IJMMS 27:9 (2001) 555–563
PII. S0161171201005737
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
DEGREE OF APPROXIMATION OF CONJUGATE OF A FUNCTION
BELONGING TO Lip(ξ(t), p) CLASS BY MATRIX SUMMABILITY
MEANS OF CONJUGATE FOURIER SERIES
SHYAM LAL and HARE KRISHNA NIGAM
(Received 21 August 2000)
Abstract. We determine the degree of approximation of conjugate of a function belonging
to Lip(ξ(t), p) class by matrix summability means of a conjugate series of a Fourier series.
2000 Mathematics Subject Classification. 42B05, 42B08.
1. Introduction. Bernstein [2], Alexits [1], Sahney and Goel [14], and Chandra [4]
have determined the degree of approximation of a function belonging to Lip α by
(C, 1), (C, δ), (N, pn ), and (N, pn ) means of its Fourier series. Working in the same
direction Sahney and Rao [15] and Khan [6] have studied the degree of approximation of functions belonging to Lip(α, p) by (N, pn ) and (N, p, q) means, respectively.
The (N, p, q) summability reduces to (N, pn ) summability for qn = 1 for all n, and to
(N, qn ) means when pn = 1 for all n. After quite a good amount of work on degree
of approximation of function by different summability means of its Fourier series,
for the first time in 1981, Qureshi [12, 13] discussed the degree of approximation of
conjugate of a function belonging to Lip α and Lip(α, p) by (N, pn ) means of conjugate Fourier series. But nothing seems to have been done so far to obtain the degree
of approximation of conjugate of a function belonging to Lip(ξ(t), p) class by matrix
means of conjugate Fourier series. The Lip(ξ(t), p) class is a generalization of Lip α
and Lip(α, p). Matrix means includes as special cases the method of (C, 1), (C, δ),
(N, pn ), (N, pn ), and (N, p, q) means. In an attempt to make an advance study in this
direction, we, in this paper, establish a theorem on degree of approximation of conjugate of a function of Lip(ξ(t), p) class by matrix summability means of conjugate
series of a Fourier series then both the results of Qureshi [12, 13] come out as particular cases of our theorem.
2. Definitions and notations. Let f be periodic with period 2π and integrable in
the Lebesgue sense. Let its Fourier series be given by
f (x) ∼
∞
1
a0 +
an cos nx + bn sin nx .
2
n=1
(2.1)
The conjugate series of (2.1) is given by
∞
∞
Bn (x).
an sin nx − bn cos nx =
n=1
n=1
(2.2)
556
S. LAL AND H. K. NIGAM
Let {pn } be a nonnegative nonincreasing generating sequence for (N, pn ) method
such that
Pn = P (n) = p0 + p1 + p2 + · · · + pn → ∞,
as n → ∞.
(2.3)
Let T = (an,k ) be an infinite triangular matrix satisfying the Silverman Toeplitz [16],
that is,
n
an,k → 1, as n → ∞,
an,k = 0, for k > n,
k=0
n
an,k ≤ M,
(2.4)
a finite constant.
k=0
Let
∞
m=0 um
be an infinite series such that
k
sk = u0 + u1 + u2 + · · · + uk =
um ,
(2.5)
m=0
that is, sk denotes the kth partial sum of the series
The sequence-to-sequence transformation
tn =
n
an,k sk =
k=0
n
∞
m=0 um .
an,n−k sn−k
(2.6)
k=0
defines the sequence {tn } of matrix means of the sequence {sn } generated by the
sequence of the coefficients (an,k ). The series un is said to be summable to the sum
“S” by matrix method if limn→∞ tn exists and equal to S (see Zygmund [17]) and we
write
tn → S(T ),
as n → ∞.
(2.7)
2.1. Particular cases. Seven important cases of matrix means are
(1) (C, 1) means when an,k = 1/(n + 1).
(2) Harmonic means when an,k = 1/(n − k + 1) log n.
n+δ / δ .
