Journal of Inequalities in Pure and
Applied Mathematics
APPROXIMATION OF ENTIRE FUNCTIONS OF TWO
COMPLEX VARIABLES IN BANACH SPACES
volume 7, issue 2, article 51,
2006.
RAMESH GANTI AND G.S. SRIVASTAVA
Department of Mathematics
Indian Institute of Technology Roorkee
Roorkee - 247 667, India.
Received 05 January, 2006;
accepted 27 January, 2006.
Communicated by: S.P. Singh
EMail: girssfma@iitr.ernet.in
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
008-06
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Abstract
In the present paper, we study the polynomial approximation of entire functions
of two complex variables in Banach spaces. The characterizations of order and
type of entire functions of two complex variables have been obtained in terms
of the approximation errors.
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
2000 Mathematics Subject Classification: 30B10, 30D15.
Key words: Entire function, Order, type, Approximation, Error.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
Ramesh Ganti and G.S.
Srivastava
3
7
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1.
Introduction
P
Let f (z1 , z2 ) =
am1 m2 z1m1 z2m2 be a function of the complex variables z1 and
z2 , regular for | zt | ≤ rt , t = 1, 2. If r1 and r2 can be taken arbitrarily large,
then f (z1 , z2 ) represents an entire function of the complex variables z1 and z2 .
Following Bose and Sharma [1], we define the maximum modulus of f (z1 , z2 )
as
M (r1 , r2 ) = max |f (z1 , z2 )| , t = 1, 2.
|zt |≤rt
The order ρ of the entire function f (z1 , z2 ) is defined as [1, p. 219]:
log log M (r1 , r2 )
= ρ.
log(r1 r2 )
r1 ,r2 →∞
lim sup
For 0 < ρ < ∞, the type τ of an entire function f (z1 , z2 ) is defined as [1,
p. 223]:
log M (r1 , r2 )
lim sup
= τ.
r1ρ + r2ρ
r1 ,r2 →∞
Bose and Sharma [1], obtained the following characterizations for order and
type of entire functions of two complex variables.
P∞
m1 m2
Theorem 1.1. The entire function f (z1 , z2 ) =
is of
m1 ,m2 =0 am1 m2 z1 z2
finite order if and only if
m2
1
log(mm
1 m2 )
−1 )
m1 ,m2 →∞ log (|am1 m2 |
µ = lim sup
is finite and then the order ρ of f (z1 , z2 ) is equal to µ.
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
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Define
q
ρ
m2
1
mm
1 m2 |am1 m2 | .
(m1 +m2 )
α = lim sup
m1 ,m2 →∞
We have
P
m1 m2
Theorem 1.2. If 0 < α < ∞, the function f (z1 , z2 ) = ∞
m1 ,m2 =0 am1 m2 z1 z2
is an entire function of order ρ and type τ if and only if α = eρτ .
Let Hq , q > 0 denote the space of functions f (z1 , z2 ) analytic in the unit
bi-disc U = {z1 , z2 ∈ C : |z1 | < 1, |z2 | < 1} such that
kf kHq =
lim
r1 ,r2 →1−0
Mq (f ; r1 , r2 ) < ∞,
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
where
Mq (f ; r1 , r2 ) =
1
4π 2
Z
π
−π
Z
π
f (r1 eit1 , r2 eit2 ) q dt1 dt2
1q
,
−π
0
and let Hq , q > 0 denote the space of functions f (z1 , z2 ) analytic in U and
satisfying the condition
1q
Z
Z
1
q
|f (z1 , z2 )| dx1 dy1 dx2 dy2
< ∞.
kf kHq0 =
π 2 |z1 |<1 |z2 |<1
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Set
0 = kf kH
kf kH∞
∞ = sup {|f (z1 , z2 )| : z1 , z2 ∈ U }.
0
Hq and Hq are Banach spaces for q ≥ 1. In analogy with spaces of functions of
0
one variable, we call Hq and Hq the Hardy and Bergman spaces respectively.
