The paper deals with dimension-controllable (tractable) embeddings of Besov spaces on n-dimensional cubes into Zygmund spaces. This can be expressed in terms of tractability envelopes.
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The paper deals with spaces $L^{s}_{p}(ℝⁿ)$ of Sobolev type where s > 0, 0 < p ≤ ∞, and their relations to corresponding spaces $B^{s}_{p,q}({ℝ}ⁿ)$ of Besov type where s > 0, 0 < p ≤ ∞, 0 < q ≤ ∞, in terms of embedding and real interpolation.
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Let $f^{j} = ∑_{k} a_{k} f(2^{j+1}x - 2k)$, where the sum is taken over the lattice of all points k in $ℝ^n$ having integer-valued components, j∈ℕ and $a_k ∈ ℂ$. Let $A^{s}_{pq}$ be either $B^{s}_{pq}$ or $F^{s}_{pq}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on $ℝ^n.$ The aim of the paper is to clarify under what conditions $∥f^{j} | A^{s}_{pq}∥$ is equivalent to $2^{j(s-n/p)} (∑_{k} |a_k|^p)^{1/p} ∥f | A^{s}_{pq}∥$.
This paper deals with wavelet frames for a large class of distributions on euclidean n-space, including all compactly supported distributions. These representations characterize the global, local, and pointwise regularity of the distribution considered.
The paper deals with quarkonial decompositions and entropy numbers in weighted function spaces on hyperbolic manifolds. We use these results to develop a spectral theory of related Schrödinger operators in these hyperbolic worlds.
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Let Γ be a closed set in $ℝ^n$ with Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure µ with supp µ ⊂ Γ and constants $c_{1} > 0$ and $c_{2} > 0$ such that $c_{1}r^{d} ≤ µ (B(x,r)) ≤ c_{2}r^{d}$ for all 0 < r < 1 and all x ∈ Γ, where B(x,r) is a ball with centre x and radius r, then Γ is called a d-set. The second aim of the paper is to provide a link between the related Lebesgue spaces $L_{p}(Γ)$, 0 < p ≤ ∞, with respect to that measure µ on the hand and the Fourier analytically defined Besov spaces $B^s_{p,q}(ℝ^n)$ (s ∈ ℝ, 0 < p ≤ ∞, 0 < q ≤ ∞) on the other hand.
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