Geodesic
A Method for Summing Fourier Integrals for Functions from $H^p(E_{2n}^+)$, $0$
Matematičeskie zametki, Tome 82 (2007) no. 5, pp. 718-728.

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Suppose that $H^p(E^+_{2n})$ is the Hardy space for the first octant $$ E_{2n}^+=\{z\in\mathbb C^n:\operatorname{Im}z_j>0,\,j=1,\dots,n\} $$ and $P^l_\varepsilon(f,x)$, $l>0$, is the generalized Abel–Poisson means of a function $f\in H^p(E^+_{2n})$. In this paper, we prove the inequalities $$ C_1(l,p)\widetilde\omega_l(\varepsilon,f)_p \le\|f(x)-P^l_\varepsilon(f,x)\|_p \le C_2(l,p)\omega_l(\varepsilon,f)_p, $$ where $\widetilde\omega_l(\varepsilon,f)_p$ and $\omega_l(\varepsilon,f)_p$ are the integral moduli of continuity of $l$th order. For $n=1$ and an integer $l$, this result was obtained by Soljanik.
Keywords: Fourier integral, Hardy space, generalized Abel–Poisson mean, modulus of continuity, holomorphic function. 
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     publisher = {mathdoc},
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     number = {5},
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S. G. Pribegin. A Method for Summing Fourier Integrals for Functions from $H^p(E_{2n}^+)$, $0

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