Translation beyond Delsarte. (English) Zbl 1540.46023
Summary: We introduce general translations as solutions to Cauchy or Dirichlet problems. This point of view allows us to handle for instance the heat-diffusion semigroup as a translation. With the given examples, Kolmogorov-Riesz characterization of compact sets in certain \(L^p_\mu\) spaces is given. Pego-type characterizations are also derived. Finally, for some examples, the equivalence of the corresponding modulus of smoothness and K-functional is pointed out.
MSC:
| 46B50 | Compactness in Banach (or normed) spaces |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 42C20 | Other transformations of harmonic type |
| 47D06 | One-parameter semigroups and linear evolution equations |
Keywords:
generalised translation; Kolmogorov-Riesz theorem; Pego’s theorem; modulus of smoothness; \(K\)-functionalReferences:
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