Henstock-Kurzweil Fourier transforms

Talvila, Erik

doi:10.1215/ijm/1258138475

Cited by 33 publications

(27 citation statements)

References 17 publications

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“…The first inequality was proved in [28,Lemma 24] for the Henstock-Kurzweil integral and the same proof works here. The second inequality is similar.…”

Section: Integration By Parts Hölder's Inequalitysupporting

confidence: 61%

The Distributional Denjoy Integral

Talvila¹

2008

Real Analysis Exchange

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Let f be a distribution (generalised function) on the real line. If there is a continuous function F with real limits at infinity such that F ′ = f (distributional derivative) then the distributional integral of f is defined as. It is shown that this simple definition gives an integral that includes the Lebesgue and Henstock-Kurzweil integrals. The Alexiewicz norm leads to a Banach space of integrable distributions that is isometrically isomorphic to the space of continuous functions on the extended real line with uniform norm. The dual space is identified with the functions of bounded variation. Basic properties of integrals are established using elementary properties of distributions: integration by parts, Hölder inequality, change of variables, convergence theorems, Banach lattice structure, Hake theorem, Taylor theorem, second mean value theorem. Applications are made to the half plane Poisson integral and Laplace transform. The paper includes a short history of Denjoy's descriptive integral definitions. Distributional integrals in Euclidean spaces are discussed and a more general distributional integral that also integrates Radon measures is proposed. 2000 subject classification: 26A39, 46B42, 46E15, 46F05, 46G12

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“…The first inequality was proved in [28,Lemma 24] for the Henstock-Kurzweil integral and the same proof works here. The second inequality is similar.…”

Section: Integration By Parts Hölder's Inequalitysupporting

confidence: 61%

The Distributional Denjoy Integral

Talvila¹

2008

Real Analysis Exchange

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“…However, it appears earlier in [18]. It is proved for a more symmetric version of the Alexiewicz norm for Henstock-Kurzweil integrals in [19,Lemma 24].…”

Section: F ′′ Henstock-kurzweil Integrablementioning

confidence: 99%

Optimal error estimates for corrected trapezoidal rules

Talvila¹,

Wiersma²

2012

Journal of Mathematical Inequalities

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Abstract. Corrected trapezoidal rules are proved for b a f (x) dx under the assumption that f ∈ L p ([a,b]) for some 1 p ∞ . Such quadrature rules involve the trapezoidal rule modified by the addition of a term k[ f (a) − f (b)] . The coefficient k in the quadrature formula is found that minimizes the error estimates. It is shown that when f is merely assumed to be continuous then the optimal rule is the trapezoidal rule itself. In this case error estimates are in terms of the Alexiewicz norm. This includes the case when f is integrable in the Henstock-Kurzweil sense or as a distribution. All error estimates are shown to be sharp for the given assumptions on f . It is shown how to make these formulas exact for all cubic polynomials f . Composite formulas are computed for uniform partitions.Mathematics subject classification (2010): Primary 26D15, 41A55, 65D30. Secondary 26A39, 46F10.

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“…For these results see, for example, [4]. See [8] for related results with the Henstock-Kurzweil integral. Using the Alexiewicz norm, all of these results have generalisations to continuous primitive integrals that are proven below.…”

Section: Introduction and Notationmentioning

confidence: 99%

Convolutions with the Continuous Primitive Integral

Talvila

2009

Abstract and Applied Analysis

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IfFis a continuous function on the real line andf=F′is its distributional derivative, then the continuous primitive integral of distributionfis∫abf=F(b)−F(a). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolutionf∗g(x)=∫−∞∞f(x−y)g(y)dyforfan integrable distribution andga function of bounded variation or anL1function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. Forgof bounded variation,f∗gis uniformly continuous and we have the estimate‖f∗g‖∞≤‖f‖‖g‖ℬ𝒱, where‖f‖=supI|∫If|is the Alexiewicz norm. This supremum is taken over all intervalsI⊂ℝ. Wheng∈L1, the estimate is‖f∗g‖≤‖f‖‖g‖1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

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Henstock-Kurzweil Fourier transforms

Cited by 33 publications

References 17 publications

The Distributional Denjoy Integral

The Distributional Denjoy Integral

Optimal error estimates for corrected trapezoidal rules

Convolutions with the Continuous Primitive Integral

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