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The function f (z1 , z2 ) analytic in U belongs to the space B(p, q, κ), where
0 < p < q ≤ ∞, and 0 < κ ≤ ∞, if
Z
1
Z
kf kp,q,κ =
0
1
{(1 − r1 )(1 − r2 )}κ(1/p−1/q)−1 Mqκ (f, r1 , r2 )dr1 dr2
κ1
< ∞,
0
0 < κ < ∞,
kf kp,q,∞ = sup {[(1 − r1 )(1 − r2 )}(1/p−1/q)−1 Mq (f, r1 , r2 ) : 0 < r1 , r2 < 1} < ∞.
The space B(p, q, κ) is a Banach space for p > 0 and q, κ ≥ 1, otherwise it is a
Fréchet space. Further, we have
q
0
(1.1)
Hq ⊂ Hq = B , q, q , 1 ≤ q < ∞.
2
Let X be a Banach space and let Em,n (f, X) be the best approximation of
a function f (z1 , z2 ) ∈ X by elements of the space P that consists of algebraic
polynomials of degree ≤ m + n in two complex variables:
(1.2)
Em,n (f, X) = inf {kf − pkx ; p ∈ P }.
To the best of our knowledge, characterizations for the order and type of entire functions of two complex variables in Banach spaces have not been obtained
so far. In this paper, we have made an attempt to bridge this gap.
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
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Notation: For reducing the length of expressions we use the following notations
in the main results.
1 1
1/κ
B
(n + 1)κ + 1; κ
−
= B[n, p, 2, κ]
p 2
1 1
1/κ
B
(m + 1)κ + 1; κ
−
= B[m, p, 2, κ]
p 2
1 1
1/κ
−
= B[n, p, q, κ]
B
(n + 1)κ + 1; κ
p q
1 1
1/κ
B
(m + 1)κ + 1; κ
−
= B[m, p, q, κ].
p q
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
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2.
Main Results
P
m n
Theorem 2.1. Let f (z1 , z2 ) = ∞
m,n=0 amn z1 z2 , then the entire function f (z1 , z2 )
∈ B(p, q, κ) is of finite order ρ, if and only if
ρ = lim sup
(2.1)
m,n→∞
ln (mm nn )
.
− ln Em,n (f, B(p, q, κ))
Proof. We prove the above result in two steps. First we consider the space
B(p, q, κ), q = 2, 0 < p < 2 and κ ≥ 1. Let f (z1 , z2 ) ∈ B(p, q, κ) be of order
ρ. From Theorem 1.1, for any > 0, there exists a natural number n0 = n0 ()
such that
|amn | ≤ m−m/ρ+ n−n/ρ+
(2.2)
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
m, n > n0 .
We denote the partial sum of the Taylor series of a function f (z1 , z2 ) by
Tm,n (f, z1 , z2 ) =
n
m X
X
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aj1 j2 z1j1 z2j2 .
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j1 =0 j2 =0
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We write
(2.3)
Em,n (f, B(p, 2, κ)) = kf − Tm,n (f )kp,2,κ
Z 1 Z 1
=
{(1 − r1 )(1 − r2 )}κ(1/p−1/2)−1
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0
! κ2
×
II
I
XX
j1
j2
r12j1 r22j2 |aj1 j2 |2
dr1 dr2
1
κ
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,
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J. Ineq. Pure and Appl. Math. 7(2) Art. 51, 2006
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where
XX
j1
S1 =
r12j1 r22j2 |aj1 j2 |2
= S1 + S2 +
j2
∞
X
∞
X
r12j1 r22j2 |aj1 j2 |2 ,
j1 =m+1 j2 =n+1
m
∞
X
X
r12j1 r22j2 |aj1 j2 |2
and
j1 =0 j2 =n+1
S2 =
∞
n
X
X
r12j1 r22j2 |aj1 j2 |2 .
j1 =m+1 j2 =0
Since S1 , S2 are bounded and r1 , r2 < 1, therefore the above expression (2.3)
becomes
Em,n (f, B(p, 2, κ))
Z 1
( X
∞
κ(1/p−1/2)−1
(s+1)κ
≤C
{(1 − r)
}r
dr
0
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
∞
X
) 12
|aj1 j2 |2
,
j1 =m+1 j2 =n+1
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where
Z
0
1
{(1 − r)κ(1/p−1/2)−1 }r(s+1)κ dr
Z 1
(m+1)κ
κ(1/p−1/2)−1
=
{(1 − r1 )
}r1
dr1
0
Z 1
κ(1/p−1/2)−1 (n+1)κ
×
{(1 − r2 )}
r2
dr2 .
