### 130 let divergentních trigonometrických Fourierových řad (2. část)

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We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${\psi (\xb7-n)}_{n=-\infty}^{\infty}$, which is equivalent to the weight function ${\sum}_{n=-\infty}^{\infty}\left|\psi \u0302(\xb7+n)\right|\xb2$ being > 0 a.e.

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 $r>\mathrm{\U0001d696\U0001d68a\U0001d6a1}\{{p}^{\text{'}},2\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

Four theorems of Ahmad [1] on absolute Nörlund summability factors of power series and Fourier series are proved under weaker conditions.

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.

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.