Let \(f(z) = \sum\limits_{k=0}^{\infty} f_k z^k\) be an entire transcendental function, and let \((\lambda_n)\) be a sequence of positive numbers increasing to \(+\infty\). Suppose that the series \(A(z) = \sum\limits_{n=1}^{\infty} a_n f(\lambda_n z)\) is regularly convergent in \(\mathbb{D} = \{ z : |z| < 1 \}\), i.e., \(\mathfrak{M}(r, A) := \sum\limits_{n=1}^{\infty} |a_n| M_f(r \lambda_n) < +\infty\) for all \(r \in [0, 1)\). For a positive function \(l\) continuous on \([0, 1)\), the function \(A\) is said to be of bounded \(l\)–\(\mathfrak{M}\)-index if there exists \(N \in \mathbb{Z}_+\) such that \[\frac{\mathfrak{M}(r, A^{(n)})}{n! \, l^n(r)} \leq \max \left\{ \frac{\mathfrak{M}(r, A^{(k)})}{k! \, l^k(r)} : 0 \leq k \leq N \right\},\] for all \(n \in \mathbb{Z}_+\) and all \(r \in [0, 1)\). The growth of bounded \(l\)–\(\mathfrak{M}\)-index functions is studied. In particular, under the conditions \(a_n \geq 0\) and \(f_k \geq 0\), it is proved that the function \(A\) is of bounded \(l\)–\(\mathfrak{M}\)-index with \(l(r) = p(1 – r)^{-(p+1)}\), \(p > 0\), if and only if \[\lim\limits_{r \uparrow 1} (1 – r)^p \ln \mathfrak{M}(r, A) < +\infty.\] This condition is satisfied if and only if \[\lim\limits_{k \to \infty} k^{-p} \left( \ln^+ (f_k \mu_D(k)) \right)^{p+1} < +\infty,\] where \(\mu_D(k) = \max\{ a_n \lambda_n^k : n \geq 1 \}\).
We introduce Littlewood Paley functions defined in terms of a reparameterization of the Ornstein-Uhlenbeck semigroup obtaining that these operators are bounded in \(L^p\), \(1<p<\infty\), with respect to the unidimensional gaussian measure, by means of singular integrals theory. In addition, we study the Abel summability of the Fourier Hermite expansions considering their pointwise convergence and their convergence in the \(L^p\) sense, obtaining a version of Tauber’s theorem.
