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Eigenvalues of Hille-Tamarkin operators and geometry of Banach function spaces

Thomas Kühn, Mieczysław Mastyło (2011)

Studia Mathematica

We investigate how the asymptotic eigenvalue behaviour of Hille-Tamarkin operators in Banach function spaces depends on the geometry of the spaces involved. It turns out that the relevant properties are cotype p and p-concavity. We prove some eigenvalue estimates for Hille-Tamarkin operators in general Banach function spaces which extend the classical results in Lebesgue spaces. We specialize our results to Lorentz, Orlicz and Zygmund spaces and give applications to Fourier analysis. We are also...

Entropy and approximation numbers of embeddings between weighted Besov spaces

Iwona Piotrowska (2008)

Banach Center Publications

The present paper is devoted to the study of the “quality” of the compactness of the trace operator. More precisely, we characterize the asymptotic behaviour of entropy numbers of the compact map t r Γ : B p ₁ , q s ( ℝ ⁿ , w ϰ Γ ) → L p ₂ ( Γ ) , where Γ is a d-set with 0 < d < n and w ϰ Γ a weight of type w ϰ Γ ( x ) d i s t ( x , Γ ) ϰ near Γ with ϰ > -(n-d). There are parallel results for approximation numbers.

Entropy numbers of general diagonal operators.

Thomas Kühn (2005)

Revista Matemática Complutense

We determine the asymptotic behavior of the entropy numbers of diagonal operators D: lp → lq, (xk) → (skxk), 0 &lt; p,q ≤ ∞, under mild regularity and decay conditions on the generating sequence (σk). Our results extend the known estimates for polynomial and logarithmic diagonals (σk). Moreover, we also consider some exotic intermediate examples like (σk)=exp(-√log k).

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