Some properties of interpolating basic functions in the unit disk (Q1108402)

Let \(L^ p(m)\) denote the set of all functions f measurable on the unit circle \(| z| =1\) and such that \[ (1)\quad \int^{2\pi}_{0}| f(e^{it})|^ p dm(t)<\infty,\quad p>0, \] where m is an increasing function such that \(\int^{2\pi}_{0}\ln m'(t)dt>-\infty.\) Let B(z) denote the Blaschke product over a sequence \(a_ k\), \(| a_ k| <1\), which can have repetitions (multiplicity). The author studies the question of approximation with respect to (1) by linear combinations of interpolating functions \(\Omega_ k\) with nodes at \(a_ k\). When the sequence has no repetition, \(\Omega_ k(z)=B(z)/B'(a_ k)(z-a_ k)\) and a more complicated expression when there are repetitions. If a function f allows such approximation, then necessarily f is the boundary function on \(| z| =1\) of two functions, one holomorphic in \(| z| <1\), the other meromorphic in \(| z| >1\) with poles at \(1/\bar a_ k\), \(k=1,2,... \). These functions are described and this gives rise to a subspace \(A\subset L^ p(m)\). When \(m(t)=t\) and \(p\geq 1\), the condition is also sufficient. Under a separation condition imposed upon the sequence \(a_ k\) and \(p>1\), the author proves that the system \(\Omega_ k\) form\(\{g(f^ n)\}\) is a linearly independent sequence of functions. The proofs follow from a consideration of the rate of growth of the functions in question.