The norm of the Fourier transform on finite abelian groups
type: Item
instance of: scholarly article
title: The norm of the Fourier transform on finite abelian groups
zbMATH Open document ID: 1202.42065
author: Yanyan Li
publication date: 2010-09-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116305
review text: Let \(G\) be a finite abelian group and \(\hat G\) its dual group. The Fourier transform on \(G\) is defined by \({\mathcal F}v(\gamma)=|G|^{\frac{1}{2}}\langle v,\gamma\rangle\) for \(\gamma\in\hat G\), where \(\langle\cdot,\cdot\rangle\) is the inner product on \({\mathbb C}^{|G|}\), and its norm is given by \(C_{p,q}=\sup_{\|v\|_p=1}\|{\mathcal F}v\|_q\) for \(1\leq p,q\leq\infty\). In this paper, the authors calculate the values of \(C_{p,q}\) and find the functions which attain the upper bound. They split the square \([0,1]^2=\{(\frac{1}{p},\frac{1}{q})\mid 1\leq p,q\leq\infty\}\) into three regions. In each region, an upper bound follows from Riesz-Thorin convexity theorem, and then it is attained by characters, delta functions, and biunimodular functions, respectively. Here, characters form a frequency basis on \(\hat G\), delta functions do a time basis on \(G\), and biunimodular functions are constructed from a time-frequency basis. They also characterize the set of extremals \(E_{p,q}=\{v\in L^p(G)\mid \|{\mathcal F}v\|_q=C_{p,q}\|v\|_p\}\). Especially, biunimodular functions appear in the above rigion and the so-called wave packets does in the classical range of \(p,q\) that corresponds to the Hausdorff-Young inequality on the real line. A survey on biunimodular functions is given.
reviewed by: Takeshi Kawazoe
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 5793934
zbMATH Keywords: Fourier transform
zbMATH Keywords: finite abelian groups
zbMATH Keywords: wave packets
zbMATH Keywords: biunimodular functions
MaRDI profile type: Publication
OpenAlex ID: W2461593154
cites work: Constructing symmetric ciphers using the CAST design procedure
cites work: Generating bent sequences
cites work: Q5734726
cites work: Inequalities in Fourier analysis
cites work: Q3350138
cites work: A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic \(n\)-roots
cites work: Q4852329
cites work: On Polynomials with Coefficients of Modulus One
cites work: Two New Classes of Bent Functions
cites work: Q4378179
cites work: Q4288301
cites work: Entropy-based algorithms for best basis selection
cites work: Q4115154
cites work: Uncertainty Principles and Signal Recovery
cites work: Pointwise convergence of Fourier series
cites work: METAPLECTIC OPERATORS ON Cn
cites work: Q4215408
cites work: A Maximum Problem in Harmonic Analysis
cites work: Q5723435
cites work: \(L^p\) estimates on the bilinear Hilbert transform for \(2<p<\infty\)
cites work: On Calderón's conjecture
cites work: Nonexistence of abelian difference sets: Lander’s conjecture for prime power orders
cites work: Finding mixed cells in the mixed volume computation
cites work: Gaussian kernels have only Gaussian maximizers
cites work: On the mean values of certain trigonometrical polynomials. II
cites work: Q4146667
cites work: Q4680159
cites work: Q4036868
cites work: Discrete-time, discrete-frequency, time-frequency analysis
cites work: On ``bent'' functions
cites work: Q3851094
cites work: Some polynomial extremal problems which emerged in the twentieth century
cites work: Cyclotomic integers and finite geometry
cites work: Q4408055
cites work: ENUMERATION OF ALL SOLUTIONS OF A COMBINATORIAL LINEAR INEQUALITY SYSTEM ARISING FROM THE POLYHEDRAL HOMOTOPY CONTINUATION METHOD
cites work: An uncertainty principle for cyclic groups of prime order
cites work: A fast algorithm for adapted time-frequency tilings
cites work: Character sums and difference sets
cites work: Discrete chirp-Fourier transform and its application to chirp rate estimation
DOI: 10.5802/AIF.2556
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5793934