4_365_srivastava.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 4, 2009, (520-531) ISSN 1307-5543 – www.ejpam.com On Approximation and Generalized Type of Entire Functions of Several Complex Variables G. S. Srivastava∗ and Susheel Kumar Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, IN- DIA. Abstract. In the present paper, we study the polynomial approximation of entire functions of several complex variables. The characterizations of generalized type of entire functions of several complex variables have been obtained in terms of approximation and interpolation errors. 2000 Mathematics Subject Classifications: 30B10, 30D20, 32K05 Key Words and Phrases: Entire function, Siciak extremal function, Generalized Type, Ap- proximation errors, Interpolation errors. 1. Introduction The concept of generalized order and generalized type for entire transcendental functions was given by Seremeta [4] and Shah [5]. Hence, let L0 denote the class of ∗Corresponding author. Email address: girssfma�iitr.ernet.in (G. Srivastava) http://www.ejpam.com 520 c© 2009 EJPAM All rights reserved. G. Srivastava and S. Kumar / Eur. J. Pure Appl. Math, 2 (2009), (520-531) 521 functions h(x) satisfying the following conditions: (i) h(x) is defined on [a,∞) and is positive, strictly increasing, differentiable and tends to∞ as x →∞, (ii) lim x→∞ h[{1+ 1/ψ(x)}x] h(x) = 1 for every function ψ(x) such that ψ(x)→∞ as x →∞. Let Λ denote the class of functions h(x) satisfying conditions (i) and (iii) lim x→∞ h(cx) h(x) = 1 for every c > 0, that is h(x) is slowly increasing. For an entire transcendental function f (z) = ∑∞ n=1 bnzn, define M(r) =max |z|=r | f (z)|. For functions α(x) ∈ Λ , β(x) ∈ L0 , the generalized order of f (z) is given by ρ(α,β , f ) = lim r→∞ sup α[log M(r)] β(log r) . Further, for α(x) , β−1(x) and γ(x) ∈ L0 , generalized type of an entire transcendental function f (z) is given as σ(α,β ,ρ, f ) = lim r→∞ sup α[log M(r)] β[{γ(r)}ρ] where 0< ρ <∞ is a fixed number. Let g : C N → C , N ≥ 1 , be an entire transcendental function. For z = (z1, z2, ..., zN) ∈ C N , we put S(r, g) = sup{|g(z)| : |z1|2+ |z2|2+ ...+ |zN |2 = r2} , r > 0. Then we define the generalized order and generalized type of g(z) as ρ(α,β , g) = lim r→∞ sup α [logS(r, g)] β (log r) G. Srivastava and S. Kumar / Eur. J. Pure Appl. Math, 2 (2009), (520-531) 522 and σ(α,β ,ρ, g) = lim r→∞ sup α [logS(r, g)] β [{γ(r)}ρ] . Let K be a compact set in C N and let ||.