Academic Journal
Approximation in the Zygmund and Hölder classes on
| Τίτλος: | Approximation in the Zygmund and Hölder classes on |
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| Συγγραφείς: | Eero Saksman, Odí Soler i Gibert |
| Συνεισφορές: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. TF-EDP - Grup de Teoria de Funcions i Equacions en Derivades Parcials |
| Πηγή: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| Στοιχεία εκδότη: | Canadian Mathematical Society, 2021. |
| Έτος έκδοσης: | 2021 |
| Θεματικοί όροι: | Funcions de diverses variables reals, BMO-Sobolev spaces, Zygmund class, Classificació AMS::26 Real functions::26B Functions of several variables, 4. Education, Functions of real variables, Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica::Funcions de variable complexa, Wavelet characterizations, 0101 mathematics, 01 natural sciences, Hölder classes, 12. Responsible consumption |
| Περιγραφή: | We determine the distance (up to a multiplicative constant) in the Zygmund class $\Lambda _{\ast }(\mathbb {R}^n)$ to the subspace $\mathrm {J}_{}(\mathbf {bmo})(\mathbb {R}^n).$ The latter space is the image under the Bessel potential $J := (1-\Delta )^{{-1}/2}$ of the space $\mathbf {bmo}(\mathbb {R}^n)$ , which is a nonhomogeneous version of the classical $\mathrm {BMO}$ . Locally, $\mathrm {J}_{}(\mathbf {bmo})(\mathbb {R}^n)$ consists of functions that together with their first derivatives are in $\mathbf {bmo}(\mathbb {R}^n)$ . More generally, we consider the same question when the Zygmund class is replaced by the Hölder space $\Lambda _{s}(\mathbb {R}^n),$ with $0 < s \leq 1$ , and the corresponding subspace is $\mathrm {J}_{s}(\mathbf {bmo})(\mathbb {R}^n)$ , the image under $(1-\Delta )^{{-s}/2}$ of $\mathbf {bmo}(\mathbb {R}^n).$ One should note here that $\Lambda _{1}(\mathbb {R}^n) = \Lambda _{\ast }(\mathbb {R}^n).$ Such results were known earlier only for $n = s = 1$ with a proof that does not extend to the general case.Our results are expressed in terms of second differences. As a by-product of our wavelet-based proof, we also obtain the distance from $f \in \Lambda _{s}(\mathbb {R}^n)$ to $\mathrm {J}_{s}(\mathbf {bmo})(\mathbb {R}^n)$ in terms of the wavelet coefficients of $f.$ We additionally establish a third way to express this distance in terms of the size of the hyperbolic gradient of the harmonic extension of f on the upper half-space $\mathbb {R}^{n +1}_+$ . |
| Τύπος εγγράφου: | Article |
| Περιγραφή αρχείου: | application/pdf |
| Γλώσσα: | English |
| ISSN: | 1496-4279 0008-414X |
| DOI: | 10.4153/s0008414x21000523 |
| Σύνδεσμος πρόσβασης: | http://arxiv.org/pdf/2009.09752 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X21000523 |
| Rights: | Cambridge Core User Agreement |
| Αριθμός Καταχώρησης: | edsair.doi.dedup.....d444ee3bfad9bcc8fd10c17b82ffc1c0 |
| Βάση Δεδομένων: | OpenAIRE |
| ISSN: | 14964279 0008414X |
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| DOI: | 10.4153/s0008414x21000523 |