Acad. Sci. Paris 228 (1949), 1549–1551 (French). MR 30590

Leopoldo Nachbin, A generalization of Whitney’s theorem on ideals of differentiable functions, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 935–937. MR 90005, DOI 10.1073/pnas.43.10.935

Leopoldo Nachbin and Seán Dineen, Entire functions of exponential type bounded on the real axis and Fourier transforms of distributions with bounded supports, Israel J. Math. 13 (1972), 321–326 (1973). MR 326379, DOI 10.1007/BF02762807

João Bosco Prolla, On polynomial algebras of continuously differentiable functions, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 57 (1974), no. 6, 481–486 (1975) (English, with Italian summary). MR 423407

João Bosco Prolla and Claudia S. Guerreiro, An extension of Nachbin’s theorem to differentiable functions on Banach spaces with the approximation property, Ark. Mat. 14 (1976), no. 2, 251–258. MR 448077, DOI 10.1007/BF02385839

M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), no. 3, 375–481. MR 1501905, DOI 10.1090/S0002-9947-1937-1501905-7

Kondagunta Sundaresan, Geometry and nonlinear analysis in Banach spaces, Pacific J. Math. 102 (1982), no. 2, 487–498. MR 686565

17.

Ch. Vallé Poussin de la, Sur l'approximation des fonctions d'une variable réele et de leurs dérivées par des polynômes et des suites finies de Fourier, Bull. Acad. Sci. Belgique (1908), 193-254.

Hassler Whitney, On ideals of differentiable functions, Amer. J. Math. 70 (1948), 635–658. MR 26238, DOI 10.2307/2372203

1.

T. Abuabara, A version of the Paley-Wiener-Schwartz theorem in infinite dimensions, Advances in Holomorphy, North-Holland Math. Studies 34 (1979), 1-29. MR 0520652

2.

R. M. Aron, C. Herves, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983), 189-203. MR 707203

3.

R. M. Aron and J. B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew Math. 313 (1980), 195-216. MR 552473

4.

R. M. Aron and R. M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, J. Funct. Anal. 21 (1976), 7-30. MR 402504

5.

S. Bernstein, Leçons sur les propriétés extremales de la meilleure approximation des fonctions analytiques d'une variable réele, Paris, 1926.

6.

F. Bombal and J. Llavona, La propiedad de approximación en espacios de funciones differenciables, Rev. R. Acad. Ciencias de Madrid 70 (1976), 337-346. MR 428036

7.

M. M. Day, Normed linear spaces, Third ed. Springer-Verlag, New York, 1973. MR 344849

8.

J. Lesmes, On the approximation of continuously differentiable functions in Hilbert spaces, Rev. Colombiana Mathemáticas 8 (1974), 217-223. MR 358346

9.

J. Llavona, Approximation of differentiable functions, Adv. in Math. Supplementary Studies 4 (1979), 197-221. MR 546807

10.

L. Nachbin, Sur les algèbres denses de fonctions différentiables sur une variété, C. R. Acad. Sci. Paris 228 (1949), 1549-1551. MR 30590

11.

L. Nachbin, A generalization of Whitney's theorem on ideals of differentiable functions, Proc. Nat. Acad. Sci. U. S. A. 43 (1957), 935-937. MR 90005

12.

L. Nachbin and S. Dineen, Entire functions of exponential type bounded on the real axis and Fourier transform of distributions with bounded supports, Israel J. Math. 13 (1972), 321-326. MR 326379

13.

J. B. Prolla, On polynomial algebras of continuously differentiable functions, Rend. Accad. Naz. Lincei. 57 (1974), 481-486. MR 423407

14.

J. B. Prolla and C. S. Guerreiro, An extension of Nachbin's theorem to differentiable functions on Banach spaces with approximation property, Ark. Math. 14 (1976), 251-258. MR 448077

15.

M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375-381. MR 1501905

16.

K. Sundaresan, Geometry and nonlinear analysis in Banach spaces, Pacific J. Math. 102(1982), 487-498. MR 686565

17.

Ch. Vallé Poussin de la, Sur l'approximation des fonctions d'une variable réele et de leurs dérivées par des polynômes et des suites finies de Fourier, Bull. Acad. Sci. Belgique (1908), 193-254.

18.

H. Whitney, On ideals of differentiable functions, Amer. J. Math. 70 (1948), 635-658. MR 26238