We consider dilation operators Tk : f → f(2·) in the framework of Besov spaces B p,q(R ) when 0 < p ≤ 1. If s > n ` 1 p − 1 ́ , Tk is a bounded linear operator from B p,q(R ) into itself and there are optimal bounds for its norm. We study the situation on the line s = n `

We consider a dilation operator on Besov spaces $$B^s_{r,t}(K)$$ over local fields and estimate an norm such field for $$s > \sigma _r = \text {max}\big (\frac{1}{r} -1,\,0\big )$$ which depends the constant k unlike case of Euclidean spaces. In $${\mathbb {R}}^n,$$ it is independent k, appears limiting $$s=0$$ $$s=\sigma _r.$$ fields, still open.

We consider dilation operators Tk : f → f(2 k ·) in the framework of Triebel-Lizorkin spaces F s p,q(R ). If s > n max ` 1 p − 1, 0 ́ , Tk is a bounded linear operator from F s p,q(R ) into itself and there are optimal bounds for its norm. We study the situation on the line s = n max ` 1 p − 1, 0 ́ , an open problem mentioned in [ET96, 2.3.1]. It turns out that the results shed new light upon the...

In this paper we construct a special sort of dilation for an arbitrary polynomially bounded operator. This enables us to show that the problem whether every polynomially bounded operator is similar to a contraction can be reduced to a subclass of it.

The regularity of refinable functions has been studied extensively in the past. A classical result by Daubechies and Lagarias [6] states that a compactly supported refinable function in R of finite mask with integer dilation and translations cannot be in C∞. A bound on the regularity based on the eigenvalues of certain matrices associated with the refinement equation is also given. Surprisingly...

Mathematical morphology (MM) is a theory of non-linear operators used for the processing and analysis images. Morphological neural networks (MNNs) are whose neurons compute morphological operators. Dilations erosions elementary MM. From an algebraic point view, dilation erosion that commute respectively with supremum infimum operations. In this paper, we present linear dilation-erosion perceptr...

We construct a weak dilation of a not necessarily unital CP-semigroup to an E–semigroup acting on the adjointable operators of a Hilbert module with a unit vector. We construct the dilation in such a way that the dilating E–semigroup has a pre-assigned product system. Then, making use of the commutant of von Neumann correspondences, we apply the dilation theorem to proof that covariant represen...