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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 Singular integral operators on closed curves ： ウィキペディア英語版 Singular integral operators on closed curves In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a simple algebraic formula. The Hilbert transform is an involution and the Cauchy transform an idempotent. The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan curve. The theory for the original curve can be deduced from that on the unit circle, where, because of rotational symmetry, both operators are classical singular integral operators of convolution type. The Hilbert transform satisfies the jump relations of Plemelj and Sokhotski, which express the original function as the difference between the boundary values of holomorphic functions on the region and its complement. Singular integral operators have been studied on various classes of functions, including Hőlder spaces, L''p'' spaces and Sobolev spaces. In the case of L2 spaces—the case treated in detail below—other operators associated with the closed curve, such as the Szegő projection onto Hardy space and the Neumann–Poincaré operator, can be expressed in terms of the Cauchy transform and its adjoint. ==Operators on the unit circle== (詳細はHardy space H2(T) consists of the functions for which the negative coefficients vanish, ''a''''n'' = 0 for ''n'' < 0. These are precisely the square-integrable functions that arise as boundary values of holomorphic functions in the unit disk |''z''| < 1. Indeed, ''f'' is the boundary value of the function :$\backslash displaystyle$ in the sense that the functions :$\backslash displaystyle,$ defined by the restriction of ''F'' to the concentric circles |''z''| = ''r'', satisfy :$\backslash displaystyle$ The orthogonal projection ''P'' of L2(T) onto H2(T) is called the Szegő projection. It is a bounded operator on L2(T) with operator norm 1. By Cauchy's theorem :$\backslash displaystyle\; \backslash ,d\backslash zeta=\; \backslash int\_^\; \backslash ,\; d\backslash theta.\}$ Thus :$\backslash displaystyle\; \backslash int\_^$ When ''r'' equals 1, the integrand on the right hand side has a singularity at θ = 0. The truncated Hilbert transform is defined by :$\backslash displaystyle\; \backslash int\_$ where δ = |1 – ''e''''i''ε|. Since it is defined as convolution with a bounded function, it is a bounded operator on L2(T). Now :$\backslash displaystyle\backslash int\_\backslash varepsilon^\backslash pi\; 2\; \backslash Re\; (1-e^)^\; \backslash ,\; d\backslash theta\; =\backslash int\_\backslash varepsilon^\backslash pi\; 1\; \backslash ,\; d\backslash theta\; =\; i\; -\; .\}$ If ''f'' is a polynomial in ''z'' then :$\backslash displaystyle\; \backslash int\_\; \backslash ,\; d\backslash zeta.\}$ By Cauchy's theorem the right hand side tends to 0 uniformly as ε, and hence δ, tends to 0. So :$\backslash displaystyle$ uniformly for polynomials. On the other hand, if ''u''(''z'') = ''z'' it is immediate that :$\backslash displaystyle\; H^\backslash varepsilon(\; u\; \backslash overline).\}$ Thus if ''f'' is a polynomial in ''z''−1 without constant term :$\backslash displaystyle$ uniformly. Define the Hilbert transform on the circle by :$\backslash displaystyle$ Thus if ''f'' is a trigonometric polynomial :$\backslash displaystyle$ uniformly. It follows that if ''f'' is any L2 function :$\backslash displaystyle$ in the L2 norm. This is a consequence of the result for trigonometric polynomials since the ''H''ε are uniformly bounded in operator norm: indeed their Fourier coefficients are uniformly bounded. It also follows that, for a continuous function ''f'' on the circle, ''H''ε''f'' converges uniformly to ''Hf'', so in particular pointwise. The pointwise limit is a Cauchy principal value, written :$\backslash displaystyle\; \backslash int$ The Hilbert transform has a natural compatibility with orientation-preserving diffeomorphisms of the circle.〔See: * * *〕 Thus if ''H'' is a diffeomorphism of the circle with :$\backslash displaystyle,\backslash ,\backslash ,\backslash ,\; h(\backslash theta+2\backslash pi)=h(\backslash theta)+2\backslash pi,\}$ then the operators :$\backslash displaystyle\backslash int\_|\backslash ge\; \backslash varepsilon\}$ -e^} \, e^\, d\theta,} are uniformly bounded and tend in the strong operator topology to ''H''. Moreover, if ''Vf''(''z'') = ''f''(''H''(''z'')), then ''VHV''−1 – ''H'' is an operator with smooth kernel, so a Hilbert–Schmidt operator. 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「Singular integral operators on closed curves」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.