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The surprising almost everywhere convergence of Fourier–Neumann series

✍ Scribed by Óscar Ciaurri; Juan Luis Varona

Publisher
Elsevier Science
Year
2009
Tongue
English
Weight
340 KB
Volume
233
Category
Article
ISSN
0377-0427

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✦ Synopsis

a b s t r a c t

For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in L p requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L 2 is the celebrated Carleson theorem, proved in 1966 (and extended to L p by Hunt in 1967).

In this paper, we take the system

. . (with J µ being the Bessel function of the first kind and of the order µ), which is orthonormal in L 2 ((0, ∞), x 2α+1 dx), and whose Fourier series are the so-called Fourier-Neumann series. We study the almost everywhere convergence of Fourier-Neumann series for functions in L p ((0, ∞), x 2α+1 dx) and we show that, surprisingly, the proof is relatively simple (inasmuch as the mean convergence has already been established).

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✍ Terence Tao 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 124 KB

Let ψ be a rapidly decreasing one-dimensional wavelet. We show that the wavelet expansion of any L p function converges pointwise almost everywhere under the wavelet projection, hard sampling, and soft sampling summation methods, for 1 < p < ∞. In fact, the partial sums are uniformly dominated by th