On approximation and interpolation of entire functions with index-pair (p, q)

Publicacions Matemátiques, Vol 38 (1994), 255®267 . ON APPROXIMATION AND INTERPOLATIO N OF ENTIRE FUNCTION S WITH INDEX -PAIR (p, q ) H . S . KASANA * AND D . KUMA R A b stract  In this paper we have studied the Chebyshev and interpolatio n errors for functions in C(E) , the normed algebra of analytic functions on- a compact set E of positive transfinite diameter . Th e (p, q)-order and generalized (p, q )-type have been characterized i n terms of , these approximation errors . Finally, we have obtaine d a saturation theorem for f E C(E) which can be extendedto a n entire function of (p, q) -order 0 or 1 and for entire functions o f minimal generalized (p, q )-type . Introductio n Let E be a compact set in complex plane and 1' (n) = {U,  . . . , ~ nn } be a system of (n + 1) points of the set E such that n H Inj - 'nk 1 and q (j)Wn)}-- fi Ir i - Ink l , °Ç jÇkÇn  k= q j = o,l , . . ., n . Again, let n(n) -- {7ni, 7 ]n2, . . . , ?i nri } be the system of (n + 1) points i n E such tha t Tl n ; v (r] ( n) ) sup V(1 ( rz) ) and q ° (n( n) )  á(j ) (n(n) ) for j = 1, 2, . . . , n . 1 (n) C E *This work was supported by the Swedish Institute, Stockholm . 256  H . S . KASANA, D . FUMA R Such a system always exists and is called the nth extrema] system of E . The polynomials n L (i) (z a 77( n )) = H  z — 71nk  j ~ O1 . . >  >  a  ~ —  ,n , r in , ry  r ]n k k= 0 are called Lagrange extremal polynomials and the limit d  d(E) = limn_ ›oo Vñ /n(n+1) is called the transfanite diameter of E . Let C(E) denote the algebra of analytic functions on the set E . Le t us define the approximation errors as follows : Pn,l U ; E} l u n,I( .f ) = ~ f If— g H , 9 where ii  is the sup norm and 7r n (z) denotes the set of all polynomial s of degree n . Further, we also defin e p n,2( .Íi F % ) = m n,2( .f) = Ii L n — L n,—1 11, n > 2 mn,3(f ; E ) z mn,3(f) = Il L n — f 11 , where n E N and n L n (z) = E L(3)(z,n(n)) f(77 ni ) j= 0 is the Lagrange interpolation polynomial of degree n . Reddy [$], [9] connected classical order and type with polynomial approximation error of an entire function which is an extension of a continu - ous function defined on [—1,1] . Contemporarily, Rice [10] and Winiarsk i [13] studied these results for different approximation errors of a continu - ous function on the arbitrary domains . Later on, Massa [0] developed a simpler proof of Reddy's results . It has been noticed that these author s fail to compare the approximation errors of those entire functions whic h have the same positive finite arder but their types are infinity . In orde r to include this important class of entire functions we shall utilise th e concept of proximate order due to Levin [5] . Moreover, for the inclusio n of entire functions of slow growth and fast growth their results will als o be extended to the (p, q)-scale introduced by Juneja et al . ([2], [3]) . It i s significant to mention that Shah [11] and Kapoor and Nautiyal [3] hav e studied in this direction far continuous functions on the domain [—1, 1] . They studied the results for (a, ) 3}-orders . However they have to stud y separately the entire functions of slow and fast growth . That is why i n our studies the (p, q )-growth has been preferred to the (a, ,C3 )-growth . APPROXIMATION AND INTERPOLATION OF ENTIRE FUNCTIONS 25 7 1 . Definitions and auxiliary result s Before stating the auxiliary results it will be justifiedto introduce wit h the concept of (p, q )-scale, p ~ q ~ 1, and certain notations which wil l be frequently used in the text : exp ími x == log {-m] x = exp(exp [m-11 x) = log(log Hm-13 x), m = ±1,±2, . . . , r A[,, ] (x) = fliog [zi x for r = 0,1, . . . z = o L(p,q)  if q C p C oo , 1+ L(p,q)  ifp =q=2 ,  P(L(p, q)) =  • max(1, L(p, q)} 1f 3 c _ p = q , oo  ifp — q = oo . 