
The article's main result concerns the Ca\-m\-po\-na\-to-Mor\-rey spaces \(\mathcal L^{(q,\lambda)}_k\), defined here, for \(1\le q0\), and \(k\in\mathbb N\cup\{0\}\) as the space of all \(L^q\)-functions \(f\) on \(\mathbb R\) such that the value \[ \sup_{x\in\mathbb R,\,r>0}\,\frac1{r^\lambda}\inf_{P\in\mathcal P_k}\int_{x-r}^{x+r}\!|f-P|^q \] is finite; \(\mathcal P_k\) is the class of all polynomials of degree \(\le k\). The author, as a consequence of a series of lemmas, arrives to the main result, Theorem 8, which says that for \(01\), every member of \(\mathcal L^{(q,\lambda)}_k\) is a.e.\ equal to some element of the Lipschitz (Hölder) \(\Lambda_\alpha\) space. Moreover, the corresponding injection \(\mathcal L^{(q,\lambda)}_k\to\Lambda_\alpha\) is a continuous mapping. This result is offered as a part of a collection of various theorems that subordinate to one leitmotiv: all describe smoothness/differentiability without derivation terms. The topics covered include approximation theory, Fourier transform, Poisson integral, and Landau's inequalities.
Laplace transform, finite differences, Taylor expansion, Lipschitz (Hölder) classes, Fourier transform, Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems, derivative, Poisson integral, approximation theory, Campanato-Morrey theory, Fourier series, 26A24
Laplace transform, finite differences, Taylor expansion, Lipschitz (Hölder) classes, Fourier transform, Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems, derivative, Poisson integral, approximation theory, Campanato-Morrey theory, Fourier series, 26A24
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