The two-dimensional classical Hardy spaces $H_p\left(ℝ\times ℝ\right)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_p\left(ℝ\times ℝ\right)$ to $L_p\left({ℝ}^2\right)$ (1/2 < p ≤ ∞) and is of weak type $(H_1^{}\left(ℝ\times ℝ\right),L_1\left({ℝ}^2\right))$ where the Hardy space $H_1\left(ℝ\times ℝ\right)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_1^{(}ℝ\times ℝ)$ ⊃ $LlogL\left({ℝ}^2\right)$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_p\left(ℝ\times ℝ\right)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_p\left(ℝ\times ℝ\right)$, the Fejér means...

We give a complete characterization of the positive trigonometric polynomials $Q(\theta ,\varphi )$ on the bi-circle, which can be factored as $Q(\theta ,\varphi )=|p(e^{i\theta },e^{i\varphi }){|}^2$ where $p(z,w)$ is a polynomial nonzero for $\left|z\right|=1$ and $\left|w\right|\le 1$. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac{1}{4{\pi }^{2}Q(\theta ,\varphi )}$ on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating...

We consider Boolean functions defined on the discrete cube $-\gamma ,{\gamma }^{-1}ⁿ$ equipped with a product probability measure ${\mu }^{\otimes n}$, where $\mu =\beta {\delta }_{-\gamma }+\alpha {\delta }_{{\gamma }^{-1}}$ and γ = √(α/β). This normalization ensures that the coordinate functions ${\left(x_i\right)}_{i=1,...,n}$ are orthonormal in $L₂(-\gamma ,{\gamma }^{-1}ⁿ,{\mu }^{\otimes n})$. We prove that if the spectrum of a Boolean function is concentrated on the first two Fourier levels, then the function is close to a certain function of one variable. Our theorem strengthens the non-symmetric FKN Theorem due to Jendrej, Oleszkiewicz and Wojtaszczyk. Moreover, in the symmetric...

The main topic of this paper is the study of compactly supported functions in a multi-resolution analysis and especially of the minimally supported ones. We will show that this class of functions is stable under differentiation and integration and how to compute basic quantities with them.

The theory of convergence for (non-stationary) scaling functions and the approximation of interpolating scaling filters by means of Bernstein polynomials, allow us to construct a non-stationary interpolating scaling function with interesting approximation properties.

Soit $a$ un réel de $]0,1[$. Nous étudions le système d’équations de convolution suivant$$(*)\forall x\in {\mathbf{R}}^2,f\left(x\right)=\frac{1}{4}\sum _{\genfrac{}{}{0pt}{}{ϵ=\pm 1}{{ϵ}^{\text{'}}=\pm 1}}f(x+(ϵ,{ϵ}^{\text{'}}\left)\right)=\frac{1}{4}\sum _{\genfrac{}{}{0pt}{}{ϵ=\pm 1}{{ϵ}^{\text{'}}=\pm 1}}f(x+a(ϵ,{ϵ}^{\text{'}}\left)\right).$$Nous démontrons que les exponentielles polynômes solutions de $(*)$ sont denses dans l’espace des solutions $C^{\infty }$ du système d’équations; l’idéal de ${ℰ}^{\text{'}}({\mathbf{R}}^2)$ engendré par les transformées de Fourier des deux mesures intervenant ici est “slowly decreasing” au sens de Berenstein-Taylor. Lorsque $a$ n’est pas un nombre de Liouville, nous montrons qu’il existe un ouvert relativement compact telle que toute solution distribution de $(*)$ régulière...

Clearly, one of the most basic contributions to the fields of real variables, partial differential equations and Fourier analysis in recent times has been the celebrated theorem of Calderón and Zygmund on the boundedness of singular integrals on Rn [1].

We derive new upper bounds for the densities of measurable sets in ${ℝ}^n$ which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions $2,\cdots ,24$. This gives new lower bounds for the measurable chromatic number in dimensions $3,\cdots ,24$. We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg,...