0 o Let f(z) — E Qnzn be an entire function . We set M ( r, f ) rz= o max iziT , . ~  M(r, f) is called the maximum modulus of f(z) on th e circle ~z! = r . Definition 1 . An entire function f(z) is said to be of (p, g)-orde r p(p, q) if it is of index -pair (p, g) such tha t  lim  M(r, f) _ m sup  [ g ]  — p(p ' q) ' r—~  log r and the function f ( z ) having (p,q)-order p(p, g )(b < p(p,q) C oo} is sai d to be of (p,q)-type T(p,q) i f lim sup log[pIIM(r' f ,q ) ) = T(p, q) , r~~ (iog_ r }~~~~ g ) where b = 1 ifp =q, b=0 if p ~ q . Nandan et al. [7] has extended the idea of proxim at e order to entir e functions with (p, q )-growth a s Definition 2 . A positive function p p,q (r) defined on [r o , oo}, r o > exp[q-I] 1, is said to be of the proximate order of an entire function wit h index -pair (p,q) i f (i) p p,q (r) —> p(p,q) as r ~ oo, b < p(p,q) C oo , (u) A[q ] (r)p p ' yq (r) —> 0 as r ~ oo ; pP ,q (r) denotes the derivative o f p p,q ( r ) . o T(p,q) oo, 258  H . S . KASANA, D . KUMA R It is known [7, Thm . 4] that (logíg - 11 )PPq(T)_4 is a monotonicall y increasing function of r for r ~ ro, where A = 1 if (p, q) = (2, 2) an d A = o otherwise . Hence we can define the function O(x) far x ~ xo to be the uniqu e solution of the equation , x  (1og1' r)nP,4(r)-A  O( x) = log [q-1] r . Definition 3 . Let f (z) be an entire function of (p, q)-order p(p, q)( b p(p,q) < oo) such tha t log ~ - ~~  lim   m ( r, f)  T*  * sup  [g-=  ~ p a g } a ~ T(p, q) Ç oo . r~vo (1og 11 r )PP>q (T ) If the quantity T* (p, q) is different from zero and infinite then p p,q (r) i s said to be the proximate order of a given entire function f (z) and T* (p, q ) as its generalized (p, q)-type . Clearly, proximate order and the corresponding generalized (p, q)-type of an entire function are not uniquel y determined . For example, if we add c/ lo g Iql r, U C c < oo to the proxi - mate arder p p,q (r) then p p,q (r) + c / lo g Iql r is also a proximate order sat - isfaying (i) and (u) and consequently, the generalized (p, q)-type turn s out to be e c T*(p, q) . Definition 4 . An entire function with inde x-pair (p, q) is said to be o f minimal, normaland maximal (p, q)-type with respect to a proximat e order according as T* (p, g) as zero, positive finite and infinite respec - tively . Let E r be the curve defined b y = {z C : 10(z)Id = r} ; where w = ~ ~ z } is holomorphic and maps the unbounded component o f the complement of E on Iw 1 > 1 such that «oc) = oo and 7 P '(oc) > O . Also, we set M(r, f) = sup zEEr ~  for r ~ 1 . The following auxiliary results will be utilised in the sequel : Lemma 1 . lf f(z) is an entire function of (p,q)-orderp(p,q), the n log~ ] lim   ~ r, f } sup  ~V I [q]  = p (p, q ) ' r—oo  log r APPROXIMATION AND INTERPOLATION OF ENTIRE FUNGTIONS 25 9 and for p(p, g )(b < p(p,q) C Do), T*(p,q) is given b y log [P–1 ] lim sup  m ( r ' f) = T * (p, q) . r -,oa (1og 1 ' r ) P P,g(T ) Pro a : Let z o be a fixed point of the set E, and