Fourier Series with the Continuous Primitive Integral

Talvila, Erik

doi:10.1007/s00041-011-9183-4

Cited by 2 publications

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“…The integral was applied to Fourier series [53] and a type of Salem-Zygmund-Rudin-Cohen factorization was proved there. See also [43].…”

Section: Introductionmentioning

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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“…The integral was applied to Fourier series [53] and a type of Salem-Zygmund-Rudin-Cohen factorization was proved there. See also [43].…”

Section: Introductionmentioning

confidence: 99%