Kernel function

Japanese: 核関数 - かくかんすう（英語表記）kernel function

The theory of orthogonal functions, which originated from the trigonometric functions that form the basis of Fourier series, has been developed for a long time, supported by eigenfunctions of various linear differential equations that are important in practical applications. The theory of orthogonal functions was applied to complex function theory by G. Szegö, and the concept of kernel function was first established by S. Bergman in his study of multivariate complex functions. Generally speaking, in a Hilbert space H consisting of holomorphic functions in an n -dimensional complex domain D ( n ≧ 1, if n = 1, it is the domain of the complex plane ), when a linear functional H ∋ f → f (ζ)∈ C is bounded for each point ζ∈ D , a K _ζ ∈ H that satisfies f (ζ)＝( f , K _ζ ) is determined, and this K _ζ is called a kernel function.

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

フーリエ級数の基底をなす三角関数系をモデルとして起こった直交関数系の理論は，応用上重要な種々の線形微分方程式の固有関数系に裏づけられながら，古くから展開されてきている。直交関数系の理論をとくに複素関数論に応用したセゲーG.Szegöの研究に端を発し，ベルクマンS.Bergmanが多変数複素関数の研究において，はじめて核関数の概念を確立した。　一般的にいうと，n次元複素領域D(n≧1,n＝1ならば複素平面の領域)における正則関数からなるヒルベルト空間Hで，点ζ∈Dごとに線形汎関数H∋f→f(ζ)∈Cが有界であるとき，f(ζ)＝(f,K_ζ)を満たすK_ζ∈Hが決まるが，このK_ζを核関数という。

出典　株式会社平凡社世界大百科事典第２版について　情報

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