GENERALISASI KLAS MONOTON BARISAN-BARISAN BILANGAN DAN FUNGSI DARI KLAS MONOTON UMUM; GENERALIZATION OF MONOTONE CLASS OF NUMBER SEQUENCES AND FUNCTION FROM CLASS GENERAL MONOTONE

Imron, Moch Aruman, Rini Indrati

2015 | Tesis | FMIPA

At first the Fourier series was introduced by Joseph Fourier (1768-1830) to solve the problem of heat equation on a metal plate . Recently the results of this problem were developed and discussed about the monotonicity of the Fourier series coefficients are decreasing monotone and converges to zero because it is one sufficient condition that the series converges . Sequences of sinus Fourier series coefficients with decreasing monotone properties and converge to zero are called class of MS (monotone sequences) . Furthermore, class of MS is developed into a class of general monotone and finally into a class that is called supremum bounded variation sequences. In this paper, we discuss a generalization of monotone class into a class of supremum bounded p-variation sequences. First, it will be discussed a construction of a class of supremum bounded p-variation sequences and generalization of a difference sequence from that class. Futhermore, considering the class studied is difference sequence which is related to class of bounded variation sequence. Class of sequence o which is bounded variation and decreasing monotone, is guaranteed that Fourier series converge, then it is necessary to study on bonded variation of the class. It is proved that the class of sequence is a Banach space to certain norm, so that the sinus Fourier series is proved converges uniformly. It can be shown that class of supremum bounded p-variation sequences is more general than class of General Monotone and class of Supremum Bounded Variation Sequences. Since, the class of supremum bounded variation sequences has developed into a class of supremum bounded variation double sequences, we also discuss the construction of a class of supremum bounded p-variation double sequences and generalization of difference sequence from that class. Class of general monotone has been developed into the class of general monotone functions, before monotone class is developed into class of supremum bounded variation sequences. Now, we also discuss the construction class of supremum bounded p-variation functions, its properties and sufficient condition of that class, so that the sine integral converges uniformly. Furthermore, we discuss some application of the classes of supremum bounded p-variation. For the class of supremum bounded p-variation sequences, we apply it to the approximation problem and for a class of supremum bounded p-variation functions, we apply it to integrality of Fourier transform near the zero .

Kata Kunci : general monotone; supremum bounded p-variation; Banach space; approximation problem; Fourier transform