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A note on rearrangements of Fourier coefficients

Hugh L. Montgomery (1976)

Annales de l'institut Fourier

Let f ( x ) ∼ Σ a n e 2 π i n x , f * ( x ) ∼ ∑ n = 0 ∞ a * n cos 2 π n x , where the a * n are the numbers | a n | rearranged so that a n * ↘ 0 . Then for any convex increasing ψ , ∥ ψ ( | f | 2 ∥ 1 ≤ ∥ ψ ( 20 | f * | 2 ∥ 1 . The special case ψ ( t ) = t q / 2 , q ≥ 2 , gives ∥ f ∥ q ≤ 5 ∥ f * ∥ q an equivalent of Littlewood.

A note on some expansions of p-adic functions

Grzegorz Szkibiel (1992)

Acta Arithmetica

Introduction. Recently J. Rutkowski (see [3]) has defined the p-adic analogue of the Walsh system, which we shall denote by ( ϕ ₘ ) m ∈ ℕ ₀ . The system ( ϕ ₘ ) m ∈ ℕ ₀ is defined in the space C(ℤₚ,ℂₚ) of ℂₚ-valued continuous functions on ℤₚ. J. Rutkowski has also considered some questions concerning expansions of functions from C(ℤₚ,ℂₚ) with respect to ( ϕ ₘ ) m ∈ ℕ ₀ . This paper is a remark to Rutkowski’s paper. We define another system ( h ₙ ) n ∈ ℕ ₀ in C(ℤₚ,ℂₚ), investigate its properties and compare it to the system defined by Rutkowski. The system...

A note on the convolution theorem for the Fourier transform

Charles S. Kahane (2011)

Czechoslovak Mathematical Journal

In this paper we characterize those bounded linear transformations T f carrying L 1 ( ℝ 1 ) into the space of bounded continuous functions on ℝ 1 , for which the convolution identity T ( f * g ) = T f · T g holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.

A note on the strong maximal operator on ℝⁿ

Jiecheng Chen, Xiangrong Zhu (2004)

Studia Mathematica

We prove that for f ∈ L ln⁺L(ℝⁿ) with compact support, there is a g ∈ L ln⁺L(ℝⁿ) such that (a) g and f are equidistributed, (b) M S ( g ) ∈ L ¹ ( E ) for any measurable set E of finite measure.

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