0
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Therefore
(2.4) Em,n (f, B(p, 2, κ))
(
≤ CB[m, p, 2, κ]B[n, p, 2, κ]
∞
X
∞
X
) 21
|aj1 j2 |2
,
j1 =m+1 j2 =n+1
where C is a constant and B(a, b) (a, b > 0) denotes the beta function.
By using (2.2), we have
∞
∞
∞
∞
2j
2j
X
X
X
X
− 1 − 2
2
|aj1 j2 | ≤
j1 ρ+ j2 ρ+
j1 =m+1 j2 =n+1
≤
j1 =m+1 j2 =n+1
∞
∞
2j1
2j
X
X
− ρ+
− 2
j1
j2 ρ+
j1 =m+1
j2 =n+1
≤ O(1)(m + 1)−2(m+1)/ρ+ (n + 1)−2(n+1)/ρ+ .
Using the above inequality in (2.4), we have
Em,n (f, B(p, 2, κ))
≤ CB[m, p, 2, κ]B[n, p, 2, κ](m + 1)−(m+1)/ρ+ (n + 1)−(n+1)/ρ+ .
⇒ ρ+ ≥
ln [(m + 1)(m+1) (n + 1)(n+1) ]
.
− ln {Em,n (f, B(p, 2, κ))} + ln {B[m, p, 2, κ]} + ln {B[n, p, 2, κ]}
Now
Γ((n + 1)κ + 1)Γ κ 1 − 1
p
2
1 1
−
=
.
B (n + 1)κ + 1; κ
p 2
Γ n + 12 + p1 κ + 1
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
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Hence
1 1
B (n + 1)κ + 1; κ
−
p 2
e−[(n+1)κ+1] [(n + 1)κ + 1](n+1)κ+3/2 Γ
'
1
p
−
1
2
e[(n+1/2+1/p)κ+1] [(n+ 12 + p1 )κ + 1](n+1/2+1/p)κ+3/2
.
Thus
(2.5)
1
(n+1)
1 1
∼
B (n + 1)κ + 1; κ
−
= 1.
p 2
Now proceeding to limits, we obtain
(2.6)
ρ ≥ lim sup
m,n→∞
ln (mm nn )
.
− ln {Em,n (f, B(p, 2, κ))}
For the reverse inequality, since from the right hand side of the inequality (2.4),
we have
(2.7)
|am+1n+1 |B[m, p, 2, κ]B[n, p, 2, κ] ≤ Em,n (f, B(p, 2, κ)),
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
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we have
ln (mm nn )
− ln Em,n (f, B(p, 2, κ))
≥
ln (mm nn )
.
− ln |am+1n+1 | + ln {B[m, p, 2, κ]} + ln {B[n, p, 2, κ]}
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Now proceeding to limits, we obtain
lim sup
(2.8)
m,n→∞
ln (mm nn )
≥ ρ.
− ln Em,n (f, B(p, 2, κ))
From (2.6) and (2.8), we get the required result.
In the second step, for the general case B(p, q, κ), q 6= 2, we have
(2.9)
Em,n (f, B(p, q, κ)) ≤ kf − Tm,n (f )kp,q,κ
Z 1 Z 1
=
{(1 − r1 )(1 − r2 )}κ(1/p−1/q)−1
0
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
0
! κq
×
XX
j1
r1qj1 r2qj2 |aj1 j2 |q
dr1 dr2
1
κ
Ramesh Ganti and G.S.
