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Projection from the spaces \(E^ p\) on a convex polygon onto subspaces of periodic functions. (English. Russian original) Zbl 0722.30021
Math. USSR, Izv. 33, No. 2, 373-390 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 5, 1051-1069 (1988).
Let \(1<p<\infty\) and let \(H^ p_+\) be the subspace of \(L^ p(- \pi,\pi)\) generated with functions \(e^{inw}\), \(n>0\), i.e. the space of trigonometric series \(\sum^{\infty}_{n=1}c_ ne^{inw}\) convergent in \(L^ p(-\pi,\pi)\). Put \(H^ p_ -=\{f:\) \(f(-w)\in H^ p_+\}\). The well known M. Riesz theorem on boundedness in \(L^ p(-\pi,\pi)\) of the projection \(\sum^{\infty}_{n=-\infty}c_ ne^{inw}\mapsto \sum^{\infty}_{n=1}c_ ne^{inw}\) can be reformulated as an assertion on a decomposition of \(L^ p(-\pi,\pi)\) into the direct sum \({\mathbb{C}}\oplus H^ p_+\oplus H^ p_ -\). The spaces \(H^ p_+\) can be equivalently defined as the space of traces on the real line of \(2\pi\)-periodic functions f analytic in the half-plane Im w\(>0\) and such that \(\sup_{v>0}\int^{\pi}_{-\pi}| f(u+iv)|^ pdu<\infty\) and f(w)\(\to 0\) as \(w\to \infty\) uniformly in the stripe \(| Re w| \leq \pi\), Im w\(>0.\)
Now, let D be a convex polygon in \({\mathbb{C}}\) with vertices \(a_ 1,...,a_ m\), \(a_{m+1}=a_ 1\), \(m\geq 3\), numbered in the positive direction. Let \(P_ k\) be the half-plane given by the line \((a_ k,a_{k+1})\) and containing D. The space \(H^ p(P_ k)\) is obtained from \(H^ p_+\) by means of the linear transform \(z=w(a_{k+1}-a_ k)/(2\pi)+(a_{k+1}+a_ k)/2\) of the half-plane Im w\(>0\) onto \(P_ k\). The Hardy space \(E^ p(D)\) consists of all functions f analytic in D such that sup\(\int_{\gamma_ g}| f(z)|^ p| dz| <\infty\) for some sequence of closed rectifiable Jordan curves \(\gamma_ n\subset D\) tending to \(\partial D\). The author proves an analogue of the M. Riesz theorem for the spaces \(E^ p(D):\) Let m be an odd natural number. Put \(s=(m-3)/2\) if \(1<p\leq 2\), \(s=(m-1)/2\) if \(p>2\), and denote \(Q_ s\) the space of polynomials of degrees not greater than s. If \(p\in (1,2)\cup (2,\infty)\), then \(E^ p(D)=Q_ s\oplus H^ p(P_ 1)\oplus...\oplus H^ p(P_ m)\). If \(p=2\), then such decomposition does not exist. The case of m even (with \(s=m/2-1)\) was solved by the author in Izv. Akad Nauk SSSR, Ser. Mat. 42, 1101-1119 (1978; Zbl 0412.30034).
Reviewer: J.Rákosnik
MSC:
30D55 \(H^p\)-classes (MSC2000)
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