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OSCILLATORY PROPERTY AND DIMENSIONS OF RADEMACHER SERIES
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Let [Formula: see text] be the Rademacher series, where [Formula: see text] is the classical Rademacher function system and [Formula: see text] is an arbitrary real number sequence. In this paper, we first show that the value range of the Rademacher series at any subinterval of [Formula: see text] is [Formula: see text] when [Formula: see text]. This result provides us with the basic facts that when [Formula: see text], the Rademacher series cannot converge to an approximate continuous function, and there is no approximate limit at any point of [Formula: see text]. Further, when [Formula: see text], we show various dimensions of the level set of Rademacher series on any subinterval of [Formula: see text]. Finally, we give the relationship between the box dimension and the coefficient of Rademacher series when [Formula: see text], and the exact values of box dimension, packing dimension and Hausdorff dimension are obtained in some special cases.
Title: OSCILLATORY PROPERTY AND DIMENSIONS OF RADEMACHER SERIES
Description:
Let [Formula: see text] be the Rademacher series, where [Formula: see text] is the classical Rademacher function system and [Formula: see text] is an arbitrary real number sequence.
In this paper, we first show that the value range of the Rademacher series at any subinterval of [Formula: see text] is [Formula: see text] when [Formula: see text].
This result provides us with the basic facts that when [Formula: see text], the Rademacher series cannot converge to an approximate continuous function, and there is no approximate limit at any point of [Formula: see text].
Further, when [Formula: see text], we show various dimensions of the level set of Rademacher series on any subinterval of [Formula: see text].
Finally, we give the relationship between the box dimension and the coefficient of Rademacher series when [Formula: see text], and the exact values of box dimension, packing dimension and Hausdorff dimension are obtained in some special cases.
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