OSCILLATORY PROPERTY AND DIMENSIONS OF RADEMACHER SERIES

摘要

Let Sigma(infinity)(i=1) c(i)R(i)(x) be the Rademacher series, where {R-i(x)}(i=1)(infinity) is the classical Rademacher function system and {c(i)}(i=1)(infinity) is an arbitrary real number sequence. In this paper, we first show that the value range of the Rademacher series at any subinterval of [0, 1] is R boolean OR {+/- 8} when {c(i)}(1)(infinity) is an element of l(2)\l(1). This result provides us with the basic facts that when {c(i)}(1)(infinity) is an element of l(2)\l(1), the Rademacher series cannot converge to an approximate continuous function, and there is no approximate limit at any point of [0,1]. Further, when {c(i)}(1)(infinity) is an element of l(2)\l(1), we show various dimensions of the level set of Rademacher series on any subinterval of [0, 1]. Finally, we give the relationship between the box dimension and the coefficient of Rademacher series when {ci}(1)(infinity) is an element of l(1), and the exact values of box dimension, packing dimension and Hausdorff dimension are obtained in some special cases.

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