(3) (C, δ) means when an,k = n−k+δ−1
δ−1
p−1
(4) (H, p) means when an,k = 1/logp−1 (n + 1) q=0 logq (k + 1).
n
(5) Nörlund means [11] when an,k = pn−k /Pn where Pn = k=0 pk , qn = 1 for all n.
(6) Riesz means (N, pn ) [5] when an,k = pk /Pn , qn = 1 for all n.
(7) Generalized Nörlund mean (N, p, q) [3] when an,k = pn−k qk /Rn where
Rn =
n
pk qn−k .
(2.8)
k=0
In particular cases (5), (6), and (7), {pn } and {qn } are two nonnegative monotonic
nonincreasing sequences of real constants.
We define the norm
2π
f p =
0
f (x)p dx
1/p
,
p≥1
(2.9)
DEGREE OF APPROXIMATION OF CONJUGATE OF A FUNCTION . . .
557
and let the degree of approximation be given by (see Zygmund [17])
En (f ) = Min f − Tn p ,
(2.10)
Tn
where Tn (x) is some nth degree trigonometric polynomial.
A function f ∈ Lip α if
f (x + t) − f (x) = O t α ,
for 0 < α ≤ 1
(2.11)
and f ∈ Lip(α, p) if
2π
0
f (x + t) − f (x)p dx
1/p
= O tα ,
0 < α ≤ 1, p ≥ 1
(2.12)
(see [10, Definition 5.38]).
Given a positive increasing function ξ(t) and an integer p > 1, then f (x) ∈
Lip(ξ(t), p) if
2π
0
f (x + t) − f (x)p dx
1/p
= O ξ(t) ,
(see [8]).
(2.13)
In case ξ(t) = t α , we notice that Lip(ξ(t), p) class coincides with known Lip(α, p)
class [10].
We use the following notations:
ψ(t) = f (x + t) − f (x − t),
An,τ =
τ
an,n−k ,
k=0
τ = Integral part of
K n (t) =
1
1
=
,
t
t
(2.14)
n
1 cos(n − k − 1/2)t
.
an,n−k
2π k=0
sin t/2
3. Known theorems. Qureshi [12] has proved the following theorem.
Theorem 3.1. If the sequence {pn } satisfies the following conditions:
npn < C Pn ,
n
kpk − pk−1 < C Pn ,
(3.1)
k=1
then the degree of approximation of a function f˜(x), conjugate to a periodic function
f with period 2π and belonging to the class Lip α, 0 < α < 1 by Nörlund means of its
conjugate series, is given by
f˜(x) − t̃n (x) = O
n
1 Pk
,
Pn k=1 kα+1
where t̃n (x) are the (N, pn ) means of series (2.2).
(3.2)
558
S. LAL AND H. K. NIGAM
Qureshi [13] has proved another theorem in the following form.
Theorem 3.2. If f (x) is periodic and belongs to the class Lip(α, p) for 0 < α ≤ 1,
and if the sequence {pn } is as defined in (2.3) with other requirements therein and if
n p(y)q
y qα+2−δq−q
1
then
=O
t̃n − f˜
= O
p
p(n)
nα−1/q−δ−1
,
(3.3)
1
nα−1/p
,
(3.4)
where t̃n are the (N, pn ) means of the series (2.2) and 1/p+1/q = 1 such that 1 ≤ p ≤ ∞.
4. Main theorem. Our object of this paper is to prove the following theorem.
Theorem 4.1. If T = (an,k ) is an infinite regular triangular matrix such that the
elements an,k is nonnegative and nondecreasing with k, then the degree of approximation of a function f˜(x), conjugate to a 2π -periodic function f belonging to Lip(ξ(t), p)
class by matrix summability means of its conjugate series is given by
t̃n (x) − f˜(x)
= O ξ 1 n1/p
n
(4.1)
provided ξ(t) satisfies the following conditions:
1/n
0
π
p 1/p
t ψ(t)
1
,
dt
=O
ξ(t)
n
1/n
t −δ ψ(t)
ξ(t)
p
1/p
dt
= O nδ ,
(4.2)
(4.3)
where δ is an arbitrary number such that q(1 − δ) − 1 > 0, conditions (4.2) and (4.3)
hold uniformly in x,
n
t̃n (x) =
an,n−k s n−k (x),
(4.4)
k=0
that is, matrix means of conjugate Fourier series (2.2), 1/p +1/q = 1, such that 1 ≤ p ≤
∞ and
1 π
1
f˜(x) = −
(4.5)
ψ(t) cot t dt.
