Open Access

A study on degree of approximation by Karamata summability method

Journal of Inequalities and Applications20112011:85

https://doi.org/10.1186/1029-242X-2011-85

Received: 25 January 2011

Accepted: 12 October 2011

Published: 12 October 2011

Abstract

Vuĉkoviĉ [Maths. Zeitchr. 89, 192 (1965)] and Kathal [Riv. Math. Univ. Parma, Italy 10, 33-38 (1969)] have studied summability of Fourier series by Karamata (K λ ) summability method. In present paper, for the first time, we study the degree of approximation of function f Lip (α,r) and f W(L r (t)) by Kλ-summability means of its Fourier series and conjugate of function f ̃ Lip ( α , r ) and f ̃ W ( L r , ξ ( t ) ) by Kλ-summability means of its conjugate Fourier series and establish four quite new theorems.

MSC: primary 42B05; 42B08; 42A42; 42A30; 42A50.

Keywords

degree of approximationLip(α,r) classW(L r (t)) class of functionsFourier seriesconjugate Fourier seriesKλ-summabilityLebesgue integral

1 Introduction

The method K λ was first introduced by Karamata [1] and Lotosky [2] reintroduced the special case λ = 1. Only after the study of Agnew [3], an intensive study of these and similar cases took place. Vuĉkoviĉ [4] applied this method for summability of Fourier series. Kathal [5] extended the result of Vuĉkoviĉ [4]. Working in the same direction, Ojha [6], Tripathi and Lal [7] have studied Kλ-summability of Fourier series under different conditions. The degree of approximation of a function f Lip α by Cesàro and Nörlund means of the Fourier series has been studied by Alexits [8], Sahney and Goel [9], Chandra [10], Qureshi [11], Qureshi and Neha [12], Rhoades [13], etc. But nothing seems to have been done so far in the direction of present work. Therefore, in present paper, we establish two new theorems on degree of approximation of function f belonging to Lip (α,r) (r ≥ 1) and to weighted class W(L r , ξ (t))(r ≥ 1) by Kλ-means on its Fourier series and two other new theorems on degree of approximation of function f ̃ , conjugate of a 2π-periodic function f belonging to Lip (α,r) (r > 1) and to weighted class W(L r (t)) (r ≥ 1) by Kλ-means on its conjugate Fourier series.

2 Definitions and notations

Let us define, for n = 0, 1, 2,..., the numbers n m , for 0 ≤ mn, by
v - 0 n - 1 ( x + ν ) = m = 0 n n m x m = Γ ( x + n ) Γ ( x ) = x ( x + 1 ) ( x + 2 ) . . . ( x + n - 1 ) .
(2.1)

The numbers n m are known as the absolute value of stirling number of first kind

Let {s n } be the sequence of partial sums of an infinite series ∑u n , and let us write
s n λ = Γ ( λ ) Γ ( λ + n ) m = 0 n [ n m ] λ m s m
(2.2)
to denote the n th Kλ-mean of order λ > 0. If s n λ s as n → ∞, where s is a fixed finite number, then the sequence {s n } or the series ∑u n is said to be summable by Karamata method (K λ ) of order λ > 0 to the sum s, and we can write
s n λ s K λ as n .
(2.3)
Let f be a 2π-periodic function and integrable in the sense of Lebesgue. The Fourier series associated with f at a point x is defined by
f ( x ) ~ a 0 2 + n = 1 ( a n cos n x + b n sin n x ) n = 1 A n ( x )
(2.4)

with n th partial sums s n (f;x).

The conjugate series of Fourier series (2.4) is given by
n = 1 ( a n sin n x - b n cos n x ) n = 1 B n ( x )
(2.5)

with n th partial sums s ̃ n ( f ; x ) .

Throughout this paper, we will call (2.5) as conjugate Fourier series of function f.

L-norm of a function f: RR is defined by
f = sup { f ( x ) : x R }
(2.6)
L r -norm is defined by
f r = 0 2 π f ( x ) r d x 1 r , r 1 .
(2.7)

The degree of approximation of a function f: RR by a trigonometric polynomial t n of degree n under sup norm || || is defined by

(Zygmund [14])
t n - f = sup { t n x - f x : x R }
(2.8)
and E n (f) of a function f L r is given by
E n ( f ) = min t n t n - f r .
(2.9)
This method of approximation is called trigonometric Fourier approximation. A function f Lip α if
f x + t - f x = O ( t α ) for 0 < α 1
(2.10)

and

f Lip (α, r) for 0 ≤ x ≤ 2π, if
0 2 π f x + t - f x r d x 1 r = O t α , 0 < α 1  and  r 1
(2.11)

(definition 5.38 of McFadden [15]).

Given a positive increasing function ξ (t) and an integer r ≥ 1, f Lip (ξ(t), r), if
0 2 π f x + t - f x r d x 1 r = O ( ξ ( t ) )
(2.12)

and that

f W (L r , ξ (t)) if
0 2 π f x + t - f x sin β x r d x 1 r = O ξ t , β 0
(2.13)

If β = 0, our newly defined weighted i.e. W (L r , ξ (t)) reduces to Lip (ξ (t), r), if ξ (t) = t α then Lip (ξ (t), r) coincides with Lip (α, r) and if r → ∞ then Lip (α, r) reduces to Lip α.

