We construct a sequence of homogeneous polynomials on the unit ball $𝔹_d$ in $ℂ^d$ which are big at each point of the unit sphere 𝕊. As an application we construct a holomorphic function on $𝔹_d$ which is not integrable with any power on the intersection of $𝔹_d$ with any complex subspace.
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We show that in $L_p$ for p ≠ 2 the constants of equivalence between finite initial segments of the Walsh and trigonometric systems have power type growth. We also show that the Riemann ideal norms connected with those systems have power type growth.
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It follows from our earlier results [Israel J. Math., to appear] that in the Gurariy space G every finite-dimensional smooth subspace is contained in a bigger smooth subspace. We show that this property does not characterise the Gurariy space among Lindenstrauss spaces and we provide various examples to show that C(K) spaces do not have this property.
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