Completeness of a dilation system ${\left(\varphi \left(nx\right)\right)}_{n\ge 1}$ on the standard Lebesgue space $L^2(0,1)$ is considered for 2-periodic functions $\varphi$. We show that the problem is equivalent to an open question on cyclic vectors of the Hardy space $H^2\left({𝔻}_2^{\infty }\right)$ on the Hilbert multidisc ${𝔻}_2^{\infty }$. Several simple sufficient conditions are exhibited, which include however practically all previously known results (Wintner; Kozlov; Neuwirth, Ginsberg, and Newman; Hedenmalm, Lindquist, and Seip). For instance, each of the following conditions implies cyclicity...

We consider harmonic Bergman-Besov spaces $b_{\alpha }^p$ and weighted Bloch spaces $b_{\alpha }^{\infty }$ on the unit ball of ${ℝ}^n$ for the full ranges of parameters $0<p<\infty$, $\alpha \in ℝ$, and determine the precise inclusion relations among them. To verify these relations we use Carleson measures and suitable radial differential operators. For harmonic Bergman spaces various characterizations of Carleson measures are known. For weighted Bloch spaces we provide a characterization when $\alpha >0$.

A simple and fast algorithm is presented for evaluating the indefinite integral of an oscillatory function ${\int }_x^{y}if\left(t\right)e^{i\omega t}dt$, -1 ≤ x < y ≤ 1, ω ≠ 0, where the Chebyshev series expansion of the function f is known. The final solution, expressed as a finite Chebyshev series, is obtained by solving a second-order linear difference equation. Because of the nature of the equation special algorithms have to be used to find a satisfactory approximation to the integral.

We study connections between the Boyd indices in Orlicz spaces and the growth conditions frequently met in various applications, for instance, in the regularity theory of variational integrals with non-standard growth. We develop a truncation method for computation of the indices and we also give characterizations of them in terms of the growth exponents and of the Jensen means. Applications concern variational integrals and extrapolation of integral operators.

On a metric measure space (X,ϱ,μ), consider the weight functions $w_{\alpha }\left(x\right)=\varrho {(x,z₀)}^{-\alpha ₀}$ if ϱ(x,z₀) < 1, $w_{\alpha }\left(x\right)=\varrho {(x,z₀)}^{-\alpha ₁}$ if ϱ(x,z₀) ≥ 1, $w_{\beta }\left(x\right)=\varrho {(x,z₀)}^{-\beta ₀}$ if ϱ(x,z₀) < 1, $w_{\beta }\left(x\right)=\varrho {(x,z₀)}^{-\beta ₁}$ if ϱ(x,z₀) ≥ 1, where z₀ is a given point of X, and let ${\kappa }_a:X\times X\to ℝ₊$ be an operator kernel satisfying ${\kappa }_a(x,y)\le c\varrho {(x,y)}^{a-d}$ for all x,y ∈ X such that ϱ(x,y) < 1, ${\kappa }_a(x,y)\le c\varrho {(x,y)}^{a-D}$ for all x,y ∈ X such that ϱ(x,y)≥ 1, where 0 < a < min(d,D), and d and D are respectively the local and global volume growth rate of the space X. We determine conditions on a, α₀, α₁, β₀, β₁ ∈ ℝ for the Hardy-Littlewood-Sobolev operator...

Le but de cet article est d’étendre les résultats classiques (inégalité de Hardy-Littlewood-Sobolev, inégalité de Hedberg) sur l’intégrale fractionnaire à deux types différents d’espaces métriques mesurés : les espaces métriques mesurés à mesure doublante d’une part, les espaces métriques mesurés à croissance polynomiale du volume d’autre part. Les deux résultats principaux que nous obtenons sont les suivants :Etant donné $(X,\rho ,\mu )$ un espace métrique mesuré de type homogène, étant donnés $p,q,\alpha \in \mathbf{R}$ tels que $1\le p\&lt;1/\alpha$, $1/q=1/p-\alpha$,...

We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have $\alpha \le {\sum }_{j=1}^{n-1}1/(n²-j²)sin\left(jx\right)\le \beta$, with the best possible constant bounds α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3. (II) The inequality $0<{\sum }_{j=1}^{n-1}(n²-j²)sin\left(jx\right)$ holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].

Theorems stating sufficient conditions for the inequivalence of the d-variate Haar wavelet system and another wavelet system in the spaces $L₁\left({ℝ}^d\right)$ and $BV\left({ℝ}^d\right)$ are proved. These results are used to show that the Strömberg wavelet system and the system of continuous Daubechies wavelets with minimal supports are not equivalent to the Haar system in these spaces. A theorem stating that some systems of smooth Daubechies wavelets are not equivalent to the Haar system in $L₁\left({ℝ}^d\right)$ is also shown.