We observe that
Lip  α Lip( α , r ) Lip ( ξ ( t ) , r ) W ( L r , ξ ( t ) ) for 0 < α 1 , r 1 .
We write
ϕ t = f x + t + f x - t - 2 f x K n t = m = 0 n n m λ m sin m + 1 2 t Γ λ + n sin t 2 ψ t = f x + t - f x - t K ̃ n t = m = 0 n n m λ m cos m + 1 2 t Γ λ + n sin t 2 f ̃ x = - 1 2 π 0 π ψ t cot t 2 d t

3 The main results

3.1 Theorem 1

If a function f, 2π-periodic, belonging to Lip (α, r) then its degree of approximation by Kλ-summability means on its Fourier series is given by
s n - f r = O 1 n + 1 α - 1 r log n + 1 e n + 1 + 1 Γ λ + n , 0 < α 1 , n = 0 , 1 , 2 . . . ,
(3.1)

where s n is Kλ-mean of Fourier series (2.4).

3.2 Theorem 2

If a function f, 2π-periodic, belonging to W (L r , ξ (t)) then its degree of approximation by Kλ-summability means on its Fourier series is given by
s n - f r = O n + 1 β + 1 r ξ 1 n + 1 log n + 1 n + 1 + 1 n + 1 2 + 1 Γ λ + n
(3.2)
provided that ξ (t) satisfies the following conditions:
ξ t t is non - increasing  i n t ,
(3.3)
0 1 n + 1 t ϕ t ξ t r sin β r t d t 1 r = O 1 n + 1 ,
(3.4)
and
1 n + 1 π t - δ ϕ t ξ t r d t 1 r = O n + 1 δ ,
(3.5)

where δ is an arbitrary positive number such that s (1 - δ) - 1 > 0, 1 r + 1 s = 1 , 1 ≤ r ≤ ∞, conditions (3.4) and (3.5) hold uniformly in x, s n is Kλ-mean of Fourier series (2.4).

3.3 Theorem 3

If a function f ̃ , conjugate to a 2π-periodic function f, belonging to Lip(α,r) then its degree of approximation by Kλ-summability means on its conjugate Fourier series is given by
s ̃ n - f ̃ r = O 1 n + 1 α - 1 r log n + 1 e n + 1 2 + 1 Γ λ + n + 1 , 0 < α 1 , n = 0 , 1 , 2 , . . . ,
(3.6)
where s ̃ n is Kλ-mean of conjugate Fourier series (2.5) and
f ̃ x = - 1 2 π 0 π ψ t cot t 2 d t .

3.4 Theorem 4

If a function f ̃ , conjugate to a 2π-periodic function f, belonging to W (L r (t)) then its degree of approximation by Kλ-summability means on its conjugate Fourier series is given by
s ̃ n - f ̃ r = O n + 1 β + 1 r ξ 1 n + 1 2 n + 1 2 + log n + 1 n + 1 2 + 1 Γ λ + n
(3.7)
provided that ξ (t) satisfies the conditions (3.3)-(3.5) in which δ is an arbitrary positive number such that s (1 - δ) - 1 > 0, 1 r + 1 s = 1 ,1≤r≤ ∞. Conditions (3.4) and (3.5) hold uniformly in x, s ̃ n is Kλ-mean of conjugate Fourier series (2.5) and
f ̃ x = - 1 2 π 0 π ψ t cot t 2 d t .
(3.8)

4 Lemmas

For the proof of our theorems, following lemmas are required.

4.1 Lemma 1

(Vuĉkoviĉ [14]). Let λ > 0 and 0 < t < π 2 , then
Γ λ e i t + n Γ λ cos t + n sin t 2 = sin λ log n + 1 . sin t sin t 2 + O 1 n uniformly in t .

4.2 Lemma 2

K n t = O λ log n + 1 + O 1 .
Proof. For 0 < t < 1 n + 1 , 1 - cos t < t 2 2 , sin ntnt and sin t 2 t π
K n t 1 Γ λ + n m = 0 n n m λ m sin m + 1 2 t sin t 2 = O I m e i t 2 Γ λ e i t + n Γ λ e i t Γ λ + n sin t 2 by (2.1) = O I m Γ λ e i t + n Γ λ + n sin t 2 + O R e Γ λ e i t + n Γ λ + n = O Γ λ cos t + n Γ λ + n I m Γ λ e i t + n Γ λ cos t + n sin t 2 + O Γ λ cos t + n Γ λ + n = O n - λ 1 - cos t I m Γ λ e i t + n Γ λ cos t + n sin t 2 + O n - λ 1 - cos t = O e - λ 1 - cos t log n I m Γ λ e i t + n Γ λ cos t + n sin t 2 + O e - λ 1 - cos t log n = O e - λ 2 t 2 log n + 1 I m Γ λ e i t + n Γ λ cos t + n sin t 2 + O e - λ 2 t 2 log n + 1 .
(4.1)
Considering first part of (4.1) and using Lemma 1,
K n t = O e - λ 2 t 2 log n + 1 sin λ log n + 1 sin t sin t 2 + O e - λ 2 t 2 log n + 1 + O e - λ 2 t 2 log n + 1 = O e - λ 2 t 2 log n + 1 sin λ log n + 1 sin t sin t 2 + O e - λ 2 t 2 log n + 1 = O λ log n + 1 sin λ log n + 1 sin t sin t 2 + O 1 = O λ log n + 1 + O 1 .

