Inspired by work of Montgomery on Fourier series and Donoho-Strak in signal processing, we investigate two families of rearrangement inequalities for the Fourier transform. More precisely, we show that the $L^2$ behavior of a Fourier transform of a function over a small set is controlled by the $L^2$ behavior of the Fourier transform of its symmetric decreasing rearrangement. In the $L^1$ case, the same is true if we further assume that the function has a support of finite measure. As a byproduct, we also give a simple proof and an extension of a result of Lieb about the smoothness of a rearrangement. Finally, a straightforward application to solutions of the free Shrödinger equation is given.
We study inequalities of the form \begin{equation*} ρ( \lvert \hat{f} \rvert) \leq C σ(f) < \infty, \end{equation*} with $f \in L_{1}(\mathbb{R}^n)$, the Lebesgue-integrable functions on $\mathbb{R}^n$ and \begin{equation*} \hat{f}(ξ) := \int_{\mathbb{R}^n} f(x) \, e^{- 2 πi ξ\cdot x} dx, \ \ \ ξ\in \mathbb{R}^n. \end{equation*} The functionals $ρ$ and $σ$ are so-called rearrangement-invariant (r.i.) norms on $M_{+}(\mathbb{R}^n)$, the nonnegative measurable functions on $\mathbb{R}^n$. Results first proved in the general context of r.i. spaces are then both specialized and expanded on in the …
The paper is related to the following question of P.~L.~Ul'yanov: is it true that for any $2π$-periodic continuous function $f$ there is a uniformly convergent rearrangement of its trigonometric Fourier series? In particular, we give an affirmative answer if the absolute values of Fourier coefficients of $f$ decrease. Also, we study a problem how to choose $m$ terms of a trigonometric polynomial of degree $n$ to make the uniform norm of their sum as small as possible.
We extend an estimate of Taibleson and Weiss, regarding Fourier transform of Hardy spaces, to the aniostropic setting. As consequences, we obtain necessary conditions for multiplier operators, and the anisotropic version of the Hardy-Littlewood inequality. This last inequality is strengthened with a rearrangement argument.
We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-Δ)^s$ with $s> 0$ and, in particular, any polyharmonic operator $(-Δ)^m$ with integer $m \geq 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for: i) Gagliardo-Nirenberg inequalities with derivatives of…
This article studies the rearrangement problem for Fourier series introduced by P.L. Ulyanov, who conjectured that every continuous function on the torus admits a rearrangement of its Fourier coefficients such that the rearranged partial sums of the Fourier series converge uniformly to the function. The main theorem here gives several new equivalences to this conjecture in terms of the convergence of the rearranged Fourier series in the strong (equivalently in this case, weak) operator topologies on $B(L_2(T))$. Additionally, a new framework for further investigation is introduced by consideri…
This paper gives a short survey of some basic results related to estimates of fractional integrals and Fourier transforms. It is closely adjoint to our previous survey papers \cite{K1998} and \cite{K2007}. The main methods used in the paper are based on nonincreasing rearrangements. We give alternative proofs of some results. We observe also that the paper represents the mini-course given by the author at Barcelona University in October, 2014.
We obtain necessary and sufficient conditions on weights for the generalized Fourier-type transforms to be bounded between weighted $L^p-L^q$ spaces. As an important example, we investigate transforms with kernel of power type, as for instance the sine, Hankel or $\mathscr{H}_α$ transforms. The obtained necessary and sufficient conditions are given in terms of weights, but not in terms of their decreasing rearrangements, as in several previous investigations.
We investigate the square variation operator $V^2$ (which majorizes the partial sum maximal operator) on general orthonormal systems (ONS) of size $N$. We prove that the $L^2$ norm of the $V^2$ operator is bounded by $O(\ln(N))$ on any ONS. This result is sharp and refines the classical Rademacher-Menshov theorem. We show that this can be improved to $O(\sqrt{\ln(N)})$ for the trigonometric system, which is also sharp. We show that for any choice of coefficients, this truncation of the trigonometric system can be rearranged so that the $L^2$ norm of the associated $V^2$ operator is $O(\sqrt{\l…
We prove that there exists a rearrangement of the first $N$ elements of the trigonometric system such that the $L^2$-norm of the square variation operator is at most $O_ε(\log^{9/22+ε}(N))$. This is an improvement over $O(\log^{1/2}(N))$ from the canonical ordering.
We consider the algebras $M_p$ of Fourier multipliers and show that every bounded continuous function $f$ on $\mathbb R^d$ can be transformed by an appropriate homeomorphic change of variable into a function that belongs to $M_p(\mathbb R^d)$ for all $p$, $1<p<\infty$. Moreover, under certain assumptions on a family $K$ of continuous functions, one change of variable will suffice for all $f\in K$. A similar result holds for functions on the torus $\mathbb T^d$. This may be contrasted with the known result on the Wiener algebra, related to Luzin's rearrangement problem.
We introduce a family of differential-reflection operators $Λ_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a Strum-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ For special pairs $(A,\varepsilon),$ we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). The spectral problem for the operators $Λ_{A, \varepsilon}$ is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of $Λ_{A, \varepsilon}$. As the operators $Λ_{A, \varepsilon}$ are mixture of $d/dx$ and reflection operators, w…
There were established the exact-order estimations of the best uniform approximations byψ the trigonometrical polynoms on the $C^ψ_{β,p}$ classes of $2π$-periodic continuous functions $f$, which are defined by the convolutions of the functions, which belong to the unit ball in $L_p$, $1\leq p <\infty$ spaces with generating fixed kernels $Ψ_β\subset|L_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$, whose Fourier coeficients decreasing to zero approximately as power functions. The exact order estimations were also established in $L_p$-metrics, $1 < p \leq\infty$ for $L^ψ_{β,1}$ classes of $2π$-periodic fu…
Let $f\in L_{2π}$ be a real-valued even function with its Fourier series $ \frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos nx,$ and let $S_{n}(f,x), n\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and $\lim\limits_{n\to \infty}a_{n}=0$, then $$ \lim\limits_{n\to \infty}\Vert f-S_{n}(f)\Vert_{L}=0 {if and only if} \lim\limits_{n\to \infty}a_{n}\log n=0. $$ We weaken the monotone condition in this classical result to the so-called mean value bounded variation ($MVBV$) condition. The generalization of the above cl…
In this work we verify the sufficiency of a Jensen's necessary and sufficient condition for a class of genus 0 or 1 entire functions to have only real zeros. They are Fourier transforms of even, positive, indefinitely differentiable, and very fast decreasing functions. We also apply our result to several important special functions in mathematics, such as modified Bessel function $K_{iz}(a),\ a>0$ as a function of variable $z$, Riemann Xi function $Ξ(z)$, and character Xi function $Ξ(z;χ)$ when $χ$ is a real primitive non-principal character satisfying $φ(u;χ)\ge0$ on the real line, we prove t…
The Howe dual pair (sl(2),O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a well-chosen element of sl(2) such that the Helmholtz relations are satisfied. In this paper we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized Fourier transforms, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We…
