Emanuel Carneiro, Jeffrey D. Vaaler
We obtain the best approximation in $L^1(\R)$, by entire functions of exponential type, for a class of even functions that includes $e^{-λ|x|}$, where $λ>0$, $\log |x|$ and $|x|^α$, where $-1 < α< 1$. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree.
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We obtain extremal majorants and minorants of exponential type for a class of even functions on $\R$ which includes $\log |x|$ and $|x|^α$, where $-1 < α< 1$. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev inequalities. A further application provides an Erdös-Turán-type inequality that estimates the sup norm of algebraic polynomials on the unit disc in term…
We prove the existence of functions that extremize the endpoint $L^2$ to $L^4$ adjoint Fourier restriction inequality on the one-sheeted hyperboloid in Euclidean space $\mathbb{R}^4$ and that, taking symmetries into consideration, any extremizing sequence has a subsequence that converges to an extremizer.
On $L^p$ extremals for Fourier extension estimate to fractional surface
本文研究分数阶曲面对应的傅里叶延拓算子，证明了α≥2时该延拓不等式在p∈[1,2]存在L^p极值函数，且p∈(1,2]时极值序列在对称下准紧。
The adjoint Fourier restriction inequality of Tomas and Stein states that the mapping $f\mapsto \widehat{fσ}$ is bounded from $\lt(S^2)$ to $L^4(\reals^3)$. We prove that there exist functions which extremize this inequality, and that any extremizing sequence of nonnegative functions has a subsequence which converges to an extremizer.
This note establishes the full range of $L^p$--$L^q$ Fourier extension estimates for the model $n$-dimensional quadratic submanifold in ${\mathbb R}^{n(n+3)/2}$ parametrized by $γ(x_1,\ldots,x_n) := (x_1,\ldots,x_n, (x_i x_j)_{1 \leq i \leq j \leq n})$. This class of submanifolds is extremal in the sense that an $n$-dimensional quadratic submanifold of ${\mathbb R}^d$ can only satisfy nontrivial Fourier extension inequalities when $d \leq \frac{n(n+3)}{2}$. The proof is via an inflation-type argument, with the unexpected twist that a significant amount of "overinflation" is necessary but in no…
Some extremal problem for martingale transforms. III
本文延续前作，研究鞅变换极值问题中新的局部叶状结构（主口袋、中口袋及双线性区域），并通过三次多项式实例分析其出现，最终给出边界值为三次多项式时所有可能叶状结构的完整概述。
The spectral theory of the Fourier operator (non-truncated) is expounded. The known construction of basis of eigenvectors consisting of the Hermite functions is presented. The detail description of the eigenspaces in the spirit of a work by Hardy and Titchmarsh is done.
This paper presents the Gaussian subordination framework to generate optimal one-sided approximations to multidimensional real-valued functions by functions of prescribed exponential type. Such extremal problems date back to the works of Beurling and Selberg and provide a variety of applications in analysis and analytic number theory. Here we majorize and minorize (on $\mathbb{R}^N$) the Gaussian ${\bf x} \mapsto e^{-πλ|{\bf x}|^2}$, where $λ>0$ is a free parameter, by functions with distributional Fourier transforms supported on Euclidean balls, optimizing weighted $L^1$-errors. By integratin…
We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann hypothesis.
We study the family of Fourier-Laplace transforms $$ F_{α,β}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^β\exp(\mathrm{i} t^α-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for $α>1$ and $β\in\mathbb{C}$, where Hadamard finite part is used to regularize the integral when $\operatorname*{Re} β\leq -1$. We prove that each $F_{α,β}$ has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a…
We study the problem of existence of extremizers for the $L^2$ to $L^p$ adjoint Fourier restriction inequalities on the hyperboloid in dimensions 3 and 4, in which cases $p$ is an even integer. We will use the method developed by Foschi to show that extremizers do not exist.
Sharp Fourier extension on fractional surfaces
本文研究分数曲面上的傅里叶延拓算子，利用缺失质量方法与双线性限制理论刻画了α-Strichartz不等式极值序列的预紧性，并在二维情形证明了α∈[2,α0)（α0>5）时极值的存在性。
Monotone extremal functions and the weighted Hilbert's inequality
本文寻找满足单调性条件的 signum 函数的指数型最优单侧控制函数，并以此给出加权 Hilbert-Montgomery-Vaughan 不等式的简单 Fourier 分析证明。
We study stability estimates for the almost extremal functions associated with the $L^p$-bound for the real and imaginary parts of the Beurling-Ahlfors operator. The proof exploits probabilistic methods and rests on analogous results for differentially subordinate martingales which are of independent interest. This allows us to obtain stability inequalities for a larger class of Fourier multipliers.
Sharp endpoint extension inequalities for the moment curve on finite fields II: an extremal property of the uniform distribution
本文确定了有限域上矩曲线傅里叶端点扩展不等式的最优常数及所有极大化子，证实了先前猜想，并证明其等价于均匀分布唯一最大化独立同分布样本不同重排期望数的概率极值命题。
We construct majorants and minorants of a Gaussian function in Euclidean space that have Fourier transforms supported in a box. The majorants that we construct are shown to be extremal and our minorants are shown to be asymptotically extremal as the sides of the box become uniformly large. We then adapt the Distribution and Gaussian Subordination methods of Carneiro-Littmann-Vaaler to the multidimensional setting to obtain majorants and minorants for a class of radial functions. Periodic analogues of the main results are proven and applications to Hilbert-type inequalities are given.
First we consider the following problem which dates back to Chebyshev, Zolotarev and Achieser: among all trigonometric polynomials with given leading coefficients $a_0,...,a_l,$ $b_0,...,b_l \in \mathbb R$ find that one with least maximum norm on $[0,2 π].$ We show that the minimal polynomial is on $[0,2 π]$ asymptotically equal to a Blaschke product times a constant where the constant is the greatest singular value of the Hankel matrix associated with the $τ_j = a_j + i b_j.$ As a special case corresponding statements for algebraic polynomials follow. Finally the minimal norm of certain linea…
Fourier optimization and pair correlation problems
本文通过傅里叶分析中的极值问题建立通用框架，研究广泛序列的对相关性界，并应用于 Selberg 类零点、Dedekind zeta 函数及黎曼 zeta 函数实虚部，进而否定 Gonek-Ki 猜想在某些情形下的成立。
The zeros of point evaluation extremals in Paley-Wiener spaces
该研究针对Paley-Wiener空间的点赋值极值函数φₚ，确定其零点的渐近展开，推导其傅里叶变换性质，解决了B. Green2001年提出的自卷积极值问题。
The Hörmander--Bernhardsson extremal function
本文研究了Hörmander--Bernhardsson极端函数的性质，通过分解和微分方程推导其结构，并利用交换关系进行高精度计算。
The smoothest average and some extremal problems for polynomials
本文研究了函数在与合适核卷积后的最平局部平均问题，通过复分析工具重新证明了k=1,2时的最优核，并首次解决了k=3,4,6时的极值问题，揭示了k与2k对应的最优常数关系。
The $L^p$ convergence of Fourier series on triangular domains
本文针对 45-90-45、等边和半等边三类三角形，证明 Dirichlet Laplacian 特征函数 Fourier 级数适当截断的 $L^p$ 范数收敛，并依据 Lamé 定理说明方法局限。
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