Abstract
We provide explicit commutative sequence space representations for classical function and distribution spaces on the real half-line. This is done by evaluating at the Fourier transforms of the elements of an orthonormal wavelet basis.
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Notes
The differential operators \(x^{n}D^{n}\) occurring in the definition of \(X_{\lambda ,n}\) are very natural in the context of the group \(\mathbb {R}_{+}\) since they commute with all dilation operators.
This can be deduced via an exponential change of variables, under which the space \(\mathcal {S}(\mathbb {R}_{+})\) corresponds to the space \(\mathcal {K}_{1}(\mathbb {R})\) of exponentially rapidly decreasing smooth functions, whose (additive) convolutor space has predual \(\mathcal {O}_{C}(\mathcal {K}_{1})=\{\phi \in \mathcal {E}(\mathbb {R}): (\exists \gamma )\) \((\forall n\in \mathbb {N}) \, (\sup _{x\in \mathbb {R}} e^{\gamma |x|} |\phi ^{(n)}(x)| <\infty )\},\) see [7, Section 6].
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L. Neyt gratefully acknowledges support by the Research Foundation–Flanders through the postdoctoral grant 12ZG921N. J. Vindas was supported by Ghent University through the BOF-grant 01J04017 and by the Research Foundation–Flanders through the FWO-grant G067621N.
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Debrouwere, A., Neyt, L. & Vindas, J. Explicit commutative sequence space representations of function and distribution spaces on the real half-line. Arch. Math. 118, 383–391 (2022). https://doi.org/10.1007/s00013-021-01701-1
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DOI: https://doi.org/10.1007/s00013-021-01701-1
Keywords
- Sequence space representations
- Valdivia–Vogt table
- Function and distribution spaces on the real half-line
- Orthonormal wavelets