On the Smoothness Properties of a Family of Bernoulli Convolutions
نویسنده
چکیده
Let L (u, a), oo < u < + oo denote the Fourier-Stieltjes transform, 00 f eluoda(x), of a distribution function u(x), oo < x < + co . Thus if 00 /3(x) is the distribution function which is 0, J, 1 according as x -1 < x ,< 1, 1 < x, then L (u,,8) = cos u ; and so, if b is a positive constant, cos (u/b) is the transform of the distribution function /3(bx) . Hence, if a is a positive constant, the infinite convolution aa(x) =/3(ax) *,R (a 2x) * f (a8x) * . . . is convergent if and only if a > 1 ; its Fourier-Stieltjes transform being 00 (1) L(u,oa) = II cos (u/a'v), (a > 1) . n=1
منابع مشابه
Smoothness of Projections, Bernoulli Convolutions, and the Dimension of Exceptions
YUVAL PERES and WILHELM SCHLAG
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