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Norm approximation by Taylor polynomials in Hardy and Bergman spaces. (English) Zbl 1475.32006

Let \(0<p< \infty\) and \(\alpha \in \mathbb R.\) Fix a non-negative integer \(k\) such that \(pk + \alpha > -1.\) Generalizing the standard weighted Bergman spaces the authors introduce the space \(A^p_\alpha\) as the space of holomorphic functions \(f\) on the unit ball \(\mathbb B_n\) such that \((1-|z|^2)^k R^kf(z) \in L^p(\mathbb B_n, dv_\alpha),\) where \(Rf(z)= z_1 \frac{\partial f}{\partial z_1} + \dots + z_n \frac{\partial f}{\partial z_n}\) and \(dv_\alpha (z) = (1-|z|^2)^\alpha \, dv(z).\) They endow \(A^p_\alpha\) with the “norm” \(\|f\|^p_{p,\alpha} = |f(0)|^p + \int_{\mathbb B_n} \left | (1-|z|^2)^k R^kf(z) \right |^p\, dv_\alpha (z),\) indicating that for \(0<p<1\) one has only an F-space. Consider the \(N\)th Taylor polynomial \(S_N f\) of \(f\) and let \(0<p<\infty\) and \(\alpha \in \mathbb R.\) It is shown that \(\|S_Nf -f \|_{p,\alpha} \to 0 \) as \(N\to \infty\) for every \(f\in A^p_\alpha\) if and only if \(p>1,\) and \(\|S_Nf -f \|_{H^p} \to 0\) as \(N\to \infty\) for every \(f\) in the Hardy space \(H^p\) if and only if \(p>1.\) In addition, they prove that for \(\beta > \alpha\) and \(f\in A^1_\alpha\) one has \(\|S_Nf -f \|_{1,\beta} \to 0 \) as \(N\to \infty.\) The last part of the paper is devoted to interesting results in the case \(0<p<1.\)

MSC:

32A36 Bergman spaces of functions in several complex variables
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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