Srivastava
,

j2
where
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XX
j1
S1 =
r12j1 r22j2 |aj1 j2 |2
= S1 + S2 +
j2
m
∞
X
X
j1 =0 j2 =n+1
∞
X
∞
X
r12j1 r22j2 |aj1 j2 |2 ,
j1 =m+1 j2 =n+1
r12j1 r22j2 |aj1 j2 |2
and
S2 =
∞
n
X
X
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r12j1 r22j2 |aj1 j2 |2 .
j1 =m+1 j2 =0
Since S1 , S2 are bounded and r1 , r2 < 1, therefore the above expression (2.9)
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becomes
Em,n (f, B(p, q, κ))
Z 1
( X
∞
0
κ(1/p−1/q)−1
(s+1)κ
≤C
{(1 − r)
}r
dr
0
where
Z
∞
X
) 1q
|aj1 j2 |q
,
j1 =m+1 j2 =n+1
1
κ(1/p−1/q)−1
{(1 − r)
}r
(s+1)κ
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
dr
0
Z
=
0
1
κ(1/p−1/q)−1
{(1 − r1 )
Z
×
1
{(1 −
(m+1)κ
}r1
dr1
Ramesh Ganti and G.S.
Srivastava
(n+1)κ
r2 )}κ(1/p−1/q)−1 r2
dr2
.
0
Contents
Therefore
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(2.10) Em,n (f, B(p, q, κ))
(
0
≤ C B[m, p, q, κ]B[n, p, q, κ]
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∞
X
∞
X
) 1q
|aj1 j2 |q
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j1 =m+1 j2 =n+1
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0
where C is constant and B[m, p, q, κ] is Euler’s integral of the first kind. By
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using (2.2), we get
∞
X
∞
X
j1 =m+1 j2 =n+1
q
|aj1 j2 | ≤
∞
X
∞
X
−qj1
(ρ+)
j1
j1 =m+1
−qj2
j2(ρ+)
j2 =n+1
≤ O(1)(m + 1)
−q(m+1)
(ρ+)
(n + 1)
−q(n+1)
(ρ+)
.
Using above inequality in (2.10), we get
Em,n (f, B(p, q, κ))
0
≤ C B[m, p, q, κ]B[n, p, q, κ](m + 1)−(m+1)/ρ+ (n + 1)−(n+1)/ρ+ .
ln [(m + 1)m+1 (n + 1)n+1 ]
⇒ ρ+ ≥
.
− ln Em,n (f, B(p, q, κ)) + ln {B[m, p, q, κ]} + ln {B[n, p, q, κ]}
Now proceeding to limits, we obtain
(2.11)
ln (mm nn )
.
ρ ≥ lim sup
m,n→∞ − ln Em,n (f, B(p, q, κ))
Let 0 < p < q < 2, and κ, q ≥ 1. Since
Approximation of Entire
Functions of Two Complex
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Ramesh Ganti and G.S.
Srivastava
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1 − 1
1 1 κ κ1
1/q−1/q1
Em,n (f, B(p1 , q1 , κ1 )) ≤ 2
κ
−
Em,n (f, B(p, q, κ)),
p q
Close
where p1 = p, q1 = 2 and κ1 = κ, and the condition (2.1) is already proved for
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the space B(p, 2, κ), we get
(2.12)
lim sup
m,n→∞
ln (mm nn )
− ln Em,n (f, B(p, q, κ))
≥ lim sup
m,n→∞
ln (mm nn )
= ρ.
− ln Em,n (f, B(p, 2, κ))
Now let 0 < p ≤ 2 < q. Since
M2 (f, r1 , r2 ) ≤ Mq (f, r1 , r2 ),
0 < r1 , r2 < 1,
therefore
(2.13)
Em,n (f, B(p, q, κ))
Z 1 Z 1
κ1
κ(1/p−1/q)−1
≥
{(1 − r1 )(1 − r2 )}
Qdr1 dr2
0
0
≥ |am+1n+1 |B[m, p, q, κ]B[n, p, q, κ],
where Q = inf
[M2κ (f
− p; r1 , r2 ) : p ∈ P ]. Hence we have
ln (mm nn )
− ln Em,n (f, B(p, q, κ))
ln (mm nn )
.
