It is shown that the integral Hausdorff mean Tp of the Fourier-Stieltjes transform of a measure on the real line is the Fourier transform of an L1 function if and only if Tp. vanishes at infinity or the kernel of T has mean value zero. Also a sufficient condition on the kernel of T and a necessary and sufficient condition on the measure is established in order for-i sï%r{x)T p(x) to be the Fourier transform of an L1-function. These results yield an improvement of Fejer's and Wiener's formulas for the inversion of Fourier-Stieltjes transforms, the uniqueness property of certain generalized Fourier transforms, and a generalization of the mean ergodic theorem for unitary operators. Let Af (/?) be the space of complex bounded regular Borel measures on the real line R and ßix) be the Fourier-Stieltjes transform of a measure p in MiR), pix)= [ e~'xt dpit), xeR. Jr We consider the integral Hausdorff mean Tp generated by a Borel measurable kernel ip in LliR), which is defined for x e R by Tpix)= / fiixs)y/is)ds =-/ tp (?-) piy)dy, Jr \x\ Jr xx/

Abstract.It is shown that the integral Hausdorff mean Tp of the FourierStieltjes transform of a measure on the real line is the Fourier transform of an L1 function if and only if Tp. vanishes at infinity or the kernel of T has mean value zero. Also a sufficient condition on the kernel of T and a necessary and sufficient condition on the measure is established in order for -i sï%r{x)T p(x) to be the Fourier transform of an L1-function. These results yield an improvement of Fejer's and Wiener's formulas for the inversion of Fourier-Stieltjes transforms, the uniqueness property of certain generalized Fourier transforms, and a generalization of the mean ergodic theorem for unitary operators.

Let {a"}^L0 be the cosine Fourier-Stieltjescoefficients of the Bore! measure /i and {a0, (a, +-• •+û«)/«i"=i = {(7a)"}^=0 be the sequence of their arithmetic means. Then ]>,íí=o (Tá)n cos nx is a Fourier-Stieltjes series. Moreover, (a) ^^=o (Ta)n cos nx is a Fourier series if and only if (7o)"->-0 at infinity 01, equivalently, the measure n is continuous at the origin, (b) 2™_i (Ta)" sin nx is a Fourier series if and only if the function x~'/i\iO. x)) is in ¿'[0,77]. These results form the best possible analogue of a theorem of G. Goes, concerning arithmetic means of Fourier-Stieltjes sine coefficients, and improve considerably the theorems of L. Fejér and N. Wiener on the inversion and quadratic variation of Fourier-Stieltjes coefficients. G. H. Hardy first studied in [6] the transformation T. He showed that if 2«-o an cos nx is the Fourier series of a function in Lp[0, tt], then so is the series 2n-o (T~a)n cos nx for l^/>

be the sequence-to-function Hausdorff transformation generated by the completely monotone function g or, what is equivalent, the Laplace transform of a finite positive measure -1, whose norm ||T|| = /o°° t-(i+«)/P do{t) = C{p,s) if and only if C(p, s) < oo, and that for 1 < p < oo, ||To||Pl4 < C(p, s)||a||p,s unless an is a null sequence.Furthermore, if 1 < p < r < oo, 0 < X < 1, 1/r = 1/p + A -1 and o is absolutely continuous with derivatives ip such that the function ipr(t) = t~l'ril>(t) belongs to L'/^JO, oo), then the transformation (T),a)(y) = y1~x(Ta)(y) is bounded from lp to Lr[0, oo) and has norm ||7\|| < ||^r IIi/a■ The transformation T includes in particular the Borel transform and that of generalized Abel means. These results constitute an improved analogue of a theorem of Hardy concerning the discrete Hausdorff transformation on V which corresponds to a totally monotone sequence, and lead to improved forms of some inequalities of Hardy and Littlewood for power series and moment sequences.Hardy showed that the discrete Hausdorff transformation (Ha)n = J2 UJ(An~V*K, Kp< oo,where pn is a totally monotone sequence or, equivalently, pn -fQ tn da(t), where a is a probability measure on [0,1], such that

Abstract. We show that the characteristic function of a unimodal probability distribution function can be inverted by the Fourier transform a.e. if and only if the distribution is absolutely continuous. The result complements Khintchine's criterion for unimodal distributions.The distribution function F (x) = Pr(X ≤ x) of a random variable X is unimodal with vertex at the origin if F (x) is convex for x < 0 and concave for x > 0. Some examples of unimodal distributions are: i) the standard normal, exponential and Cauchy probability densities; ii) the class of stable distributions; iii) the class of symmetric infinitely divisible distributions. According to Khintchine's criterion ( for 0 < x ≤ t < ∞, −t −1 for − ∞ < t ≤ x < 0, 0 otherwise.