ON THE MÜNTZ THEOREM FOR NUMERABLE COMPACT SETS

J. M. ALMIRA

Abstract

In this note we solve the Müntz problem for the space C(K) whenever K ⊂ [0, ∞) is a numerable compact set which satisfies certain additional assumptions and we propose the general case as an open question. By using a well known consequence of the Hahn-Banach theorem, we convert the problem of the density of the span of the monomials {x^{λ_{i}}}_{i=0}^{∞} in a problem about the zero sets of certain analytic functions. The open problem can thus be formulated as a basic question in the theory of complex functions.

References

[1] J. A. Clarkson and P. Erdös, Approximation by polynomials, Duke Math. J. 10 (1943) 5-11.

[2] L. Schwartz, Etude des sommes de’Exponentielles, Herman (Paris) (1959)

[3] P. Borwein, T. Erdelyi, The full Müntz theorem in C[0, 1] and L_{1}[0, 1], J. London Math. Soc. 54 (2) (1996) 102-110.

[4] P. Borwein, T. Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, Springer (1996).

[5] A. Pinkus, Weierstrass and Approximation Theory, J. Approx. Theory 107 (2000), 1-66.

[6] O. Szàsz, Über die approximation stetiger funktionen durch lineare aggregate von potezen, Math. Ann. 77 (1916), 482-496.

J. M. Almira Departamento de Matemáticas. Universidad de Jaén E. U. P. Linares. C/Alfonso X el Sabio, 28 23700 Linares (Jaén) SPAIN

email: jmalmira@ujaen.es.