Uniform approximation and explicit estimates for the prolate spheroidal wave functions

For fixed $c,$ Prolate Spheroidal Wave Functions (PSWFs), denoted by $\psi_{n, c},$ form an orthogonal basis with remarkable properties for the space of band-limited functions with bandwith $c$. They have been largely studied and used after the seminal work of D. Slepian and his co-authors. In several applications, uniform estimates of the $\psi_{n,c}$ in $n$ and $c,$ are needed. To progress in this direction, we push forward the uniform approximation error bounds and give an explicit approximation of their values at $1$ in terms of the Legendre complete elliptic integral of the first kind. Also, we give an explicit formula for the accurate approximation the eigenvalues of the Sturm-Liouville operator associated with the PSWFs

Equality cases for the uncertainty principle in finite Abelian groups

We consider the families of finite Abelian groups \ZZ/p\ZZ\times \ZZ/p\ZZ, \ZZ/p^2\ZZ and \ZZ/p\ZZ\times \ZZ/q\ZZ for $p,q$ two distinct prime numbers. For the two first families we give a simple characterization of all functions whose support has cardinality $k$ while the size of the spectrum satisfies a minimality condition. We do it for a large number of values of $k$ in the third case. Such equality cases were previously known when $k$ divides the cardinality of the group, or for groups \ZZ/p\ZZ.Comment: Mistakes have been corrected. This paper has been accepted for publication in Acta Sci. Math. (Szeged

Spectral Decay of Time and Frequency Limiting Operator

For fixed $c,$ the Prolate Spheroidal Wave Functions (PSWFs) $\psi_{n, c}$ form a basis with remarkable properties for the space of band-limited functions with bandwidth $c$. They have been largely studied and used after the seminal work of D. Slepian, H. Landau and H. Pollack. Many of the PSWFs applications rely heavily of the behavior and the decay rate of the eigenvalues $(\lambda_n(c))_{n\geq 0}$ of the time and frequency limiting operator, which we denote by $\mathcal Q_c.$ Hence, the issue of the accurate estimation of the spectrum of this operator has attracted a considerable interest, both in numerical and theoretical studies. In this work, we give an explicit integral approximation formula for these eigenvalues. This approximation holds true starting from the plunge region where the spectrum of $\mathcal Q_c$ starts to have a fast decay. As a consequence of our explicit approximation formula, we give a precise description of the super-exponential decay rate of the $\lambda_n(c).$ Also, we mention that the described approximation scheme provides us with fairly accurate approximations of the $\lambda_n(c)$ with low computational load, even for very large values of the parameters $c$ and $n.$ Finally, we provide the reader with some numerical examples that illustrate the different results of this work.Comment: arXiv admin note: substantial text overlap with arXiv:1012.388

Factorization of some Hardy type spaces of holomorphic functions

We prove that the pointwise product of two holomorphic functions of the upper half-plane, one in the Hardy space $\mathcal H^1$, the other one in its dual, belongs to a Hardy type space. Conversely, every holomorphic function in this space can be written as such a product. This generalizes previous characterization in the context of the unit disc.Comment: C. R. Math. Acad. Sci. Paris (to appear

Integral Concentration of idempotent trigonometric polynomials with gaps

We prove that for all p>1/2 there exists a constant $\gamma_p>0$ such that, for any symmetric measurable set of positive measure E\subset \TT and for any $\gamma<\gamma_p$, there is an idempotent trigonometrical polynomial f satisfying \int_E |f|^p > \gamma \int_{\TT} |f|^p. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of $\gamma_p>0$ for p>1 and conjectured that it does not exists for p=1. Furthermore, we prove that one can take $\gamma_p=1$ when p>1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when $p\neq 2$. This shows striking differences with the case p=2, for which the best constant is strictly smaller than 1/2, as it has been known for twenty years, and for which having arbitrarily large gaps with such concentration of the integral is not possible, according to a classical theorem of Wiener. We find sharper results for $0<p\leq 1$ when we restrict to open sets, or when we enlarge the class of idempotent trigonometric polynomials to all positive definite ones.Comment: 43 pages; to appear in Amer. J. Mat