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In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform. Marshall H. Stone considerably generalized the theorem and simplified the proof . His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval , an arbitrary compact Hausdorff space is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of is investigated. The Stone–Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space. Further, there is a generalization of the Stone–Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone–Weierstrass theorem and described below. A different generalization of Weierstrass' original theorem is Mergelyan's theorem, which generalizes it to functions defined on certain subsets of the complex plane. == Weierstrass approximation theorem == The statement of the approximation theorem as originally discovered by Weierstrass is as follows: :Weierstrass Approximation Theorem. Suppose is a continuous real-valued function defined on the real interval . For every , there exists a polynomial such that for all in , we have , or equivalently, the supremum norm . A constructive proof of this theorem using Bernstein polynomials is outlined on that page. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stone–Weierstrass theorem」の詳細全文を読む スポンサード リンク
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