r ~ 1 . Then fro m [13], r --2 1 Ef — I z a l ç Iz1 C r+ IEI +  z E E r . For p ~ q ~ 1, a < 1 and ~ > 1, using log [g] kx  log [g] x as x --~ oo , oC k Coo , log' M(ar) < log'' M(r) Ç log~~ m (or) log [9] r  log l9l r  1og19l r  for r > r o . Also, for p(p, q)(b < p(p, q) < oo), using 1og 19l (k+x) ' ~ log í9l x as x —> oo , log [p-1i M(r —a)  log~ -11 M(r)  log~ -11 M(r + c ) (1og_r)PPQ(T) < (1og 1I r) Ppq fr) — (Iog 1i r)PPq(r) ' where a = 21E1 + z o , c = IEI + iz o ' . After passing to the límits and taking the Definition 1 and Definition 3 finto account this lemma is established . Lemma2 . If a function f is defined and bounded on a compact se t E, then p in,I(f) 5 . llf — L nll  (n+ and Ii L n — L n_1I1 Ç 2(n + 2 }Pn-~,i ~ .f)a for n = 2, 3, . . . , whre L~ is the Lagrangeinterpolation polynomial with nades at extrema l points rin i . The proof of this lemma is available as Lemma 3 .2 in [13] . Lemma 3 . Let f E C(E) . Then f can be extended to an entire function if and only if 1/n( (f) —} o as n ---~ oo, j = 1, 2, 3 . This lemma is a direct consequence of Lemma 1, equation (4 .5) o f Winiarski [131 and an inequality due to Walsh [12, p . 771 . 260  H . S . KASANA, D . KUMA R Lemma 4 . For every f E C(E) and  (f), j = 1,2,3, there exist s an entire function g i (z) = E µ n,i (f )z n+l such tha t n= 0 M(r, f ) ao + 2g j (r/d) . Proof : Define the function 0 0 (1 .1)  f - (z) = 7ro + E{7rn+I(Z) 7rn(Z)} . k— o Obviously, J(z) = f(z) for all z E E . We prove that f(z) = f(z) fo r all z in the complex plane . For this it is enough to show that this serie s converges uniformly on every compact subset of the complex plane, sinc e I e n +l( z ) = 7r n (z)l < Il~n+l — 7rnII b .z E E ~ µn+1,1(f) +µ n ,1(f) ~ 2 µn-i(f) , using Walsh inequality [12, p . 771, we have n+ l 1— ~~,( z )I  2 µn,, I( .f) ( 7 1 r )  for z E E T . Hence 00 Is I+ E I e n +I(z) - 7rn(=) I n- 1 00 n+ 1 a o+ 2 E ~n,1(f) r  forzEE T . n = Ü The last series converges for every r and therefore, the series on the righ t of (1 .1) converges uniformly on every compact subset of C and so f(z ) is entire and f - (z) = f (z) . Consider the functio n 9J(z) = E µ n >7(f) zn+ 1 n= U Since limn,~ ~n~~ (f) = o, j = 1,2,3, by Lemma 3, it follws that eac h g i (z) is entire and further, (1 .2) implies the required inequality . ■ APPROXIMATIoN AND INTERPOLATION OF ENTIRE FUNOTIONS 26 1 2 . Main result s Theorem 1 . If f E C(E) can be extended to an entire function wit h índex-Pair (p, q), (p,q)-order p(p,q)(b < p(p,q) < oo) and generalize d (p,q)-type T* (p, q), then for every µn,j (f ), there exists an entire func - tions g j (z) = E µ n,j (f )z n+ such that n= 0 p(p,q, f)  p(p,q,g j ) and T*(p, q, f) _ / C3T * ( p , q , g j) y where / 3 = d -P(p,1 ) for q = 1 and ,3 = 1 for q ~ 1 . Proof : It has been shown in Lemma 3 and Lemma 4 that 1(z) = f(z ) far all z in the complex plain and for each p, n,j ( f ) , the function g j ( z ) = cc i E p n,j ( f )z n+1 is an entire function . Winiarski [13, p . 