AKm sup log Ж(r)

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE X IX (1976)

J .

(Kuruksehtra, India)

O n the degree oî approximation oî an entire function

1. Let f(æ) be a real-valued continuous function defined on [ —1,1], and let

E J f ) = i n f ||/ - p ||, n = 0 , 1 , 2 , . . . , penn

where the norm is the maximum norm on [ —1,1] and лп denotes the set of all polynomials with real coefficients of degree at most n. Bernstein [1], p. 118, proved th a t

lim = 0

71-5-00

if and only if f(æ) is the restriction to [ —1 ,1 ] of an entire function.

Further, let f(z) be an entire function, and M{r) max |/ ( 0 ) I,

\e\=r

lim S.U^ log log M{r)jlogr — ® ( 0 < A < £ <

),

1П1 A

Km sup log Ж (r) T

™ inf VQ ~ t

0 < Q < OO

0 < t < T < oof

where q , à and T , t denote the order, lower order and type, lower type, respectively, of an entire function f(z).

Bernstein [1], p. 114, has shown th a t there exist constants q (posi­

tive) a, T (non-negative) such th a t

lim sup nlleElln(f) = a 71— >0O

if and only if f(x) is the restriction to [ —1 ,1 ] of an entire function of order q and type T.

Becently Varga [11], Theorem 1, has proved th at lim sup {n log w /log 1 /,£/п ( / )} = q ,

n~>oo

where q is the non-negative real number if and only if f(œ) is the restriction

to [ —1 ,1 ] of an entire function of order

The results of Bernstein and Varga give us the clue th at the ia te at which E]ln(f) tends to zero depends on the order and type of an entire function f(z).

The object of this paper is to study th e relationship of growth con­

stants defined above with the rate of growth of E n{f). Some theorems of this nature have been given in [7]-[10] but we consider some further results which will improve the results of Beddy [7], [8].

2. T

1. Let f(co) be a real-valued continuous function defined on [ —1, 1]. I f f(x) is the restriction to Г —1, 1] of an entire function f(z) of order q and lower order X, there exist two sequences {k(n)}, (e(w)}

(n = 0,1, 2, ...) such that one has

k(n) =

1+0(1),

(

)- >0 as n оо, (*) Em (f)IE m + h(f) >

(n ^ n0J h = l ?2 , 3 , . . . ) r then

у = limsup \fogEn(f)jE n+l{f)}j\ogn.

We require the following lemma in the proof of Theorem 1.

L

1. I f {tqj is a sequence of real or complex numbers, then limsup log \l[an\[n\ogn < limsup log \anlan+1\llogn.

n —> oo n—>oo

P r o o f of L e m m a 1. See [9], p. 193.

P r o o f of T h e o r e m 1. First, assume th at f(x) has an analytic ex­

tension f(z), which is an entire function of order q and lower order A.

Following Bernstein’s original proof we have (cf. [5], p. 78, [6], p. 84) for each n > 0,

2B(a)

(2-1) E n( f ) ^ ~ —----—- for any u > 1 , a (и- l )

where В (a) is the maximum of the absolute value of f(z) on E a, and E a with a > 1 denotes the closed interior of ellipse with foci at ±1 and half

/ ( 72 + l \ / ( 72 - l \

major axis --- and half minor a x i s ---. The closed discs E A a )

\ 2(7 / \ 2(7 /

and D 2(<7) bound the ellipse E a in the sense that

( (72 — 1 1 1 (72 -h 1 1

Z > i((7)

= |« | \z\ < - 2 ~ [ c Е в с в г{а) = |я| \z\ J.

(

2

2

2(7 J u ' { 2(7

From this inclusion, it follows by definition th at

(72 2(7

1 \ / ( 7“ + l

for all a > 1.

From this, it can be verified th a t

(2.3) Q = lim ^ log log M{a)j\oga = lim ^ IoglogB(cr)/log(r.

' y G—>00 CO

From (2.1), we have K B {a)

<2.4) E n( f ) ^ ---- ±-L where К is some positive constant a"

since a > 1, 2/(cr —1) < K .

Consequently from (2.4), we have for any e > 0

ОСУ OQ CO

V / .

V A Y ^ jSl5 (o, + s) Y H / о

(2.5) > E n(f)a n < > ~ Г 1 г [ п = ^ ( a + e) }

H 1—i (or + e)n l - l

n = 0 n = 0 ' ' n = 0 '

ILB(<7-j- f)

We note [11] th at

OO

(2.6) JS(«

) < |Р 0( г ) | + 2 с г у £ ). ( / ) 0''.

j=0 Let us write

OO

(2.7) ’

3= 0

which represents an entire function, then we have from (2.5) and (2.6) (2.8) B(a) ^ K aJ (a) ^ K"a(a + e) В (a + s) ,

where K ' and K " are some positive constants. From (2.2) and (2.8) it follows th at

(2.9) ? = = lim ™P loglog J(a)/log(J.