≥
− ln |am+1n+1 | + ln {B[m, p, q, κ]} + ln {B[n, p, q, κ]}
Now proceeding to limits, we obtain
(2.14)
ln (mm nn )
lim sup
≥ ρ.
m,n→∞ − ln Em,n (f, B(p, q, κ))
Approximation of Entire
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Ramesh Ganti and G.S.
Srivastava
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From (2.11) and (2.14), we get the required result.
Now we prove
Theorem 2.2. Let
f (z1 , z2 ) =
∞
X
amn z1m z2n ,
m,n=0
then the entire function f (z1 , z2 ) ∈ Hq is of finite order ρ, if and only if
ln (mm nn )
.
(2.15)
ρ = lim sup
m,n→∞ − ln Em,n (f, Hq )
P∞
m n
Proof. Let f (z1 , z2 ) =
m,n=0 amn z1 z2 ∈ Hq be an entire transcendental
function. Since f is entire, we have
p
(2.16)
lim (m+n) |amn | = 0,
m,n→∞
and f ∈ Hq , therefore
and f (z1 , z2 ) ∈ B(p, q, κ), 0 < p < q ≤ ∞; q, κ ≥ 1. By (1.1) we obtain
Em,n (f, B(q/2, q, q)) ≤ ςq Em,n (f, Hq ),
Ramesh Ganti and G.S.
Srivastava
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Mq (f ; r1 , r2 ) < ∞,
(2.17)
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
1 ≤ q < ∞,
where ςq is a constant independent of m, n and f . In the case of space H∞ ,
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(2.18)
Em,n (f, B(p, ∞, ∞)) ≤ Em,n (f, H∞ ),
0 < p < ∞.
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From (2.17), we have
(2.19)
ln (mm nn )
m,n→∞ − ln Em,n (f, Hq )
ln (mm nn )
≥ lim sup
m,n→∞ − ln Em,n (f, B(q/2, q, q))
ξ(f ) = lim sup
≥ ρ,
1 ≤ q < ∞,
and using estimate (2.18) we prove inequality (2.19) for the case q = ∞.
For the reverse inequality
Ramesh Ganti and G.S.
Srivastava
ξ(f ) ≤ ρ,
(2.20)
since
Em,n (f, Hq ) ≤ O(1)
∞
X
∞
X
|aj1 j2 (f )|,
j1 =m+1 j2 =n+1
using (2.2), we have
Em,n (f, Hq ) ≤ O(1)
≤
∞
X
∞
X
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
j
j
1
− ρ+
− 2
j2 ρ+
j1
j1 =m+1 j2 =n+1
∞
∞
j1
j
X
X
− ρ+
− 2
O(1)
j1
j2 ρ+
j1 =m+1
j2 =n+1
≤ O(1)(m + 1)−(m+1)/ρ+ (n + 1)−(n+1)/ρ+ .
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⇒ρ+≥
ln [(m + 1)(m+1) (n + 1)(n+1) ]
.
− ln [Em,n (f, Hq )]
Now proceeding to limits and since is arbitrary, then we will get (2.20). From
(2.19) and (2.20) we will obtain the required result.
Now we prove sufficiency. Assume that the condition (2.15) is satisfied.
Then it follows that ln [1/Em,n (f, Hq )]1/(m+n) → ∞ as m, n → ∞.
This yields
q
lim
(m+n)
m,n→∞
Em,n (f, Hq ) = 0.
This relation and the estimate |am+1n+1 (f )| ≤ Em,n (f, Hq ) yield the relation
(2.16). This means that f (z1 , z2 ) ∈ Hq is an entire transcendental function.