2π 0
2
5. Lemmas. For the proof of our theorem the following lemmas are required.
Lemma 5.1 [9]. If an,k is nonnegative and nonincreasing with k, then for 0 ≤ a ≤ b ≤
∞, a ≤ t ≤ π and any n,
b
i(n−k)t = O An,τ .
an,n−k e
k=a
(5.1)
DEGREE OF APPROXIMATION OF CONJUGATE OF A FUNCTION . . .
559
Lemma 5.2. Under the conditions of Theorem 4.1 on (an,k ) for 0 < 1/n ≤ t ≤ π ,
K n (t) = O
An,τ
.
t
(5.2)
Proof. Since for 0 < 1/n ≤ t ≤ π , sin(t/2) < t, therefore for t > 0 and τ ≤ n, we
have,
n
1 cos(n − k − 1/2)t K(t) = an,n−k
2π
sin(t/2)
k=0
n
1
an,n−k ei(n−k−1/2)t Re
≤
2π
sin(t/2)
k=0
n
1
i(n−k)t −it/2 e
=O
an,n−k e
t k=0
(5.3)
n
1
i(n−k)t an,n−k e
=O
t
k=0
=O
An,τ
t
by Lemma 5.1.
6. Proof of the main theorem. Let s n (x) denote the nth partial sum of series (2.2),
then, following [7], we have
1
s n (x) − −
2π
π
0
1 π
1
cos(n + 1/2)t
dt.
ψ(t) cot t dt =
ψ(t)
2
2π 0
sin t/2
(6.1)
Now
n
an,n−k s n−k (x) − −
k=0
1
=
2π
π
0
1
2π
π
0
1
ψ(t) cot t dt
2
(6.2)
n
cos(n − k − 1/2)t
dt
ψ(t)
an,n−k
sin(t/2)
k=0
or
1
t̃n (x) − f˜(x) =
2π
π
ψ(t)
0
π
=
0
n
an,n−k
k=0
ψ(t)K n (t) dt
1/n
=
0
cos(n − k − 1/2)t
dt
sin(t/2)
(6.3)
π
ψ(t)K n (t) dt +
= I 1 + I2 .
1/n
ψ(t)K n (t) dt
560
S. LAL AND H. K. NIGAM
Applying Hölder’s inequality and the fact that ψ(t) = W (Lip ξ(t), p), we get
1/n
I1 =
ψ(t)K n (t) dt
0
≤O
1/n
0
1
=o
n
q 1/q
p 1/p 1/n t ψ(t)
K n (t)ξ(t)
dt
dt
t
ξ(t)
0
1/n 0
n
q 1/q
cos(n−k−1/2)t ξ(t) 1 dt
an,n−k
t 2π k=0
sin(t/2)
1/n 1
=O
n
0
n
ξ(t) an,n−k q
t
t k=0
by (4.2)
1/q
1/q
ξ(t) q
dt
t2
0
1/q
1/n
1
1
dt
O ξ
=O
by mean value theorem
n
n
t 2q
1
−2q+1 1/n 1/q
1
1
t
=O
O ξ
n
n
−2q + 1 1
=O
1
n
1/n (6.4)
ξ(1/n)
O n2−1/q
n
1
n1−1/q ,
=O ξ
n
1
1 1
n1/p
since + = 1 .
I1 = O ξ
n
p q
=O
Consider I2
I2 =
ξ(t)
1/n
=O
t −δ ψ(t) p
π
π
1/n
= O nδ · O
= O nδ · O
1/p π
dt
1/n
K n (t)ξ(t)
t −δ
1/q
q
dt
q 1/q
p 1/p π ξ(t)An,τ
t −δ ψ(t)
dt
O
dt
ξ(t)
t −δ+1
1/n
π
1/n
n 1/n
ξ(t)An,τ
t −δ+1
by Lemma 5.2
1/q
q
dt
ξ(1/y)An,[y]
y δ−1
q
by condition (4.3)
dy
y2
1/q
n 1/q
dy
1
An,n
= O nδ · O ξ
n
y q(δ−1)+2
1
by mean value theorem
DEGREE OF APPROXIMATION OF CONJUGATE OF A FUNCTION . . .
561
y −q(δ−1)−1 n 1/q
1
= O nδ ξ
n
−q(δ − 1) − 1 1
1
O n−δ+1−1/q
= O nδ ξ
n
1
=O ξ
n1−1/q
n
1
1 1
n1/p
I2 = O ξ
since + = 1 .
n
p q
(6.5)
By combining (6.3), (6.4), and (6.5) we have
t̃n (x) − f˜(x) = O ξ 1 n1/p ,
n
(6.6)
therefore
1/p p
1
n1/p
dx
ξ
n
0
1/p 2π
1
=O ξ
n1/p
dx
n
0
1
=O ξ
n1/p .
n
t̃n (x) − f˜(x)
= O
p
2π
(6.7)
This completes the proof of the theorem.
7. Applications. The following corollaries can be derived from the main theorem.
Corollary 7.1. If ξ(t) = t α , 0 < α ≤ 1, then the degree of approximation of a
function f˜(x), conjugate to 2π -periodic function f belonging to the class Lip(α, p) is
given by
1
t̃n (x) − f˜(x) = O
.
(7.1)
nα−1/p
Proof. We have
t̃n (x) − f˜(x)
= O
p
or
2π
0
t̃n (x) − f˜(x)p dx
1/p
2π
1/p
p
1
t̃n (x) − f˜(x)p dx
n1/p
=O
ξ
n
0
or
O(1) = O
Hence
2π
0
t̃n (x) − f˜(x)p dx
1/p O
1
.
ξ(1/n)n1/p
t̃n (x) − f˜(x) = O ξ 1 n1/p
n
(7.2)
(7.3)
(7.4)
(7.5)
562
S. LAL AND H. K. NIGAM
for if not the right-hand side will be O(1), therefore
α
1
1
t̃n (x) − f˜(x) = O
n1/p = O
.
α−1/p
n
n
(7.6)
This completes the proof.
Corollary 7.2. If p → ∞ in Corollary 7.1, then for 0 < α < 1,
t̃n (x) − f˜(x) = O 1 .
p
nα
(7.7)
Remark 7.3. An independent proof of Corollary 7.1 can be derived along the same
lines as the theorem.
8. Particular cases. (1) If an,k = pn−k /pn , ξ(t) = t α , 0 < α < 1, p → ∞ and using
n
1/nα ≤ 1/pn k=1 pk /kα+1 (see [14, Lemma 1]), then the result of Qureshi [12] becomes
the particular case of the main theorem.
(2) The result of Qureshi [13] becomes the particular case of our theorem if (an,k )
is defined as in case (1) and ξ(t) = t α , 0 < α ≤ 1.
Acknowledgements. The authors are grateful to Prof L. M. Tripathi and to Prof S.
N. Lal, Department of Mathematics, Banaras Hindu University, Varanasi-221005, India
to see the manuscript of this paper. Authors are also thankful to Prof A. P. Dwivedi,
Director and to Prof V. K. Jain, former Director, H. B. T. I, Kanpur-208002 (India) for
encouraging to this work.
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Shyam Lal: Department of Mathematics, Harcourt Butler Technological Institute,
Nawab Ganj, Kanpuk-208 002 (U.P.), India
Hare Krishna Nigam: Department of Mathematics, Harcourt Butler Technological
Institute, Nawab Ganj, Kanpuk-208 002 (U.P.), India
E-mail address: harekrishnan@yahoo.com