||K denote the sup norm on K . The function ΦK(z)= sup � |p(z)|1/n : p−polynomial, deg p ≤ n, ||p||K ≤ 1, n = 1, 2, .. and z ∈ C N � , is called the Siciak extremal function of the compact set K (see [2] and [3]). Given a function f defined and bounded on K , we put for n = 1, 2, ... E1 n ( f , K) = || f − tn||K; E2 n ( f , K) = || f − ln||K; E3 n+1 ( f , K) = ||ln+1 − ln||K; where tn denotes the nth Chebyshev polynomial of the best approximation to f on K and ln denotes the nth Lagrange interpolation polynomial for f with nodes at ex- tremal points of K (see [2] and [3]). Janik [1] obtained the characterizations of order of entire functions in terms of the approximation errors defined above. Later he obtained the characterizations of the generalized order [3]. In this note we obtained the characterizations of the general- ized type. For the case N = 1 this result was obtained by Shah [5]. 2. Results We first prove a lemma. Lemma 1. Let K be a compact set in C N such that ΦK is locally bounded in C N . Set G(x , t ,ρ) = γ−1{[β−1{tα(x)}]1/ρ}. Suppose that for all t , 0< t <∞, (a) If γ(x) ∈ Λ and α(x) ∈ Λ, then d[log{G(x , t ,ρ)}] d(log x) = O(1) G. Srivastava and S. Kumar / Eur. J. Pure Appl. Math, 2 (2009), (520-531) 523 as x →∞. (b) If γ(x) ∈ (L0 −Λ) or α(x) ∈ (L0 −Λ), then lim x→∞ d[log{G(x , t ,ρ)}] d(log x) = 1 ρ . Let (pn)n∈N be a sequence of polynomials in C N such that (i) deg pn ≤ n, n ∈ N . (ii) there exists n0 ∈ N such that ||pn||K ≤ en/ρ � γ−1 ¨ � β−1 � 1 t α(n/ρ) ��1/ρ «�−n , where t = t + ǫ, for small ǫ > 0. Then ∑∞ n=0 pn is an entire function and the generalized type σ(α,β ,ρ, ∑∞ n=0 pn) of this entire function satisfies σ(α,β ,ρ, ∞ ∑ n=0 pn)≤ t provided ∑∞ n=0 pn is not a polynomial. Proof. By assumption, we have ||pn||K rn ≤ rnen/ρ � γ−1 ¨ � β−1 � 1 t α(n/ρ) ��1/ρ «�−n , n ≥ n0, r > 0. If γ(x) ∈ Λ and α(x) ∈ Λ, then by assumptions of lemma, there exists a number b > 0 such that for x > a, we have � � � � d[log{G(x , t ,ρ)}] d(log x) � � � � 0. (1) Let us write Kr = {z ∈ C N : ΦK(z) < r, r > 1}, then for every polynomial p of degree ≤ n, we have (see e.g. [3] p.323) |pn(z)| ≤ ||pn||KΦn K (z), z ∈ C N . (2) So the series ∑∞ n=0 pn is convergent in every Kr , r > 1, whence ∑∞ n=0 pn is an entire function. Put M ∗(r) = sup{||pn||K rn : n ∈ N , r > 0}. On account of 1, for every r > 0, there exists a positive integer ν(r) such that M ∗(r) = ||pν(r)||K rν(r) and M ∗(r)>||pn||K rn, n > ν(r). It is evident that ν(r) increases with r. First suppose that ν(r)→∞ as r →∞. Then putting n = ν(r) in 1 we get for sufficiently large r M ∗(r)≤ exp(bρα−1[tβ({γ(re 1/ρ+b )}ρ)]). (3) Put Fr = {z ∈ C N : ΦK(z) = r}, r > 1 G. Srivastava and S. Kumar / Eur. J. Pure Appl. Math, 2 (2009), (520-531) 525 and M(r) = sup{| ∞ ∑ n=0 pn(z)| : z ∈ Fr}, r > 1. Now following Janik ( [3] p.323), we have for some positive constant k, S r, ∞ ∑ n=0 pn ! ≤ M(kr) ≤ 2M ∗(2kr). (4) Combining 3 and 4, we get S r, ∞ ∑ n=0 pn ! ≤ 2exp(bρα−1[tβ({γ(2kre 1/ρ+b )}ρ)]) or α � 1 bρ log ¦ 1 2 S(r, ∑∞ n=0 pn) © � β({γ(2kre 1/ρ+b )}ρ) ≤ t. Since α(x) and γ(x) ∈ Λ, we get on using (iii), lim sup r→∞ α � logS(r, ∑∞ n=0 pn) � β({γ(r)}ρ) ≤ t. (5) Now let α(x) ∈ (L0−Λ) or γ(x) ∈ (L0−Λ), then by the assumption of the lemma and as in [4], we have log r + o(1) = log F(x/ρ, 1/t,ρ). Hence we obtain r{1+ o(1)} = F(x/ρ, 1/t,ρ). As in [4], the maximum of the function φ(x) in this case is attained for x∗(r) = ρα−1[tβ([γ(r{1+ o(1)})]ρ)]. Further, ||pn||K rn ≤ exp({1+ o(1)}α−1[tβ([γ(r{1+ o(1)})]ρ)]), n ≥ n0, r > 0 G. Srivastava and S. Kumar / Eur. J. Pure Appl. Math, 2 (2009), (520-531) 526 and in this case we have S r, ∞ ∑ n=0 pn ! ≤ 2exp � {1+ o(1)}α−1[tβ({γ(2kr{1+ o(1)})}ρ)] � or α � {1+ o(1)}−1 log ¦ 1 2 S(r, ∑∞ n=0 pn) ©� β({γ(2kr{1+ o(1)})}ρ) ≤ t. Using the properties of the functions α,β and γ and proceeding to limits we again obtain 5. Since t = t + ǫ,ǫ > 0 being arbitrarily, we finally get σ(α,β ,ρ, ∞ ∑ n=0 pn) ≤ t . In the case when ν(r) is bounded then M ∗(r) is also bounded, whence ∑∞ n=0 pn re- duces to a polynomial. Hence the Lemma is proved. Now we give our main result. Theorem 1. Let K be a compact set in C N such that ΦK is locally bounded in C N . Set F(x , t ,ρ) = γ−1{[β−1{tα(x)}]1/ρ}. Suppose that for all t , 0< t <∞, (a) If γ(x) ∈ Λ and α(x) ∈ Λ, then d[log(F(x , t ,ρ)) d(log x) = O(1) as x →∞. (b) If γ(x) ∈ (L0 −Λ) or α(x) ∈ (L0 −Λ), then lim x→∞ d[log(F(x , t ,ρ)) d(log x) = 1 ρ . Then the function f , defined and bounded on K , is the restriction of an entire function g of the generalized type σ(α,β ,ρ, g) if and only if σ(α,β ,ρ, g) = lim n→∞ sup α(n/ρ) β{[γ(e1/ρ[Es n ( f , K)]−1/n)]ρ} ; s = 1, 2, 3. G. Srivastava and S. Kumar / Eur. J. Pure Appl. Math, 2 (2009), (520-531) 527 Proof. First we assume that f has an entire function extension g which is of generalized type σ = σ(α,β ,ρ, g). We write ηs = lim n→∞ sup α(n/ρ) β{[γ(e1/ρ[Es n ]−1/n)]ρ} ; s = 1, 2, 3. Here Es n stands for Es n � g|K , K � , s = 1, 2, 3. We show that σ = ηs, s = 1, 2, 3. It is known (see e.g. [6]) that E1 n ≤ E2 n ≤ (n∗ + 2)E1 n , n ≥ 0, (6) E3 n ≤ 2(n∗ + 2)E1 n−1 , n≥ 1, (7) where n∗ =    n+ N n    . Using Stirling formula for the approximate value of n! ≈ e−nnn+1/2 p 2π, we get n∗ ≈ nN N ! for all large values of n. Hence for all large values of n, we have E1 n ≤ E2 n ≤ nN N ! [1+ o(1)]E1 n and E3 n ≤ 2 nN N ! [1+ o(1)]E1 n . Thus η3 ≤ η2 = η1 and it suffices to prove that η1 ≤ σ ≤ η3. First we prove that η1 ≤ σ. Using the definition of the generalized type , for ǫ > 0 and r > r0(ǫ), we have S(r, g)≤ exp[α−1{σβ({γ(r)}ρ)}], where σ = σ+ ǫ provided r is sufficiently large. Without loss of generality, we may suppose that K ⊂ B = {z ∈ C N : |z1|2+ |z2|2+ ...+ |zN |2 ≤ 1}. Then E1 n ≤ E1 n (g, B). G. Srivastava and S. Kumar / Eur. J. Pure Appl. Math, 2 (2009), (520-531) 528 Now following Janik ( [3] p.324), we get E1 n (g, B) ≤ r−nS(r, g), r ≥ 2, n ≥ 0 or E1 n ≤ r−n exp[α−1{σβ({γ(r)}ρ)}]. Putting r = r(n) = F(n/ρ, 1/σ,ρ) = γ−1 ¨ � β−1 � 1 σ α(n/ρ) ��1/ρ « , we get E1 n ≤ en/ρ � γ−1 ¨ � β−1 � 1 σ α(n/ρ) ��1/ρ «�−n or [E1 n ]−1/n ≥ e−1/ργ−1 ¨ � β−1 � 1 σ α(n/ρ) ��1/ρ « or α(n/ρ) β{[γ(e1/ρ[E1 n ]−1/n)]ρ} ≤ σ. Taking limits as n→∞, we get lim n→∞ sup α(n/ρ) β{[γ(e1/ρ[E1 n ]−1/n)]ρ} ≤ σ. Since ǫ > 0 is arbitrarily small, therefore finally we get η1 ≤ σ. Now we will prove that σ ≤ η3. Suppose that η3 < σ. Then for every λ,η3 < λ < σ, α(n/ρ) β{[γ(e1/ρ[E3 n ]−1/n)]ρ} ≤ λ provided n is sufficiently large. Thus E3 n ≤ en/ρ � γ−1 ¨ � β−1 � 1 λ α(n/ρ) ��1/ρ «�−n . G. Srivastava and S. Kumar / Eur. J. Pure Appl. Math, 2 (2009), (520-531) 529 Also by previous lemma, σ ≤ λ, where σ = σ(α,β ,ρ, g) is the generalized type of g(z) as dfefined on page 2. Since λ has been chosen less than σ, we get a contradiction. Hence σ ≤ η3. Now let f be a function defined and bounded on K and such that for s = 1, 2, 3, ηs = lim n→∞ sup α(n/ρ) β{[γ(e1/ρ[Es n ]−1/n)]ρ} . So for every λ1 > ηs and for sufficiently large n, we have α(n/ρ) β{[γ(e1/ρ[Es n ]−1/n)]ρ} ≤ λ1 or Es n ≤ en/ρ � γ−1 ¨ � β−1 � 1 λ1 α(n/ρ) ��1/ρ «�−n . Proceeding to limits as n→∞, we get lim n→∞[E s n ]1/n ≤ 0. Also it is obvious that lim n→∞ [Es n ]1/n ≥ 0. Hence finally we get lim n→∞[E s n ]1/n = 0. So following Janik (see [1], Prop. 3.1), we claim that the function f can be continu- ously extended to an entire function . Let us put g = l0 + ∞ ∑ n=1 (ln − ln−1), REFERENCES 530 where {ln} is the sequence of Lagrange interpolation polynomials of f as defined earlier. Now we claim that g is the required continuation of f and σ(α,β ,ρ, g) = ηs. For every λ1 > η3 and for sufficiently large n, we have E3 n ≤ en/ρ � γ−1 ¨ � β−1 � 1 λ1 α(n/ρ) ��1/ρ «�−n or ||ln− ln−1|| ≤ en/ρ � γ−1 ¨ � β−1 � 1 λ1 α(n/ρ) ��1/ρ «�−n . So using the Lemma 1, we get σ(α,β ,ρ, g) ≤ λ1. Since λ1 > η3 is arbitrary, so finally we get σ(α,β ,ρ, g) ≤ η3. Using the inequalities 6, 7 and the proof of first part given above, we haveσ(α,β ,ρ, g) = ηs, as claimed. This completes the proof of the Theorem. ACKNOWLEDGEMENTS The authors are very thankful to the referee for his valu- able comments and observations which helped in improving the paper. References [1] Adam Janik, On approximation and interpolation of entire functions, Zeszyty Naukowe Universtytetu Jagiellonskiego, 22 (1981), 173-188. [2] Adam Janik, A characterization of the growth of analytic functions by means of polyno- mial approximation, Univ. Iagel. Acta Math., 24 (1984), 295-319. [3] Adam Janik, On approximation of entire functions and generalized order, Univ. Iagel. Acta Math., 24 (1984), 321-326. REFERENCES 531 [4] M. N. Seremeta, On the connection between the growth of the maximum modulus of an entire function and the moduli of the coefficients of its power series expansion, Amer. Math. Soc. Transl., 88 (2) (1970), 291-301. [5] S. M. Shah, Polynomial approximation of an entire function and generalized order, J. Approx. Theory, 19 (1977), 315-324. [6] T. Winiarski, Application of approximation and interpolation methods to the examina- tion of entire functions of n complex variables, Ann. Pol. Math, 28 (1973), 97-121.