The one-dimensional heat equation in the Alexiewicz norm

Talvila, Erik

doi:10.1515/apam-2014-0038

Cited by 2 publications

(2 citation statements)

References 15 publications

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“…This can occur, for example, in a convolution product. See [55] for an application on the real line.…”

Section: Convergence Theoremsmentioning

confidence: 99%

The continuous primitive integral in the plane

Talvila¹

2019

Preprint

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An integral is defined on the plane that includes the Henstock-Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions F (x, y) defined on the extended real plane [−∞, ∞] 2 that vanish when x or y is −∞. With usual pointwise operations this is a Banach space under the uniform norm. The integrable functions and distributions (generalised functions) are those that are the distributional derivative ∂ 2 /(∂x∂y) of this space of primitives.The definition then builds in the fundamental theorem of calculus. The Alexiewicz norm is f = F ∞ where F is the unique primitive of f . The space of integrable distributions is then a separable Banach space isometrically isomorphic to the space of primitives. The space of integrable distributions is the completion of both L 1 and the space of Henstock-Kurzweil integrable functions. The Banach lattice and Banach algebra structures of the continuous functions in • ∞ are also inherited by the integrable distributions. It is shown that the dual space are the functions of bounded Hardy-Krause variation. Various tools that make these integrals useful in applications are proved: integration by parts, Hölder inequality, second mean value theorem, Fubini theorem, a convergence theorem, change of variables, convolution. The changes necessary to define the integral in R n are sketched out.

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“…This can occur, for example, in a convolution product. See [55] for an application on the real line.…”

Section: Convergence Theoremsmentioning

confidence: 99%

The continuous primitive integral in the plane

Talvila¹

2019

Preprint

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“…For initial data in the Sobolev space H s , see [12]. Another paper studying the heat equation with distributions in Banach spaces is [17], in which the initial data is taken to be the distributional derivative of a continuous function.…”

Section: Introductionmentioning

confidence: 99%

The L p primitive integral

Talvila

2014

Mathematica Slovaca

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For each 1 ≤ p < ∞, a space of integrable Schwartz distributions L′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 ∈ L′p such that f is the distributional derivative of F ∈ L p, then the integral is defined as $\int\limits_{ - \infty }^\infty {fG} = - \int\limits_{ - \infty }^\infty {F(x)g(x)dx} $, where g ∈ L q, $G(x) = \int\limits_0^x {g(t)dt} $ and 1/p + 1/q =1. A norm is ‖f‖p′ = ‖F‖p. The spaces L′p and L p are isometrically isomorphic. Distributions in L′p share many properties with functions in L p. Hence, L′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′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.

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The one-dimensional heat equation in the Alexiewicz norm

Cited by 2 publications

References 15 publications

The continuous primitive integral in the plane

The continuous primitive integral in the plane

The L p primitive integral

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