Uniform polynomial approximation and generalized growth of entire functions on arbitrary compact sets (Q2389169)

Let \(K\) be a compact subset of the complex plane, and let \(u_1, u_2, \dots , u_n \in K\). Set \[ V(u_1,u_2,\dots, u_n) = \prod_{k,l\;(k < l)}(u_k - u_l) \] and \[ V_n = \max\big\{|V(u_1,u_2,\dots, u_n)|: u_j \in K, 1 \leq j \leq n \big\}. \] For a continuous function \(f\) on \(K\), the best uniform approximation by polynomials is defined by \[ E_n : = E_n(f, K) = \inf\big \{ \max_{z \in K} |f(z) - P_n(z)| : \deg P_n \leq n \big\} . \] If \(\text{card}\,K = \infty\), let \(\mu_n = z^n + a_{n-1}z^{n-1}+ \dots a_1z + a_n \) be the Chebyshev polynomial for \(K\) such that all zeros of \(\mu_n\) belong to \(K\), and let \(m_n^* = \max\{|\mu_n(z)| : z \in K \}\). The author considers the classes \(L^0\) and \(\Lambda\) of functions defined as follows. Let \(\varphi : [a,\infty) \to \mathbb{R}\) be a real valued function which is positive, differentiable in each point, strictly increasing and satisfying \(\lim_{x \to \infty} \varphi(x) = \infty\). Such a function belongs to the class \(L^0\) if \[ \lim_{x \to \infty}\frac{\varphi[(1 + \eta(x))x]}{\varphi(x)} = 1\tag{1} \] for every real function \(\eta(x)\) such that \(\eta(x) \to 0\). The function \(\varphi(x) \) is said to belong to the class \(\Lambda\) if \(\varphi\in L^0\) and, instead of (1), satisfies the stronger condition \[ \lim_{x \to \infty} \frac{\varphi(cx)}{\varphi(x)} = 1 \] for all \(c\), \(0 < c < \infty\). For an entire function \(f(z)\), its maximum modulus function is \(M(r, f) = \max_{|z| = r} |f(z)|\). Now, given functions \(\alpha (t) \in L^0 \) and \(\beta(t) \in \Lambda\), the generalized order is defined as \[ \rho(\alpha, \beta, f) : = \limsup_{r \to \infty} \frac{\alpha \left(\log M(r,f)\right)}{\beta (\log r)} , \] while the generalized lower order is defined by \[ \lambda(\alpha, \beta, f) : = \liminf_{r \to \infty} \frac{\alpha \left(\log M(r,f)\right)}{\beta (\log r)} . \] Theorem 1 in the paper states that, if \(K\) is a compact set with \(\text{card}\, K = \infty\), \(\alpha \in \Lambda\), and \(\beta \in \L^0\), then the generalized order \(\rho(\alpha, \beta, f) \) of an entire function \(f\) is \[ \limsup_{n \to \infty} \frac{\alpha(n)}{\beta\left[ \log\left( \frac{E_n(f,K)}{m^*_{n +1}}\right)^{-1/n} \right]}. \] Now, if \(\alpha (x), \beta(x), \gamma(x)\) belong to the class \(L^0\) and \(0 < \sigma, \rho < \infty\), then let \(F(x,\sigma, \rho) : = \gamma^{-1}((\beta^{-1}(\sigma \alpha(x)))^{1/\rho})\). Suppose the function \(F\) satisfies the following two conditions: (a) \(\gamma(x), \alpha(x) \in \Lambda\) and \( \frac{d F(x, \sigma, \rho)}{d \log x} = O(1)\); (b) if \(\gamma(x) \in L^0 \setminus \Lambda\) or \(\alpha(x) \in L^0 \setminus \Lambda\), then \(\lim_{x \to \infty} \frac{d \log F(x, \sigma, \rho)}{d \log x} = 1/\rho\). In this setting, Theorem 2 states that an entire function \(f\) is of generalized type \(\sigma\) if and only if \[ \limsup_{n \to \infty} \frac{\alpha(n/\rho)}{\beta\left( \left(\gamma\left( e^{1/\rho}\left(\frac{E_n(f,K)}{m^*_{n+1}} \right)^{-1/n} \right) \right)^{\rho} \right)} = \sigma . \] It would be desirable to find in the paper some examples and further comments illustrating these generalized orders and their properties in terms of the parameter functions.