Let d be a positive integer and μ a generalized Cantor measure satisfying $\mu ={\sum }_{j=1}^m{a}_j\mu \circ S_j^{-1}$, where $0<a_j<1$, ${\sum }_{j=1}^m{a}_j=1$, $S_j=\rho R+b_j$ with 0 < ρ < 1 and R an orthogonal transformation of ${ℝ}^d$. Then ⎧1 < p ≤ 2 ⇒ ⎨$su{p}_{r>0}{r}^{d(1/{\alpha }^{\text{'}}-1/p^{\text{'}})}({\int }_{J_x^r}{\left|\mu ̂\left(y\right)\right|}^{p^{\text{'}}}{dy)}^{1/p^{\text{'}}}\le D₁{\rho }^{-d/{\alpha }^{\text{'}}}$, $x\in {ℝ}^d$, ⎩ p = 2 ⇒ infr≥1 rd(1/α’-1/2) (∫J₀r|μ̂(y)|² dy)1/2 ≥ D₂ρd/α’$,$where $J_x^r={\prod }_{i=1}^d(x_i-r/2,x_i+r/2)$, α’ is defined by ${\rho }^{d/{\alpha }^{\text{'}}}={\left({\sum }_{j=1}^m{a}_j^p\right)}^{1/p}$ and the constants D₁ and D₂ depend only on d and p.

A two-sided sequence ${\left(cₙ\right)}_{n\in \mathbb{Z}}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m}+c_{n-m}=2cₙcₘ$ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence ${\left(cₙ\right)}_{n\in \mathbb{Z}}$ is bounded if $su{p}_{n\in \mathbb{Z}}\left|\right|cₙ\left|\right|<\infty$. A (bounded) group decomposition for a cosine sequence $c={\left(cₙ\right)}_{n\in \mathbb{Z}}$ is a representation of c as $cₙ=(bⁿ+b^{-n})/2$ for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying $su{p}_{n\in \mathbb{Z}}\left|\right|bⁿ\left|\right|<\infty$, respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, here referred...

This paper deals with atomic decomposition and factorization of functions in the holomorphic Hardy space $H^1$. Such representation theorems have been proved for strictly pseudoconvex domains. The atomic decomposition has also been proved for convex domains of finite type. Here the Hardy space was defined with respect to the ordinary Euclidean surface measure on the boundary. But for domains of finite type, it is natural to define $H^1$ with respect to a certain measure that degenerates near Levi-flat points...

We consider multi-dimensional Hartman almost periodic functions and sequences, defined with respect to different averaging sequences of subsets in ${ℝ}^d$ or ${\mathbb{Z}}^d$. We consider the behavior of their Fourier-Bohr coefficients and their spectrum, depending on the particular averaging sequence, and we demonstrate this dependence by several examples. Extensions to compactly generated, locally compact, abelian groups are considered. We define generalized Marcinkiewicz spaces based upon arbitrary measure spaces...