Norms of lacunary polynomials in functional spaces (Q1311109)

Let \(\Lambda\) be the space of functions \(f\) of the form \(f(x)= a_ 0+ \sum^ \infty_{k=1} a_ k\cos(n_ k x+ \psi_ k)\), where \(a_ k\in\mathbb{R}\), \(n_ k\in\mathbb{R}\), \(\psi_ k\in\mathbb{R}\) and \(n_{k+1}/n_ k\geq \lambda>1\) for some \(\lambda\), \(k=1,2,\dots,\) \((a_ k)\in l_ 2\). It is stated that \(\Lambda\subset\text{BMO}\) and the norm \(\| f\|_{\text{BMO}}\) is equivalent to the norm \((a_ k)_{l_ 2}\) for \(f\in\Lambda\). This result is extended to some symmetric spaces.