Я .-.о® m f l o g

a

Now, applying Lemma 1 to {En(f)} we have

lim sup log 1 jEn (f)lnlogn < lin is u p lo g { F ^ (/)/^ +1(/)}/logw.

( 2 . 10 )

By known result [7], Theorem 2A, we have 1 < lim sup log {En(f)jE n+1{f)}llogn.

If X = 0 , the theorem is an obvious consequence of this inequality, while, if X > 0 it is sufficient to prove th at

(

2

11

- ^ lim su p log {En(f) /E n+1(f)} /log n .

Л n— *oo

Let us pu t for simplicity

En(f)IEn+ ¹ (f) =V»(«).

The sequence y)(n) cannot be bounded (in fact, yj(n)< Tc for each value of n it follows th a t El[n(f) > 1/Jc, which is absurd). Therefore limsup^(w) = oo.

П-УОО

Then denoting the sequence of the indices by {nh} for which ip{nh)

> ip(n) for each n<. nh.

Allowing y)(n) < y>(nh) for nh ^ n < nh+1. One has

( ² . ¹² ) logy(n) lo gy>(nh)

lirn sup —--- — lim sup

logn «-oo lo gnh

OO

If 9(H) = I bkzk is an entire function of lower order A and v'(r) is

*=о

the rank of the maximum term, then logv'(r)

l = liminf -- - - -- - [12].

r^oo logr Applying this to J(a), we see th at

l — liminf cr— M X?

logr (<r) logo- [v(a) is the rank of maximum term of J (a).]

Choose X' with 0 < A'< Я for each a > o-0(A'); then we get

(2.13) logv( or) > A'logcr.

Let oh = {fp{nn)}1 and let us evaluate v(ah). For each value of n > &(%), we have according to condition (*)

Therefore,

E H„j(f)<#n* > K ( f X - Hence v(oh) < k(nh).

From (2.13), it can be derived th a t

log 4 n h) ^ logv(<rft) > A '{ l-e (w A)}logy>(%), which leads to

logv4% )/log% <

As h(nn) — wA+0(1) and e(nh)

1 A'{1 - s ( n h)}

->0 for h ->

lo gk(nh) lo gnh

oo. One then obtains

(2.14) lo g y ( n h) 1

limsup —---< —

&~>oo log

A for each A' with 0 < A' < Я.

Eow the desired conclusion follows from (2.10), (2.11), (2.12) and

(2.14). ,

E e m a r k . For s(n) = 1 /n, k(n) — n, Theorem 1 gives a result of Eeddy [8] (Theorem 3, к = 1) as a particular case.

3. T

2. Let f(x) be a real valued continuous function defined on [ —1,1], which is the restriction of an entire function f(z) of order q and lower order X. Further, i f there exists a séquence {e(n)} such that one has:

s(n)-+ 0 as n-^oo, E m(f) E m+i(f) then

(3.1)

> (Д Л Я /Д и-Л Я )1 £(n) ( n > n 0, m = n, n + 1, ...),

lim inf log {Fn(f) /E n+1(f)} /log n = —,

n->-co Q

(3.2) lim sup log {Fn(f) /En+x ( / )} /log n = —.

Г7—VOO /

P r o o f of T h e o r e m 2. We omit the proof since it is based on th e same lines as the proof of Theorem 1.

E e m a r k s . (i) If the sequence F n(f)fF n+1(f) is non-decreasing for n ^ n 0, condition (**) is satisfied with e(n) = 1/n, hence our Theorem 2 contains a result of Eeddy [8] (Theorem 3, к = 1 ) as a particular case.

(ii) Condition (*) is not sufficient to guarantee the sign of equality in (3.1).

4. T

3. L etf(x) be a real valued continuous function on [ —1 ,1 ];

then

(4.1) у = m inlim suplog{F%_i ( /) /F nJ/)} /(w A- % _ 1)log% _1, л {nh)

where {%} is an increasing sequence of positive integers and X is a non­

negative real number, i f f ( x ) is the restriction to [ —1,1] of an entire function f{z) of lower order X.

P r o o f of T h e o r e m 3. From (2.2) and (2.8) it follows th at X = lim inf

<r—>oo

loglogE(o-)

loger = lim inf

<J-> 00

log log J ( a) loger

oo

If д(г) = 2 ’ h zk is an entire function of lower order X (0 < X < oo), k=0

then

X = max lim inf (% - n ^ J l o g n ^ J l o g [31> P- 310*

{ n jf n-> oo

Applying this to we get the required result, i.e.

5. To deal with functions of infinite order, we have the following classification [7]. There exists a positive integer h > 2 for which

are finite and positive, where lk(x) = loglog.. . (&-times)æ (Jc = 1, 2, 3, ...) and lkx > 0 for all sufficiently large positive x. An entire function with Qk_ l =

and Qk <

is called an entire function of index h. Thus, Qk and Àk extend the definitions of q and Я, which correspond to h — 1.

