The structure of regulated functions
The structure of regulated functions
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监管职能的结构
DOI：
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发表时间：
1976
期刊：
影响因子：
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通讯作者：
D. Waterman
中科院分区：
文献类型：
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作者：
C. Goffman;G. Moran;D. Waterman
It is shown that there is a nontrivial class of regulated functions each of which is a representable as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function. The set of regulated functions is the union of the classes of functions of bounded -variation for convex $. The regulated functions on a closed interval are those functions whose right and left limits exist at each point. Every regulated function is bounded, has a countable set of discontinuities, and is the limit of a uniformly convergent sequence of step functions. The regulated functions are of importance in the theory of stochastic processes and in the theory of everywhere convergence of Fourier series. Functions of bounded variation are regulated, as are the functions of bounded 4>-variation (1) and the functions of bounded A-variation (3). In §1 we shall show that each regulated function is of bounded -variation for some 4>. A function of bounded variation has a canonical representation as the sum of a continuous function of bounded variation and a sum of jump functions, the jumps being those of the given function of bounded variation. In §2 we shall investigate the possibility that a regulated function have a representation as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function. We shall see that, although we can characterize a nontrivial class of functions for which this representation is possible, we can construct functions which have no such representation. 1. Let (0) = 0, $(x) > 0 for x > 0. A function / defined on an interval / is said to be of bounded $- variation ( - BV) if the «^-variation of/,
It is shown that there is a nontrivial class of regulated functions each of which is a representable as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function. The set of regulated functions is the union of the classes of functions of bounded -variation for convex $. The regulated functions on a closed interval are those functions whose right and left limits exist at each point. Every regulated function is bounded, has a countable set of discontinuities, and is the limit of a uniformly convergent sequence of step functions. The regulated functions are of importance in the theory of stochastic processes and in the theory of everywhere convergence of Fourier series. Functions of bounded variation are regulated, as are the functions of bounded 4>-variation (1) and the functions of bounded A-variation (3). In §1 we shall show that each regulated function is of bounded -variation for some 4>. A function of bounded variation has a canonical representation as the sum of a continuous function of bounded variation and a sum of jump functions, the jumps being those of the given function of bounded variation. In §2 we shall investigate the possibility that a regulated function have a representation as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function. We shall see that, although we can characterize a nontrivial class of functions for which this representation is possible, we can construct functions which have no such representation. 1. Let (0) = 0, $(x) > 0 for x > 0. A function / defined on an interval / is said to be of bounded $- variation ( - BV) if the «^-variation of/,