A uniform approximation theorem and its application to moment problems

order that a (real) sequence #, may have the representation /~,=S f dx(t), 0 where X(t) is a function of bounded variation in [0, 1]. HARDY [3] outlines the proof of HAUSDORFF; several alternate proofs of the result of HAUSDORFF are known (see for instance WIDOER [14] and LORENTZ [8]). One of these alternate proofs makes use of the uniform approximation of functions continuous in [0, 1 ] by their Bernstein polynomials and is, to a large extent, due to HILDEBRANDT [5]. His proof is what is outlined by LORENTZ. An alternate set of necessary and sufficient conditions for the same representation has been given by RAMANUJAN [10] in his investigation on the quasi-Hausdorff methods. The results of HAUSDORrF and RAMANOJAN have been generalised by JAKIMOVSKI [6], [7]. Also, the uniform approximation through the Bernstein polynomials enabled LORENTZ to determine the solution of the moment problem in the function spaces of the K6the-Toeplitz type; for the same function spaces, alternate solutions of the moment problems of LORENTZ have been provided by RAMANUJAN [12] and his solutions make use of the uniform approximation of continuous functions in [0,1] by certain power series, a result demonstrated by MEYER-K~NIG and ZELLER [9]. Among the modifications of the moment problem, the one concerning 1 the representation of #, in the form It, = S ta" dz (t), where 0 = 2 0 < 21 < . . . < 2,-.. 0 and Y" 1/2n diverges and Z(t) is a function of bounded variation in [0, 1] is due to HAUSDORFF himself. For an account of this, see SHOHAT and TAMARKIN [13]. Recently ENDL [2] and JAKIMOVSrd [7] were led to consider the represen1 tation P,=S t'+~ dz(t), where e is real and Z(t) is, as before, a function of o bounded variation. Evidently this case is not covered by HAUSDOgFF'S results. They were led to this problem while considering the regularity conditions governing the summability matrix (H ~, p)= (h~,), where