If 0 < Qk <

then there exists a proximate order Qk(r) (may be named as &-th proximate order of f(z)) satisfying the following conditions:

(i) Qk{r) is a non-negative, continuous function of r for r > r0 (ii) lim ^ (r) = Qk.

(iii) lim r^ (r)lo g r = 0, where Qk(r) is either the right-hand or the

T k will be called Jc-th proximate type. In this section we study the relationship of &-th proximate order Qk{r) and its corresponding T k with the rate of growth of ЕЦп(/). To state and prove the theorem precisely, we introduce a function Ф(х) which is defined as an unique solution (when x > x Q) of the equation

ek< .r)

X — Г .

We prove the following

T

4. Let f(x) be a real valued continuous function on [ —1,1]

which is the restriction to [ —1, 1] of an entire function f(z) of index h with order Qk ( 0 < Qk < oo) and h-th proximate order Qk{r)\ then

-are finite.

Before we start with the actual pi oof, we first consider the following lemmas which will be needed in the proof of the theorem.

limSUp li‘+'M{r'> = e*

r-*co logr \

left-hand derivative at the points where they are different, (iv) lim suplkM(r)lvQk{r ) = T*k

< T*k <

(5.1)

and if h > 1 (5.2)

lim s u p { 0 (n )^ /n(/)}

lim sup {Ф{1к- гп)Е]1п (/)}

L

2. The type T* (proximate type) of an entire function

OO

m = T “•**”

n = 0

with the proximate order g(r) is given by the equation Имтгр{Ф(п) \an\1/n} — (T*eg)lle.

oo

, 4

P r o o f of L e m m a 2. See [4], Theorem 2 , p. 42.

OO

L

3. Let f(z) be an entire function f (z) = anzn of k-th proximate

n=0

order Qk(r) and k-th proximate type T k . Then Tk is given by the equation ]imsup{#(Zfc_1« ) |a J 1/n} = (T*k)lleK

П-+ 00

We omit the proof since it can be proved on the same lines as given by Jain [2] for entire Dirichlet Series.

P r o o f of T h e o r e m 4. (2.2) and (2.8) lead to 2 Q]~ lim sup

£7—> 0 0

1кЩ ° )

Qk(c) = lim sup

a—>oo

h B ( a )

ееь{а)(а) — lim sup

<7—> 0 0

hJ(a)

*

On applying Lemma 2 to J(a) we obtain (5.1). Similarly on applying Lemma 3 to J (a) we get (5.2). This completes the proof of the theorem.

R e m a r k . Taking ok(r) = gk in Theorem 4 we get the result of Reddy [7], Theorem 3, which in tu rn includes the result of Bernstein [1] on finiteness.

My thanks are due to Dr. S. H. Dwivedi for his constant help in the preparation of this paper and to Dr. Y. B. Goyal for his encouragement.

I also thank CSIR Yew Delhi for the award of a Senior Research fellowship.

References

[1] S. N. B ern ste in , Leçons sur les propriétés extrémales et la meilleure approxi­

mation des fonctions analytiques d'une variable réelle, Gauthier-Villars, Paris 1926.

[2] P. K. Jain, Proximate order of an entire Dirichlet Series of order (P) infinity, Rev. Roumaine Math. Pures Appl. 15 (1970), p. 367-371.

[3] О. P. Ju ne) a, On the lower order of entire functions, J. London Math. Soc. (2) 5 (1972), p. 310-312.

[4] B. J a. L evin , Distribution of zeros o f entire functions, Vol. 5, Amer. Math. Soc.

Trans., Providence 1964.

[5] G. G. L oren tz, Approximation of functions, Holt, Rinehart and Winston, New York 1966.

10 — Boczniki PTM — Prace Matematyczne XIX

[6] Gi. M einardus, Approximation of functions, Theory and Numerical Methods, Springer-Verlag, New York 1967.

[7] A. R. R edd y, Approxim ation of an entire function, J. Approximation Theory 3 (1970), p. 128-137.

[8] — Best polynomial approximation to certain entire functions, ibidem 5 (1972),.

p. 97-112.

[9] D. R oux, Sul divario fra Vordine e Vordine inferiore délie funsioni intere, Riv.

Mat. Univ. Parma (2) 4 (1963), p. 191-210.

[10] J. P. Singh, Approxim ation of an entire function, Yokohama Math. J. 19 (1971), p. 105-108.

[11] R. S. Varga, On an extension o f a result of S. N . Bernstein, J. Approximation Theory 1 (1968), p. 176-179.

[12] J. M. W h itta k er, The lower order o f integral functions, J. London Math. Soc*

8 (1933), p. 20-27.

DEPARTMENT OF MATHEMATICS KURUKSHETRA UNIVERSITY KURUKSHETRA, INDIA and

43 BRAHMAN PURI ALIGARH, U. P.

INDIA