Abstract
Assume X is a compact Hausdorff space and C(X) is the space of real-valued continuous functions on X. A version of the Stone–Weierstrass theorem states that a closed subalgebra \(A\subset C(X)\), which contains a nonzero constant function, coincides with the whole space C(X) if and only if A separates points of X. In this paper, we generalize this theorem to the case in which two subalgebras of C(X) are involved.
Résumé
Soit X un espace de Hausdorff compact et soit C(X) l’espace des fonctions continues sur X à valeurs réelles. Selon une version du théorème de Stone–Weierstrass, si A est une sous-algèbre fermée de C(X) qui contient une fonction constante non nulle, alors A coïncide avec l’espace entier C(X) si et seulement si A sépare les points de X. Dans cet article, on généralise ce théorème au cas où l’on considère deux sous-algèbres de C(X).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V.I.: On functions of three variables. (Russian) Dokl. Akad. Nauk SSSR 114, 679–681 (1957) [English transl. Am. Math. Soc. Transl. 28, 51–54 (1963)]
Asgarova, A.Kh., Ismailov, V.E.: On the Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras. Proc. Indian Acad. Sci. Math. Sci. (to appear). arXiv:1603.07073
Bilalov, B.T.: On the Stone and Bishop approximation theorems. (Russian) Mat. Zametki 81, 660–665 (2007) [translation in Math. Notes 81, 590–595 (2007)]
Bishop, E.: A generalization of the Stone–Weierstrass theorem. Pac. J. Math. 11, 777–783 (1961)
Boel, S., Carlsen, T.M., Hansen, N.R.: A useful strengthening of the Stone–Weierstrass theorem. Am. Math. Mon. 108, 642–643 (2001)
Chernoff, P.R., Rasala, R.A., Waterhouse, W.C.: The Stone–Weierstrass theorem for valuable fields. Pac. J. Math. 27, 233–240 (1968)
Cowsik, R.C., Klopotowski, A., Nadkarni, M.G.: When is \(f(x, y)=u(x)+v(y)\)? Proc. Indian Acad. Sci. Math. Sci. 109, 57–64 (1999)
Diliberto, S.P., Straus, E.G.: On the approximation of a function of several variables by the sum of functions of fewer variables. Pac. J. Math. 1, 195–210 (1951)
Ismailov, V.E.: Alternating algorithm for the approximation by sums of two compositions and ridge functions. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 41(1), 146–152 (2015)
Ismailov, V.E.: On the theorem of M. Golomb. Proc. Indian Acad. Sci. Math. Sci. 119, 45–52 (2009)
Ismailov, V.E.: Representation of multivariate functions by sums of ridge functions. J. Math. Anal. Appl. 331, 184–190 (2007)
Ismailov, V.E.: On error formulas for approximation by sums of univariate functions. Int. J. Math. Math. Sci. 2006, Article ID 65620 (2006)
Ismailov, V.E.: Methods for computing the least deviation from the sums of functions of one variable. (Russian) Sibirskii Mat. Zhurnal 47, 1076–1082 (2006) [translation in Sib. Math. J. 47, 883–888 (2006)]
Khavinson, S.Ya.: Best approximation by linear superpositions (approximate nomography). Translated from the Russian manuscript by D. Khavinson. In: Translations of Mathematical Monographs, vol. 159. American Mathematical Society, Providence (1997)
Klopotowski, A., Nadkarni, M.G.: Shift invariant measures and simple spectrum. Colloq. Math. 84(85), 385–394 (2000)
Light, W.A., Cheney, E.W.: On the approximation of a bivariate function by the sum of univariate functions. J. Approx. Theory 29, 305–322 (1980)
Light, W.A., Cheney, E.W.: Approximation theory in tensor product spaces. Lecture Notes in Mathematics, vol. 1169, p 157. Springer, Berlin (1985)
Marshall, D.E., O’Farrell, A.G.: Approximation by a sum of two algebras. The lightning bolt principle. J. Funct. Anal. 52, 353–368 (1983)
Marshall, D.E., O’Farrell, A.G.: Uniform approximation by real functions. Fund. Math. 104, 203–211 (1979)
Medvedev, V.A.: On the sum of two closed algebras of continuous functions on a compact space. (Russian) Funktsional. Anal. i Prilozhen. 27, 33–36 (1993) [translation in Funct. Anal. Appl. 27, 28–30 (1993)]
Prolla, J.B.: Weierstrass Stone, the Theorem. Peter Lang, Frankfurt am Main (1993)
Srikanth, K.V., Yadav, R.B.: On an extension of the Stone–Weierstrass theorem. Math. Commun. 19, 391–396 (2014)
Sternfeld, Y.: Uniformly separating families of functions. Isr. J. Math. 29, 61–91 (1978)
Stone, M.H.: Applications of the theory of Boolean rings to general topology. Trans. Am. Math. Soc. 41, 375–481 (1937)
Stone, M.H.: The generalized Weierstrass approximation theorem. Math. Mag. 21(167–184), 237–254 (1948)
Timofte, V.: Stone–Weierstrass theorems revisited. J. Approx. Theory 136, 45–59 (2005)
Wenjen, C.: Generalizations of the Stone–Weierstrass approximation theorem. Proc. Jpn. Acad. 44, 928–932 (1968)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Asgarova, A.K. On a generalization of the Stone–Weierstrass theorem. Ann. Math. Québec 42, 1–6 (2018). https://doi.org/10.1007/s40316-017-0081-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40316-017-0081-2