Random Dirichlet functions, multipliers and smoothness,


W. George Cochran, Joel H. Shapiro,

and David C. Ullrich


Canadian J. Math. 45 (1993) 255--268


Abstract: We show that if one inserts random plus and minus signs before the coefficients of the power series of a function in the Dirichlet space, then the resulting series is almost surely a multiplier  of the Dirichlet space. This parallels a known result for lacunary power series, which also has a version for smoothness classes: every lacunary Dirichlet series lies in the Lipschitz class Lip_{1/2} of functions obeying a Lipschitz condition with exponent 1/2. However, unlike the lacunary situation, no corresponding ``almost sure'' Lipschitz result is possible for random series: we exhibit a Dirichlet function with no randomization in Lip_{1/2}. We complement this result with a ``best possible'' sufficient condition for randomizations to belong almost surely to Lip_{1/2}. Versions of our results hold for weighted Dirichlet spaces, and much of our work is carried out in this more general setting.


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