Best Polynomial Approximation in L-Norm and (p, q)-Growth of Entire Functions

and Applied Analysis 3 which vanishes on K except perhaps for a pluripolar subset and satisfies the complex Monge-Ampère equation (see [12]): (dd c V K ) n = 0 on Cn \ K. (16) If n = 1, the Monge-Ampère equation reduces to the classical Laplace equation. For this reason, the functionV K is considered as a natural counterpart of the classical Green function with logarithmic pole at infinity and it is called the pluricomplex Green function associated with K. Definition 1 (Siciak [10]). The function V K = sup { 1 d log 󵄨󵄨󵄨󵄨Pd 󵄨󵄨󵄨󵄨 , Pd polynomial of degree ≤ d, 󵄩󵄩󵄩󵄩Pd 󵄩󵄩󵄩󵄩K ≤ 1} (17) is called the Siciak’s extremal function of the compactK. Definition 2. A compactK in Cn is said to be L-regular if the extremal function, V K , associated to K is continuous on C. Regularity is equivalent to the following BernsteinMarkov inequality (see [9]). For any ε > 0, there exists an open U ⊃ K such that for any polynomial P ‖P‖ U ≤ e ε⋅deg(P) ‖P‖ K . (18) In this case we take U = {z ∈ Cn; V K (z) < ε}. Regularity also arises in polynomial approximation. For f ∈ C(K), we let ε k (K, f) = inf {󵄩󵄩󵄩󵄩f − P 󵄩󵄩󵄩󵄩K, P ∈ Pk (C n )} , (19) where P k (Cn) is the set of polynomials of degree at most d. Siciak showed that (see [10]). If K is L-regular, then lim sup k→+∞ (ε k (K, f)) 1/k = 1 R < 1 (20) if and only if f has an analytic continuation to {z ∈ Cn; V K (z) < log( 1 R )} . (21) It is known that if K is a compact L-regular of C, there exists a measure μ, called extremal measure, having interesting properties (see [9, 10]), in particular, we have the following properties. (P 1 ) Bernstein-Markov inequality: for all ε > 0, there exists a constant C = C ε such that (BM) : 󵄩󵄩󵄩󵄩Pd 󵄩󵄩󵄩󵄩K = C(1 + ε) sk 󵄩󵄩󵄩󵄩Pd 󵄩󵄩󵄩󵄩L2(K,μ), (22) for every polynomial of n complex variables of degree at most d. (P 2 ) Bernstein-Walsh (BW) inequality: for every set L-regular K and every real r > 1 we have 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩K ≤ Mr deg(f) (∫ K 󵄨󵄨󵄨󵄨f 󵄨󵄨󵄨󵄨 p dμ) 1/p