Untersuchungen über Fouriersche Reihen

Fejér, Leopold

doi:10.1007/bf01447779

Cited by 124 publications

(36 citation statements)

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“…The uniformity of the limit goes back to the original theorems of Fejer [2]. Now for the remainder of this note let/(x) be continuous in 0 ^ x ^ 1 except for at most N points at which it may have a finite jump.…”

mentioning

confidence: 91%

On the Convergence of a Solution of a Difference Equation to a Solution of the Equation of Diffusion

Juncosa¹,

Young²

1954

Proceedings of the American Mathematical Society

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mentioning

confidence: 91%

On the Convergence of a Solution of a Difference Equation to a Solution of the Equation of Diffusion

Juncosa¹,

Young²

1954

Proceedings of the American Mathematical Society

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“…It was proved by Lebesgue [17] that the Fejér means [5] of the trigonometric Fourier series of an integrable function converge almost everywhere to the function, i.e., [15] considered the so called strong summability and verified that the strong means 1 n + 1 n k=0 |s k f (x) − f (x)| q tend to 0 at each Lebesgue-point of f , as n → ∞, whenever f ∈ L p (T) (1 < p < ∞) and 0 < q < ∞ (for Fourier transforms see Giang and Móricz [10]). This result does not hold for p = 1 (see Hardy and Littlewood [16]).…”

Section: Introductionmentioning

confidence: 99%

Multi-dimensional Fourier Transforms, Lebesgue Points and Strong Summability

Weisz

2016

Mediterr. J. Math.

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A general summability method of multi-dimensional Fourier transforms, the so called θ-summability is investigated. Under some conditions on θ we show that the Marcinkiewicz-θ-means of a function f ∈ W (L1, ∞)(R d ) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of f ∈ W (Lp, ∞)(R d ), whenever 1 < p < ∞. As an application we generalize the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya for f ∈ W (L1, ∞)(R) and for strong θ-summability.Mathematics Subject Classification. Primary 42B08; Secondary 42A38, 42A24, 42B25.

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“…Richards [5] shows that the overshot tends to the typical 8.95% of shock magnitude as the degree of spline approximation approaches infinity. Almost all existing treatments for the Gibbs phenomenon reduction fall in the direction of summability (or averaging) methods, such as of Fejér [6] or Lanczos [7]. In particular, some suppression of oscillations can be achieved combining two or more periodic approximating functions having different phases.…”

Section: Suppression Of the Gibbs Phenomenonmentioning

confidence: 99%

Hopping numerical approximations of the hyperbolic equation

Tkalich

2007

Numerical Methods in Fluids

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SUMMARYPolynomial functions can be used to derive numerical schemes for an approximate solution of hyperbolic equations. A conventional derivation technique requires a polynomial to pass through every node values of a continuous computational stencil, leading to severe manifestation of the Gibbs phenomenon and strict time-step limitation. To overcome the problem, this paper introduces polynomials that skip regularly ('hop' over) one or more nodes from the computational grid. Polynomials hopping over odd and even nodes yield a series of explicit numerical schemes of a required accuracy, with Lax-Friedrichs method being a particular simplest case. The schemes have two times wider stability interval compared to conventional continuous-stencil explicit methods. Convex combinations of odd-and even-node-based updates improve further accuracy and stability of the method. Out of considered combinations (up to third-order accuracy), derived odd-order methods are stable for the Courant number ranging from 0 to 3, and even-order ones from 0 to 5. A 2-D extension of the hopping polynomial method exhibits similar properties.

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Untersuchungen über Fouriersche Reihen

Cited by 124 publications

References 1 publication

On the Convergence of a Solution of a Difference Equation to a Solution of the Equation of Diffusion

On the Convergence of a Solution of a Difference Equation to a Solution of the Equation of Diffusion

Multi-dimensional Fourier Transforms, Lebesgue Points and Strong Summability

Hopping numerical approximations of the hyperbolic equation

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