NURMA WADDAH FITRIA Primary Author mixed material bibliography Banda Aceh Universitas Syiah Kuala 2018 id Indonesia

Null

ABSTRAK Barisan fungsi memiliki dua jenis kekonvergenan yaitu kekonvergenan titik demi titik dan kekonvergenan seragam. Penelitian ini menyelidiki sifat-sifat kekonvergenan titik demi titik pada barisan fungsi kontinu di C[a; b] berdasarkan sifat-sifat yang berlaku untuk kekonvergenan seragam. Berdasarkan hasil penelitian, sifatsifat kekonvergenan barisan fungsi kontinu di C[a; b] yaitu sifat ketunggalan limit barisan fungsi, sifat operasi barisan fungsi, dan kriteria kekonvergenan Cauchy terpenuhi pada kekonvergenan titik demi titik. Namun sifat keterbatasan limit barisan fungsi tidak terpenuhi pada kekonvergenan titik demi titik. Terkait dengan fungsi kontinu di C[a; b], terdapat sebuah teorema mengenai aproksimasi fungsi-fungsi kontinu di C[a; b] dengan suatu polinomial yaitu Teorema Aproksimasi Weierstrass. Tulisan ini membahas pembuktian Teorema Aproksimasi Weierstrass yang dilakukan secara konstruktif dengan menggunakan Polinomial Bernstein. Selain itu juga diberikan sebuah ilustrasi pendekatan Polinomial Bernstein untuk fungsi kontinu f(x) = e untuk x 2 [0; 1]. Kata Kunci: Konvergen Titik Demi Titik, Fungsi Kontinu di C[a; b], Teorema Aprok- x simasi Weierstrass, Polinomial Bernstein ABSTRACT There are two types of convergence for the sequences of functions which are pointwise and uniform convergence. The purpose of this study is to investigate the properties of pointwise convergence for the sequences of continuous functions in C[a; b]; based on the uniform convergences properties. Based on this study, the convergences properties of sequences for continuous functions in C[a; b] are the uniqueness of the limit of functions, the nature of the operation of the function sequences, and the Cauchy Convergence Criteria are met at a pointwise convergence. But the limitations nature of the function sequence limit is not satis?ed at the pointwise convergence. Associated with the continuous function in C[a; b], there is a theorem about approximating continuous functions in C[a; b] with a polynomial that is Weierstrass Approximation Theorem. This paper discusses the constructive proof of Weierstrass Approximation Theorem by using Bernstein Polynomial. The illustration of Bernstein’s Polynomial Approach for continuous function f(x) = e for x 2 [0; 1] will also given in this paper. Keywords: Pointwise Corvergence, Continuous functions in C[a; b], Weierstrass Approximation Theorem, Bernstein’s Polynomial x 0 ELECTRONIC THESES AND DISSERTATION Universitas Syiah Kuala 51341 2018-12-18 15:24:37 2018-12-18 15:43:14 machine generated