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
Now we prove
P
m n
Theorem 2.3. Let f (z1 , z2 ) = ∞
m,n=0 amn z1 z2 , then the entire function f (z1 , z2 )
∈ B(p, q, κ) of finite order ρ, is of type τ if and only if
(2.21)
τ=
1
1
ρ
lim sup {mm nn Em,n
(f, B(p, q, κ))} m+n .
eρ m,n→∞
Proof. We prove the above result in two steps.
First we consider the space B(p, q, κ), q = 2, 0 < p < 2 and κ ≥ 1. Let
f (z) ∈ B(p, q, κ) be of order ρ. From Theorem 1.2, for any > 0, there exists
a natural number n0 = n0 () such that
(2.22)
|amn | ≤ m
−m/ρ −n/ρ
n
[eρ(τ + )]
m+n
ρ
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We denote the partial sum of the Taylor series of a function f (z1 , z2 ) by
Tm,n (f, z1 , z2 ) =
m X
n
X
aj1 j2 z1j1 z2j2 ,
j1 =0 j2 =0
we write
Em,n (f, B(p, 2, κ)) = kf − Tm,n (f )kp,2,κ
Z 1 Z 1
=
{(1 − r1 )(1 − r2 )}κ(1/p−1/2)−1
(2.23)
0
0
! κ2
×
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
XX
j1
r12j1 r22j2 |aj1 j2 |2
dr1 dr2
1
κ
,
Ramesh Ganti and G.S.
Srivastava

j2
Title Page
where
Contents
XX
j1
S1 =
r12j1 r22j2 |aj1 j2 |2 = S1 + S2 +
j1 =0 j2 =n+1
∞
X
r12j1 r22j2 |aj1 j2 |2 ,
j1 =m+1 j2 =n+1
j2
m
∞
X
X
∞
X
r12j1 r22j2 |aj1 j2 |2
and
S2 =
∞
n
X
X
r12j1 r22j2 |aj1 j2 |2 .
j1 =m+1 j2 =0
Since S1 , S2 are bounded, and r1 , r2 < 1 therefore the above expression (2.23)
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becomes
(2.24) Em,n (f, B(p, 2, κ))
(
∞
X
≤ DB[m, p, 2, κ]B[n, p, 2, κ]
∞
X
) 21
|aj1 j2 |2
,
j1 =m+1 j2 =n+1
where D is a constant and B(a, b) (a, b > 0) denotes the beta function. By
using (2.22), we have
∞
X
∞
X
|aj1 j2 |2
j1 =m+1 j2 =n+1
∞
X
≤
≤
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
∞
X
−
j1
2j1
ρ
j1 =m+1 j2 =n+1
∞
2j
X
− 1
j1 ρ [eρ(τ
j1 =m+1
−
j2
2j2
ρ
+ )]
[eρ(τ + )]
2j1
ρ
∞
X
Ramesh Ganti and G.S.
Srivastava
2(j1 +j2 )
ρ
Title Page
2j
− 2
j2 ρ+ [eρ(τ
+ )]
2j2
ρ
Contents
j2 =n+1
≤ O(1)(m + 1)−2(m+1)/ρ (n + 1)−2(n+1)/ρ [eρ(τ + )]
2(m+n+2)
ρ
.
Using the above inequality in (2.24), we get
ρ
Em,n
(f, B(p, 2, κ)) ≤ Dρ B ρ [m, p, 2, κ]B ρ [n, p, 2, κ]Y [eρ(τ + )](m+n+2) ,
where Y = (m + 1)−(m+1) (n + 1)−(n+1) .
Now proceeding to limits and since is arbitrary, we have
(2.25)
1
1
ρ
lim sup {mm nn Em,n
(f, B(p, 2, κ))} m+n ≤ τ.
eρ m,n→∞
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For the reverse inequality, since from the right hand side of (2.24),
|am+1n+1 |B[m, p, 2, κ]B[n, p, 2, κ] ≤ Em,n (f, B(p, 2, κ))
we have
ρ
ρ
mm/(m+n) nn/(m+n) |am+1n+1 |ρ/(m+n) B (m+n) [m, p, 2, κ]B (m+n) [n, p, 2, κ]
ρ
≤ {Em,n
mm nn }1/(m+n) .
