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 terms of power sums in the roots of the polynomials.
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.
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.
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…
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.
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
Indian Institute of Technology Bombay（印度理工学院孟买分校）；ICTP - The Abdus Salam International Centre for Theoretical Physics（阿卜杜斯·萨拉姆国际理论物理中心）；Macalester College（麦克莱斯特学院）；Virginia Polytechnic Institute and State University（弗吉尼亚理工大学）；Instituto Superior Técnico Universidade de Lisboa（里斯本大学高级技术学院）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）；University of Pennsylvania（宾夕法尼亚大学）
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.
We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials $P_{i_1}(x),\ldots ,P_{i_m}(x)$ (where $P_j(x)$ is a polynomial of degree $j$) under the restriction that the sup-norm of $\sum_{j=1}^m P_{i_j}^2(x)$ is bounded on the interval $[-b,b]$ ($b>0$). A complete solution of the problem is presented using duality theory of convex analysis and the theory of canonical moments. It turns out, that contrary to many other extremal problems the structure of the solution will depend heavily on the size of the interval $[-b,b]$.
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…
Our main result shows that if a lower-semicontinuous kernel K satisfies some mild additional hypotheses, then asympotitically polarization optimal configurations are precisely those that are asymptotically distributed according to the equilibrium measure for the corresponding minimum energy problem.
