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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Titchmarsh convolution theorem ： ウィキペディア英語版 Titchmarsh convolution theorem The Titchmarsh convolution theorem is named after Edward Charles Titchmarsh, a British mathematician. The theorem describes the properties of the support of the convolution of two functions. == Titchmarsh convolution theorem == E.C. Titchmarsh proved the following theorem in 1926: :If $\backslash phi\backslash ,(t)$ and $\backslash psi(t)\backslash ,$ are integrable functions, such that ::$\backslash int\_^\backslash phi(t)\backslash psi(x-t)\backslash ,dt=0$ :almost everywhere in the interval This result, known as the Titchmarsh convolution theorem, could be restated in the following form: :Let $\backslash phi,\backslash ,\backslash psi\backslash in\; L^1(\backslash mathbb)$. Then $\backslash inf\backslash mathop\backslash ,\backslash phi\backslash ast\; \backslash psi$ =\inf\mathop\,\phi+\inf\mathop\,\psi if the right-hand side is finite. :Similarly, $\backslash sup\backslash mathop\backslash ,\backslash phi\backslash ast\backslash psi=\backslash sup\backslash mathop\backslash ,\backslash phi+\backslash sup\backslash mathop\backslash ,\backslash psi$ if the right-hand side is finite. This theorem essentially states that the well-known inclusion :$$ \,\phi\ast \psi \subset \mathop\,\phi +\mathop\,\psi is sharp at the boundary. The higher-dimensional generalization in terms of the convex hull of the supports was proved by J.-L. Lions in 1951: : ''If $\backslash phi,\backslash ,\backslash psi\backslash in\backslash mathcal\text{'}(\backslash mathbb^n)$, then $\backslash mathop\backslash mathop\backslash ,\backslash phi\backslash ast\; \backslash psi=\backslash mathop\backslash mathop\backslash ,\backslash phi+\backslash mathop\backslash mathop\backslash ,\backslash psi.$'' Above, $\backslash mathop$ denotes the convex hull of the set. $\backslash mathcal\text{'}(\backslash mathbb^n)$ denotes the space of distributions with compact support. The theorem lacks an elementary proof. The original proof by Titchmarsh is based on the Phragmén–Lindelöf principle, Jensen's inequality, Theorem of Carleman, and Theorem of Valiron. More proofs are contained in (Theorem 4.3.3 ) (harmonic analysis style), (Chapter VI ) (real analysis style), and (Lecture 16 ) (complex analysis style). 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「Titchmarsh convolution theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.