Now proceeding to limits, we obtain
τ≤
(2.26)
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
1
1
ρ
lim sup {mm nn Em,n
(f, B(p, 2, κ))} m+n .
eρ m,n→∞
Ramesh Ganti and G.S.
Srivastava
From (2.25) and (2.26), we get the required result.
In the second step, for the general case B(p, q, κ), q 6= 2, we have
(2.27)
Title Page
Em,n (f, B(p, q, κ)) ≤ kf − Tm,n (f )kp,q,κ
Z 1 Z 1
=
{(1 − r1 )(1 − r2 )}κ(1/p−1/q)−1
0
JJ
J
0
! κq
×
Contents
XX
j1
r1qj1 r2qj2 |aj1 j2 |q
dr1 dr2
1
κ

j2
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where
XX
j1
j2
r12j1 r22j2 |aj1 j2 |2 = S1 + S2 +
∞
X
∞
X
j1 =m+1 j2 =n+1
r12j1 r22j2 |aj1 j2 |2 ,
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S1 =
m
∞
X
X
r12j1 r22j2 |aj1 j2 |2
and
j1 =0 j2 =n+1
∞
n
X
X
S2 =
r12j1 r22j2 |aj1 j2 |2 .
j1 =m+1 j2 =0
Since S1 , S2 are bounded, therefore the above expression (2.27) becomes
Em,n (f, B(p, q, κ))
Z 1
( X
∞
≤G
{(1 − r)κ(1/p−1/q)−1 }r(s+1)κ dr
0
where
Z
0
∞
X
) 1q
|aj1 j2 |q
,
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
j1 =m+1 j2 =n+1
1
Ramesh Ganti and G.S.
Srivastava
{(1 − r)κ(1/p−1/q)−1 }r(s+1)κ dr
Z 1
(m+1)κ
κ(1/p−1/q)−1
=
{(1 − r1 )
}r1
dr1
0
Z 1
κ(1/p−1/q)−1 (n+1)κ
×
{(1 − r2 )}
r2
dr2 .
0
Since r1 , r2 < 1, therefore we have
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(2.28) Em,n (f, B(p, q, κ))
(
≤ GB[m, p, q, κ]B[n, p, q, κ]
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∞
X
∞
X
) 1q
q
|aj1 j2 |
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,
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j1 =m+1 j2 =n+1
J. Ineq. Pure and Appl. Math. 7(2) Art. 51, 2006
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where G is constant and B[m, p, q, κ] is Euler’s integral of the first kind. Using
(2.22), we get
∞
X
∞
X
|aj1 j2 |q
j1 =m+1 j2 =n+1
∞
X
≤
≤
∞
X
−
j1
qj1
ρ
j1 =m+1 j2 =n+1
∞
qj
X
− ρ1
j1 [eρ(τ
j1 =m+1
−
j2
qj2
ρ
+ )]
[eρ(τ + )]
qj1
ρ
∞
X
q(j1 +j2 )
ρ
qj
− 2
j2 ρ+ [eρ(τ
+ )]
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
qj2
ρ
j2 =n+1
≤ O(1)(m + 1)−q(m+1)/ρ (n + 1)−q(n+1)/ρ [eρ(τ + )]
q(m+n+2)
ρ
Ramesh Ganti and G.S.
Srivastava
.
Using the above inequality in (2.28), we get
ρ
Em,n
(f, B(p, q, κ))
ρ
ρ
ρ
Title Page
(m+n+2)
≤ G B [m, p, q, κ]B [n, p, q, κ]Y [eρ(τ + )]
,
where Y = (m + 1)−(m+1) (n + 1)−(n+1) . Now proceeding to limits, since is
arbitrary, we have
(2.29)
1
1
ρ
lim sup {mm nn Emn
(f, B(p, q, κ))} m+n ≤ τ.
