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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)
##### Keywords:
decomposition into subspaces; convex polygon
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