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The $Lp$ primitive integral

In: Mathematica Slovaca, vol. 64, no. 6
Erik Talvila
Detaily:
Rok, strany: 2014, 1497 - 1524
Kľúčové slová:
Lebesgue space, Banach space, Schwartz distribution, generalised function, primitive, integral, Fourier transform, convolution, Banach lattice, Hilbert space, Poisson integral
O článku:
For each $1\leq p<\infty$, a space of integrable Schwartz distributions $L^{\prime p}$, is defined by taking the distributional derivative of all functions in $L^p$. Here, $L^p$ is with respect to Lebesgue measure on the real line. If $f\in L^{\prime p}$ such that $f$ is the distributional derivative of $F\in L^p$, then the integral is defined as $\int^\infty_{-\infty} fG =-\int^\infty_{-\infty} F(x)g(x) dx$, where $g\in L^q$, $G(x)=\int_0^x g(t) dt$ and $1/p+1/q=1$. A norm is $\lVert f\rVert'_p=\lVert F\rVert_p$. The spaces $L^{\prime p}$ and $L^p$ are isometrically isomorphic. Distributions in $L^{\prime p}$ share many properties with functions in $L^p$. Hence, $L^{\prime p}$ is reflexive, its dual space is identified with $L^q$, there is a type of Hölder inequality, continuity in norm, convergence theorems, Gateaux derivative. It is a Banach lattice and abstract $L$-space. Convolutions and Fourier transforms are defined. Convolution with the Poisson kernel is well defined and provides a solution to the half plane Dirichlet problem, boundary values being taken on in the new norm. A product is defined that makes $L^{\prime 1}$ into a Banach algebra isometrically isomorphic to the convolution algebra on $L^1$. Spaces of higher order derivatives of $L^p$ functions are defined. These are also Banach spaces isometrically isomorphic to $L^p$.
Ako citovať:
ISO 690:
Talvila, E. 2014. The $Lp$ primitive integral. In Mathematica Slovaca, vol. 64, no.6, pp. 1497-1524. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0288-5

APA:
Talvila, E. (2014). The $Lp$ primitive integral. Mathematica Slovaca, 64(6), 1497-1524. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0288-5
O vydaní: