ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE FUNCTIONS ON ALGEBRAIC VARIETIES

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1 International Journal of Analysis and Applications ISSN Volume 12, Number 1 (2016), ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE FUNCTIONS ON ALGEBRAIC VARIETIES DEVENDRA UMAR Abstract Let X be a complete intersection algebraic variety of codimension m > 1 in C m+n In this paper we characterized the classical growth parameters der and type f transcendental entire functions f (X), the space of holomphic functions on the complete intersection algebraic variety X, in terms of the best polynomial approximation err in L p -nm, 0 < p, on a L regular non-pluripolar compact subset of C m+n 1 Introduction The growth of transcendental entire functions in one complex variable case is well represented in the wk of BJa Levin [11] and Boas [2] In several complex variables the standard reference is the wk of PLelong and LGruman [10] and Ronkin s book [14] Einstein-Matthews and asana [3] studied the growth parameters (p, q) der and (p, q) type introduced by Juneja et al([6],[7]) of transcendental entire functions f : C n C Einstein-Matthews and Clement Lutterodt [4] extended the results studied in [3] to transcendental entire functions f : X C, defined on a complete intersection algebraic variety X in C m+n of codimension m > 1, and obtained the growth parameters in terms of the sequence of extremal polynomials occurring in the development of f It has been noticed that the growth parameters of f : X C in terms of approximation errs is not studied so far The aim of this paper is to bridge this gap and to study the results obtained in [4] in terms of the best approximation errs in L p -nm, 0 < p AR Reddy ([13],[14]) characterized the growth parameters in terms of approximation errs f a function continuous on [-1,1] T Winiarski ([21],[22]) studied the growth of entire functions in terms of Lagrange polynomial approximation errs with respect to sup nm on a compact subset (positive capacity) of C and C n, n > 1 asana and umar [8] generalized the results of Winiarski [22] by using the concept of index-pair (p, q) Adam Janik [5] characterized the generalized der of entire functions by means of polynomial approximation and interpolation on compact subsets of C n, using the Siciak extremal function ([18],[19]) In [5] Adam Janik extended the results of SM Shah [17] in the case n = 1, = [ 1, 1] and Winiarski [22] Srivastava and umar [20] extended and improved the results of Adam Janik [5] But our wk is different from all these auths The text has been divided into four parts Section 1 consists of an introducty exposition of the topic and Section 2 contains some definitions and notations In Section 3, we have given Zeriahi s Bernstein-Markov type inequality with two lemmas in which first one is due to Zeriahi extending the classical Cauchy inequality and second is concerned with a sequence of extremal polynomials Finally, in Section 4, we prove two theems f a transcendental entire function f (C m+n ), the space of holomphic functions on the complete intersection algebraic variety X and studied the growth parameters der and type in terms of L p -approximation err on a L regular non-pluripolar compact subset of C m+n 2010 Mathematics Subject Classification 41A20, 30E10 ey wds and phrases algebraic variety; approximation err; extremal polynomials and plurisubharmonic functions c 2016 Auths retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License 22

2 ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE FUNCTIONS 23 2 Definitions and Notations Following the definition of Einstein and asana [3], we have Let v : C m+n R + := r R : r > 0 be a real-valued function such that the following properties hold: (i) v(z + w) v(z) + v(w) : z, w C m+n, (ii)v(bz) = b v(z) : z C m+n, b C, (iii) v(z) = 0 z = 0 Here v is a nm on C m+n and it exhausts the complex space C m+n by a family of sublevel sets {Ω c } c 1 which are defined by Ω c = {z C m+n : v(z) c, c R} Let ϕ : C m+n R + Define M ϕ,v (r) = sup v(z) r ϕ(z), the maximum modulus of ϕ with respect to the nm v f each r R + We say that the transcendental entire function f : C m+n C is of der ρ, if log f is of der ρ, where (21) ρ = lim sup log M f,v (r) log r If ρ < +, f is said to have maximal, nmal minimal type if (22) σ = lim sup is infinite, finite zero M f,v (r) r ρ, Let be a compact subset of C m+n, which is nonpluripolar on each irreducible component of a complete intersection variety X The Siciak extremal function V associated to has been studied extensibly by Siciak [18] and Sadullaev ([15],[16]) and is defined as: where the subcone (X) is given by V = sup{u(z) : u (X); u(ζ) 0, ζ, z X} (X) = {u(z) : u P SH(X); u(z) log( z +1) + C u, z X} here C u is a constant depending only on the cone of plurisubharmonic function (P SH)u and is the Euclidean nm on C m+n The upper semi-continuous regularization of V is defined on X by V(z) = lim sup V (ζ), ζ, z X, ζ z V (z) is P SH(X) and satisfies V(z) log( z +1) + O(1), as z + If V is continuous on C m+n, then V = V It is given in [18] that if, f all z, V is continuous, then V is continuous on X In this case we say that is L regular in X We define the sublevel sets of the extremal function V by setting Ω α = {z X : V (z) α}, α > 1, α R, and sublevel sets of the upper semi-continuous regularization V of V by Ω r = {z C m+n : expv (z) r}, r > 1 It has been observe that the sequence of sublevel sets {Ω r } r>1 exhausts the complex space C m+n F f : C m+n C a transcendental entire function, set M,f (r) = sup z Ω r f(z), r > 1 It can be easily shown that log + M,f (r) and log + M,v (r) give the same der given by (23) ρ ρ(f) = lim sup log log + M,f (r) log r

3 24 DEVENDRA UMAR If 0 < ρ < +, the type of f : C m+n C is defined by (24) σ σ(f) = lim sup log + M,f (r) r ρ Zeriahi [23] constructed an thogonal polynomial basis {A k } k 1 f the space (X) The basis is thogonal in the Hilbert space L 2 (X, µ), essentially by means of the Hilbert-Schmidt process, here µ be the extremal capacity measure on given by µ = (dd c V Further details on this positive Bel measure µ suppted on can be obtained from the paper of E Bedfd and BA Tayl [1] Let P d (C m+n ) denote the C-vect space of polynomials π d : C m+n C of degree d f d 1 Let L 2 P (, µ) denote the closed subspace of the Hilbert space L2 (, µ) generated by the restriction to of polynomials π d P d (C m+n ) of degree (π d ) d, f d 1 Then every function f L 2 P (, µ) has a power series expansion of the fm (25) f = Σ k 1 f k A k, with 1 f k = 2 k () fa k dµ, k () = ( A k 2 dµ) 1 2, k 1, here is the dot product of vects Let L p (, µ), p 1 denote the class of all functions such that f L p (,µ)= ( f p dµ) 1 p <, then we define the best polynomial approximation err in L p -nm, p 1, by (26) E p d (, f) = inf{ f π d L p (,µ), π d P d (C m+n )} If the extremal function V associated with is continuous f every z, then V is continuous on X and L regular, so instead of defining sublevel sets f the upper semi-continuous regularization, we define the same f V by setting Then we have where Ω r = {z X : V (z) < log r, r R, r > 1} V (z) 1 log( A k a k () ), a k () = max z A k(z), A k Ωr a k ()r, = degree(a k ) Following the Siciak [18] we observe that if is L regular then lim sup(e p d (, f)) 1 1 d = d R < 1 if and only if f has an analytic continuous to {z C m+n ; V (z) < log( 1 R )} 3 Auxiliary Results In this section we shall state some preliminary results which will be used in the sequel First we state Zeriahi s Bernstein-Markov type inequality [23]: BM:F all ɛ > 0, there exists a constant C ɛ > 0 such that (31) sup f(z) C ɛ (1 + ɛ) deg(f) ( f 2 dµ) 1 2 z f every holomphic function f with polynomial growth on the complete intersection algebraic variety X and is a nonpluripolar compact subset of X Now we state the following lemmas of Zeriahi extending the classical Cauchy inequality

4 ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE FUNCTIONS 25 Lemma 31 Let f = Σ k 0 f k A k be a holomphic function on X Then f every θ > 1, there exist an integer N θ and a constant C θ > 0 such that (32) f k r k () C θ (r + 1) N θ (r 1) 2n 1 f k Ωrθ, f every r > 1, k 1, where C θ and N θ are independent of r, k and f Lemma 32 If is an L regular, then the sequence of extremal polynomials {A k } k 1 satisfies (33) lim ( A k(z) ) 1 k ν k f every z C m+n and = exp(v (z)), ν k = A k L 2 (,µ), (34) lim ( A k(z) ) 1 = 1 k ν k 4 Main Results In this section we shall prove our main theems Meover, we shall characterize the classical growth parameters der and types of transcendental entire function in terms of L p -approximation err defined by (26) Theem 41If f : X C is a transcendental entire function on X with a series expansion (25) with respect to the thogonal polynomial basis {A k } k 1, then f L p (, µ), 1 < p is of finite der if and only if (41) ρ = lim sup k log log(e p (, f)) < +, and ρ = ρ 1, where E p (, f) is defined by (26) Proof First we have to prove that ρ ρ 1 If ρ 1 =, then nothing to be prove Assume that ρ 1 < and let ɛ > 0 F a sufficiently large k, from (41) we have 0 log log(e p (, f)) ρ 1 + ɛ (42) E p (, f) ( ) ρ 1 +ɛ Since adding a polynomial will not change the der of a function Thus, f r 1 and a 0 () = 0, we can assume that following inequality holds f every k 0, (43) M,f (r) Σ k 1 f k a k ()r Now we will proceed the proof in two steps (p 2) and (1 < p < 2) Let f = Σ k 0 f k A k be an element of L p (, µ) Step 1 If f L p (, µ) with p 2, then f = Σ k=0 f ka k with convergence in L 2 (, µ), It gives f k = 1 νk 2 fa k dµ, k 1, ν k k (), = 1 νk 2 (f P sk 1)A k dµ f k 1 νk 2 (f P sk 1) A k dµ,

5 26 DEVENDRA UMAR now using Bernstein W alsh inequality and Hölder s inequality we have f any ɛ > 0 (44) f k ν k C ɛ (1 + ɛ) E p 1 (, f), k 0 Step 2 If 1 p < 2, let p such that 1 p + 1 p = 1 then p 2 By Hölder s inequality we get f k νk 2 f P sk 1 L p (,µ) A k L p (,µ) But A k L p (,µ) C A k = Ca k (), now by Bernstein Markov inequality we have f k νk 2 CC ɛ (1 + ɛ) f P sk 1 L p (,µ), it gives (45) f k νk 2 C ɛ(1 + ɛ) Es p k (, f) From (44) and (45), we get f p 1 (46) f k ν k A ɛ (1 + ɛ) Es p k (, f), where A ɛ is a constant depends only on ɛ Now using Zeriahi sbernstein Markov type inequality in (43) and (46), we obtain M,f (r) Σ k 0 f k C ɛ (1 + ɛ) ν k r Σ k 0 A ɛ C ɛ (1 + ɛ) 2 Es p k (, f)r, using inequality (42) in above, we get where and M,f (r) C ɛσ k 0 (1 + ɛ) 2 ( ) ρ 1 +ɛ r = Σ 1 + Σ 2, Σ 1 = Σ 1 k (2r(1+ɛ) 2 ) (ρ 1 +ɛ)(1 + ɛ)2 ( ) (ρ 1 +ɛ) r Σ 1 = Σ k (2r(1+ɛ)2 ) (ρ 1 +ɛ)(1 + ɛ)2 ( ) (ρ 1 +ɛ) r In Σ 2, we have (r(1 + ɛ) 2 k) 1 (ρ 1 +ɛ) 1 2, so that Σ 2 1, and where F (r, ɛ) = (r(1 + ɛ) 2 ) (2r(1+ɛ)2 ) Σ 1 (F (r, ɛ)) (ρ1+ɛ) Σ k 1 ( ) (ρ 1 +ɛ) Σ 1 1 exp((2r(1 + ɛ) 2 ) (ρ1+ɛ) log(r(1 + ɛ) 2 )) 2 exp(r (ρ1+ɛ) ), f some constants 1 > 0, 2 > 0 Hence it follows from definition of der given by (23) that ρ ρ 1 + ɛ, since ɛ > 0 is arbitrary, it gives (47) ρ ρ 1 In der to prove the reverse inequality ie, ρ 1 ρ we consider the polynomial of degree as then P sk (z) = Σ k j=0f j A j, (48) E p 1 (, f) Σ s j= f j A j L p (,µ) C 0 Σ s j= +1 f j A j, k 0, p 1 In the consequences of Lemmas 31 and 32, we obtain the following inequality (49) f k a k () M,f (r), r > 0 Using (49)in (48), we get (410) E p 1 (, f) C 0Σ s j= +1M,f (r)r sj = C 0 M,f (r) (r /r) (+1 ) 1 (r /r) r

6 ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE FUNCTIONS 27 f all sufficiently large and all r > r, r > 1 Here C 0 is some fixed number F all sufficiently large and r > 2r, (410) gives (411) E p 1 (, f) γm,f (r)(r /r), where γ is a constant independent of and r If ρ 1 = 0, then nothing to be prove Let us assume that 0 < ρ 1 < If ρ 1 <, define ρ = ρ 1 ɛ, f small ɛ > 0, so that ρ 1 > 0 Let ρ > 0 be arbitrary if ρ 1 = + Then f infinitely many indices k 1, from (41) we have log ρ log(e p (, f)) 1 (412) log E p (, f) log ρ Using (412) in (411) we get (413) log M,f (r) log ρ + log( r ) log γ r The minimum value of right hand side of (413) is obtained at r r = (es k ) 1 ρ value of ( r r ) in (413) we obtain the following inequality log M,f (r) ρ log γ log log M,f (r sk ) log r sk log r ρ ( log log ρ ) log + 1 Proceeding the limits and taking the definition (23) into account, we get (414) ρ = lim sup log log M,f (r) log r lim sup log log M,f (r sk ) log r sk ρ and substituting the Since ρ is arbitrary real number, smaller than ρ, it gives that ρ ρ 1 Now in view of (47) the result is immediate Hence the proof is completed Theem 42 If f : X C is a transcendental entire function on X with a series expansion (25) with respect to the thogonal polynomial basis {A k } k 1, then f L p (, µ) with a finite der ρ(0 < ρ < ) has finite type σ(0 < σ < ) if and only if eρσ = lim sup (Es p k (, f)) ρ k < +, and σ 1 = σ, where E p (, f) is given by (26) Proof Let δ 1 = eρσ 1 F given ɛ > 0 and δ 1 > 0, we have f sufficiently large k (415) (E p (, f)) ρ δ 1 + ɛ log log(es p k (, f)) ρ 1 log( δ1+ɛ ) Now it follows from Theem 41 that the der of f is at most ρ Now let us consider that 0 < δ 1 < and we have to show that σ δ1 eρ = σ 1 From (415) we get (416) E p (, f) ( δ 1 + ɛ ) sk ρ Consider (417) f(z) Σ k 1 f k A k Ωr Σ k 1 f k a k ()r Further we will proceed the proof by considering two cases:

7 28 DEVENDRA UMAR Case 1 Let p 2, then we have f = Σ k 0 f k Ak because f L 2 (, µ), L p (, µ) L 2 (, µ) and {A k } k is a basis of L 2 (, µ) Consider the series Σ k 0 f k Ak in C m+n and it can be easily seen that this series converges unifmly on every compact subsets of C m+n to an entire function Using the Bernstein Markov inequality (BM) in (417) we get it gives from (46) that Now in view of (416) we have Let us assume the function f(z) C ɛ Σ k 1 f k (1 + ɛ) ν k r, f(z) C ɛ Σ k 1 f k (1 + ɛ) 2 E p (, f)r f(z) C ɛσ k 1 ((1 + ɛ) 2ρ ( δ 1 + ɛ )r ρ ) sk ρ = C ɛ Σ 1 + Σ 2 φ(s) = ((r(1 + ɛ) 2 ) ρ ( δ 1 + ɛ )) s ρ, s > 0 s This function attains its maximum value at s = ( δ 1 + ɛ )(r(1 + ɛ) 2 ) ρ e and the value is equal to exp(( δ1+ɛ eρ )(r(1 + ɛ)2 ) ρ ) Hence f any constant 1 > 0, and C ɛσ 1 = C ɛσ 1 k 2(δ1+ɛ)(1+ɛ) 2ρ r ρ((δ 1 + ɛ )(1 + ɛ) 2ρ r ρ ) sk ρ 2(δ 1 + ɛ)(1 + ɛ) 2ρ r ρ exp(( δ 1 + ɛ eρ )(r(1 + ɛ)2 ) ρ ) 1 exp(( δ 1 + ɛ eρ )(r(1 + ɛ)2 ) ρ ), Thus from above discussion we get σ δ1 eρ C ɛσ 2 = C ɛσ k>2(δ1+ɛ)(1+ɛ) 2ρ r ρ((δ 1 + ɛ )(1 + ɛ) 2ρ r ρ ) sk ρ C ɛσ 1 k 1 2 k = 2 < Case 2 F the case 1 p < 2 and f L p (, µ) by (BM) inequality and Hölder s inequality we get again the inequality (46) Now proceeding on the lines of proof of Case 1, the result is immediate k In der to prove the reverse inequality, we note that if δ 1 > ɛ > 0, then f infinitely many indices (418) E p (, f) ( δ 1 ɛ ) sk ρ Now using (418) in (411) we obtain log M,f (r sk ) ρ log(δ 1 ɛ ) + log(r sk /r ) log γ The minimum value of right hand side is attains at r r = es k (δ 1 ɛ) Thus we get M,f (r sk ) e ρ δ 1 ɛ = exp(( eρ )rρ ) + O(1) Proceeding to limits and using the definition (24) of type of f L p (, µ), we get σ δ 1 eρ

8 ON THE GROWTH AND APPROXIMATION OF TRANSCENDENTAL ENTIRE FUNCTIONS 29 This completes the proof of theem Remark 43 Theem 41 and 42 also holds f (0 < p < 1) (see[9]) References [1] E Bedfd and BA Tayl, The complex equilibrium measure of a symmetric convex set in R n, Trans Amer Math Soc 294(1986), [2] RP Boas, Entire Functions, Academic Press, New Yk, 1954 [3] SMEinstein-Matthews and HS asana, Proximate der and type of entire functions of several complex variables, Israel Journal of Mathematics 92(1995), [4] SM Einstein-Matthews and Clement H Lutterodt, Growth of transcendental entire functions on algebraic varieties, Israel Journal of Mathematics 109(1999), [5] Adam Janik, On approximation of entire functions and generalized der,univ Iagel Acta Math 24(1984), [6] OP Juneja, GP apo and S Bajpai, On the (p, q)-der and lower (p, q)-der of an entire function, J Reine Angew Math 282(1976), [7] OP Juneja, GP apo and S Bajpai, On the (p, q)-type and lower (p, q)-type of an entire function, J Reine Angew Math 290(1977), [8] HS asana and D umar, On approximation and interpolation of entire functions with index-pair (p, q), Publicacions Matematique 38 (1994), No 4, [9] D umar, Generalized growth and best approximation of entire functions in L p-nm in several complex variables, Annali dell Universitá di Ferrara VII, 57 (2011), [10] P Lelong and L Gruman, Entire Functions of Several Complex Variables, Grundlehren der Mathematischen Wissenschaften 282, Springer-Verlag, Berlin, 1986 [11] B Ja Levin, Distributions of Zeros of Entire Functions, Translations of Mathematics Monographs 55, Amer Math Soc, Providence, RI, 1994 [12] AR Reddy, Approximation of an entire function, J Approx They 3(1970), [13] AR Reddy, Best polynomial approximation to certain entire functions, J Approx They 5(1972), [14] LI Ronkin, Introduction to the They of Entire Functions of Several Variables, Amer Math Soc, Providence, RI, 1974 [15] A Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Russian Mathematical Surveys 36(1981), [16] A Sadullaev, An estimate f polynomials on analytic sets, Mathematics of the USSR-Izvestiya 20(1980), [17] SM Shah, Polynomial approximation of an entire function and generalized ders, J Approx They 19(1977), [18] J Siciak, Extremal plurisubharmonic functions and capacities in C n, Lectures in Mathematics 14, Sofia University, Tokyo, 1982 [19] J Siciak, On some extremal functions and their applications in the they of analytic functions of several complex variables, Trans Amer Math Soc 105(1962), [20] GS Srivastava and Susheel umar, On approximation and generalized type of entire functions of several complex variables, European J Pure Appl Math 2, no 4(2009), [21] TN Winiarski, Application of approximation and interpolation methods to the examination of entire functions of n complex variables, Ann Polon Math 28(1973), [22] TN Winiarski, Approximation and interpolation of entire functions, Ann Polon Math 23(1970), [23] A Zeriahi, Meilleure approximation polynomiale et croissante des fonctions entieres sur certaines varievalgebriques affines, Annales Inst Fourier (Grenoble) 37(1987), Department of Mathematics, Al-Baha University, POBox-1988, Alaqiq, Al-Baha-65431, Saudi Arabia, SA Cresponding auth: d kumar001@rediffmailcom