eρ m,n→∞
Let 0 < p < q < 2, and κ, q ≥ 1. Since
Em,n (f, B(p1 , q1 , κ1 )) ≤ 21/q−1/q1 [κ(1/p − 1/q)]1/κ−1/κ1 Em,n (f, B(p, q, κ)),
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where p1 = p, q1 = 2 and κ1 = κ, and the condition (2.21) has already been
proved for the space B(p, 2, κ), we get
1
ρ
lim sup {mm nn Em,n
(f, B(p, q, κ))} m+n
m,n→∞
1
ρ
≥ lim sup {mm nn Em,n
(f, B(p, 2, κ))} m+n = τ.
m,n→∞
Now let 0 < p ≤ 2 < q. Since, in this case we have
M2 (f, r1 , r2 ) ≤ Mq (f, r1 , r2 ),
0 < r1 , r2 < 1,
therefore
Ramesh Ganti and G.S.
Srivastava
1
ρ
(f, B(p, q, κ))} m+n
lim sup{mm nn Em,n
(2.30)
m,n→∞
1
≥ lim sup {mm nn |amn |ρ } m+n
m,n→∞
= eρτ.
From (2.29) and (2.30), we get the required result.
P∞
Theorem 2.4. Let f (z1 , z2 ) = m,n=0 amn z1m z2n , then the entire function f (z1 , z2 )
∈ Hq having finite order ρ is of type τ if and only if
τ=
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Lastly we prove
(2.31)
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
1
1
ρ
lim sup {mm nn Em,n
(f, Hq )} m+n .
eρ m,n→∞
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Proof. Since f (z1 , z2 ) =
we have
P∞
m,n=0
amn z1m z2n is an entire transcendental function,
p
|amn | = 0.
m+n
lim
(2.32)
m,n→∞
Therefore f (z1 , z2 ) ∈ B(p, q, κ), 0 < p < q ≤ ∞; q, κ ≥ 1. We have
(2.33)
1
1
ρ
lim sup {mm nn Em,n
(f, Hq )} m+n
eρ m,n→∞
n
q
o 1
1
ρ
m+n = τ
≥
lim sup mm nn Em,n
f, B , q, q
eρ m,n→∞
2
ξ(f ) =
for 1 ≤ q < ∞. Using the estimate (2.18) we prove inequality (2.33) in the
case q = ∞. For the reverse inequality
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
Title Page
ξ(f ) ≤ τ,
(2.34)
we have
Em,n (f, Hq ) ≤
∞
X
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∞
X
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|aj1 j2 (f )|.
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Using (2.22), we get
Close
ρ
Em,n
(f, Hq )
−(m+1)
−(n+1)
(m+n+2)
≤ O(1)(m + 1)
(n + 1)
[eρ(τ + )]
1
1
ρ
(f, Hq )} (m+n+2) .
⇒ τ + ≥ {(m + 1)(m+1) (n + 1)(n+1) Em,n
eρ
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Now proceeding to limits, since is arbitrary, we get
(2.35)
τ≥
1
1
ρ
lim sup {mm nn Em,n
(f, Hq )} m+n .
eρ m,n→∞
From (2.33) and (2.35), we obtain the required result.
Now we prove sufficiency. Assume that the condition (2.31) is satisfied.
ρ
Then it follows that {Em,n
(f, Hq )}1/(m+n) → 0 as m, n → ∞. This yields
lim
m,n→∞
q
(m+n)
Em,n (f, Hq ) = 0.
This relation and the estimate |am+1n+1 (f )| ≤ Em,n (f, Hq ) yield the inequality
(2.32). This implies that f (z1 , z2 ) ∈ Hq is an entire transcendental function.
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
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References
[1] S.K. BOSE, AND D. SHARMA, Integral functions of two complex variables, Compositio Math., 15 (1963), 210–226.
[2] S.B. VAKARCHUK AND S.I. ZHIR, On some problems of polynomial approximation of entire transcendental functions, Ukrainian Mathem. J., 54(9)
(2002), 1393–1401.
Approximation of Entire
Functions of Two Complex
Variables in Banach Spaces
Ramesh Ganti and G.S.
Srivastava
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