266] has prove d n= o thatforanye> 0, ( — de e (2 .1)  1~n,3(f) ~ kM(, f)  r ~ where k is a constant and d > 0 is the transfinite diameter of E . Using (2 .1) in the power series expansion of g j (z) for j = 3, it i s inferred that ( de2f ) r n+ l = >2/~fl,3(f) (de 2 E n= 0 krm(r, f ) E °° 1 Ç  n E d 2 E e  n=o e <  l~rN~ (r, f ) or log g 3 . de2 E r  Ç 0(1) + log 1V1( r, f) + log r . Thus, in view of the aboye inequality and Lemma 1, for p > 2 and q = 1 , p ( p , 1, g 3)  P( p a 1, f) and T * ( p 7 1 , g3) Ç e 2EP(P,1) d P(P ' I) T * (p, 1, f) , andforp> 2,andq> 1 , p(p, q, g 3 ) Ç p(p, q, f) and T * ( p a q, g 3 ) Ç T* (p, q, f) . deE(eE — 1) 262  H . S . KASANA, D. FUMA R Since E > o is arbitrary, both inequalities imply that for all (p, q) , (2 .2)  p(p,q,g 3 )  p(p,qy f) and í 3T* ( p 7 q, g 3 ) Ç T * (p, q, f) • Further, using the inequality M( r, f) Ç a o + 2gj (), we observe that fo r q = 1, p(py 1,f)  p(p, 1, 9j ) and T * ( p a 1 ,f)  d -P(p'1)T* ( p s 1 ,9j ) , andforq~1 , p ( p , q 5 f) Ç p( p , q ,9j) and T * ( p a q, f) Ç T* ( p , q ,9j ) • Hence, for all index-pairs (p, q) , (2 .3)  p(p,ga f)  p ( p ? q ,9j ) and T * ( p , q , f)  fi T* ( p a q ,9j ) . Combining (2 .2) and (2 .3), we hav e p ( p , q , f) = p(p,q,g 3 ) and T * ( p a g 7 f) = OT* ( p , g a 93 ) and further, application of Lemma 2 makes this result valid for j  1 and j = 2 also . ■ Theorem 2 . Let f(z) E C (E ) . Then f(z) can be extended to a n entire function of (p, g )-order p(p, g )(b < p(p, g) C oc) if and only i f p ( p , q) = P(L(p, q)} , where L(p,q) = lim sup  [4-11 _1/ n n~ °° log  u n -1/n ( (f ) Proof : In Lemma 3 and Lemma 4 we have concluded that f E C(E ) can be extended to an entire function if and only if g j (z) is an entir e function. Moreover, by Theorem 1, f (z) and g j (z) have the same (p,q) - order . Applying Corollary 1 by Juneja et al . [1, p . 62] to the functio n 0 0 9(Z) _ ~ P n, j (f )z n+l Theorem 2 follows . ■ log 1 p- 11 n n=o APPROXIMATION AND INTERPOLATION OF ENTIRE FUNCTIONS 26 3 Remarks . (a) For (p, q) = (2,1 ), this theorem includes Theorem 2 by Winiarsk i [13] and further for (p,q) _ (2,1), j = 1 and E = [-1,1] this als o contains a result by Massa [61 . (b) On setting (p, g) = (2, 2), j = 1 and E = [-1,1] this theorem ma y also be considered as an alternative proof of Theorem 5 by Reddy [7] . Theorem 3 . Let f(z) E C(E). Then f(z) can be extended to a n entire function of (p,q)-order p(p,q)(b < p(p,q) < oo) and generalize d (p, q)-type T*(p, q)(0 < T*(p, q) G oo) if and only i f ~*  lo [~-2~ n  p(p , q)— A  ~ p ' g } = lim sup  ) ~~ ~pa q)  n~~ p [ iog_ 1 ] 1/n ( ~n~~ f } wherefi is defined in Theorem 1 an d (P( 2 , 2) - 1)n(2,2)-1/(P(2, 2))P(2'2)  if (p, q ) = ( 2 , 2 ), M(p, q) =  1/ep(2, 1)  if (p, q) = (2,1) , 1  otherwise . Proof : To prove this theorem we apply Theorem 1 by Kasana [4] to th e function g j (z) defined in Lemma 4 and the resulting characterisation o f T* (p, q, g j ) in terms of , u n,i ( f) and the relation T* (p, q ; f) = í3T* (p, q, g 3 ) taking together prove the theorem . ■ Taking p p,q (r) = p(p, g)dr > r o and O(x) = x1/(P(psg)-A), we have th e following corollary which gives a formula for (p, g)-type T(p, g) in term s of the approximation errors of an entire function f (z) . Corollary . Let f(z) E C(E) . Then f(z) is the restriction of an en - tire function having (p, q)-order p(p, q)(b < p(p, q) < oo) and (p, q)-typ e T(p, q)(0 < T(p, q) < oo) if and only i f  T ~ p s q}  log~ -2 j n ¡¡ q) = lim sup  _ (f))p(pq) — A ~ M 1 p a q }  n~oo (log — ~ (a) for ( p, q) =(2, 1) this corollary gives the Theorem 1 of Winiarsk i [13] as a particular case . (b) On the domain E = [-1,1]and for the approximation error P l n, 3 this corollary also includes some of the theorems of Massa [6] and Redd y [7], respectively for (p, g) _ (2,1) and (p,q) = (2,2) .