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A note on integer translates of a square integrable function on ℝ

Maciej Paluszyński (2010)

Colloquium Mathematicae

We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ψ ( · - n ) n = - , which is equivalent to the weight function n = - | ψ ̂ ( · + n ) | ² being > 0 a.e.

A variation norm Carleson theorem

Richard Oberlin, Andreas Seeger, Terence Tao, Christoph Thiele, James Wright (2012)

Journal of the European Mathematical Society

We strengthen the Carleson-Hunt theorem by proving L p estimates for the r -variation of the partial sum operators for Fourier series and integrals, for r > 𝚖𝚊𝚡 { p ' , 2 } . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

Accelerating the convergence of trigonometric series

Anry Nersessian, Arnak Poghosyan (2006)

Open Mathematics

A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed.

Algebrability of the set of non-convergent Fourier series

Richard M. Aron, David Pérez-García, Juan B. Seoane-Sepúlveda (2006)

Studia Mathematica

We show that, given a set E ⊂ 𝕋 of measure zero, the set of continuous functions whose Fourier series expansion is divergent at any point t ∈ E is dense-algebrable, i.e. there exists an infinite-dimensional, infinitely generated dense subalgebra of 𝓒(𝕋) every non-zero element of which has a Fourier series expansion divergent in E.

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