4.3 Lemma 3

K ̃ n t = O e - λ 2 t 2 log n + 1 t + O λ log n + 1 sin λ log n + 1 sin t + O e - λ 2 t 2 log n + 1 sin t 2
Proof. For 0 < t < 1 n + 1 , 1 - cos t < t 2 2 , sin ntnt and sin t 2 t π
K n t 1 Γ λ + n m = 0 n n m λ m cos m + 1 2 t sin t 2 = O R e e i t 2 Γ λ e i t + n Γ λ e i t Γ λ + n sin t 2 by (2.1) = O R e Γ λ e i t + n Γ λ + n sin t 2 + O I m Γ λ e i t + n Γ λ + n = O Γ λ cos t + n Γ λ + n sin t 2 + O Γ λ cos t + n Γ λ + n I m Γ λ e i t + n Γ λ cos t + n = O n - λ 1 - cos t sin t 2 + O n - λ 1 - cos t I m Γ λ e i t + n Γ λ cos t + n = O e - λ 1 - cos t log n sin t 2 + O e - λ 1 - cos t log n I m Γ λ e i t + n Γ λ cos t + n = O e - λ 2 t 2 log n sin t 2 + O e - λ 2 t 2 log n I m Γ λ e i t + n Γ λ cos t + n = O e - λ 2 t 2 log n + 1 t + O e - λ 2 t 2 log n + 1 I m Γ λ e i t + n Γ λ cos t + n .
Using Lemma 1,
K n t = O e - λ 2 t 2 log n + 1 t + O e - λ 2 t 2 log n + 1 sin λ log n + 1 sin t + O e - λ 2 t 2 log n + 1 sin t 2 = O e - λ 2 t 2 log n + 1 t + O λ log n + 1 sin λ log n + 1 sin t + O e - λ 2 t 2 log n + 1 sin t 2 .

4.4 Lemma 4

(McFadden [15]), Lemma 5.40) If f(x) belongs to Lip(α,r) on [0,π], then φ(t) belongs to Lip(α,r) on [0,π].

5 Proof of the theorems

5.1 Proof of Theorem 1

Following Titchmarsh [16] and using Riemann-Lebesgue theorem, the m th partial sum s m (x) of series (2.4) at t = x is given by
s m x - f x = 1 2 π 0 π ϕ t sin m + 1 2 t sin t 2 d t
Therefore,
Γ ( λ ) Γ ( λ + n ) m = 0 n [ n m ] λ m { s m ( x ) f ( x ) } = 1 2 π 0 π ϕ ( t ) Γ ( λ ) Γ ( λ + n ) m + 0 n [ n m ] λ m sin ( m + 1 2 ) t sin t 2 d t s m ( x ) f ( x ) = Γ ( λ ) 2 π 0 π ϕ ( t ) K n ( t ) d t = Γ ( λ ) 2 π [ 0 1 n + 1 + 1 n + 1 π ] ϕ ( t ) K n ( t ) d t = O ( I 1.1 ) + O ( I 1.2 ) ( say ) .
(5.1)
Now we consider,
I 1 . 1 = 0 1 n + 1 ϕ t K n t d t .
Using Lemma 2,
I 1 . 1 = O λ log n + 1 0 1 n + 1 ϕ t d t + O 0 1 n + 1 ϕ t d t .
Using Hölder's inequality and Lemma 4,
I 1 . 1 = O λ log n + 1 + 1 0 1 n + 1 t ϕ ( t ) t α r d t 1 r 0 1 n + 1 t α - 1 s d t 1 s = O λ log n + 1 + 1 1 n + 1 t α s - s + 1 α s - s + 1 0 1 n + 1 1 s = O log n + 1 e n + 1 1 n + 1 α s - s + 1 1 s = O log n + 1 e n + 1 1 n + 1 α - 1 + 1 s = O log n + 1 e n + 1 1 n + 1 α - 1 r since 1 r + 1 s = 1 .
(5.2)
Since, for 1 n + 1 t π , sin t 2 t π
K n ( t ) = O 1 Γ λ + n sin t 2 = O 1 Γ λ + n t .
(5.3)
Next we consider,
I 1 . 2 1 n + 1 π ϕ t K n t d t .
Using Hölder's inequality, (5.3) and Lemma 4,
I 1 . 2 = O 1 Γ λ + n 1 n + 1 π t - δ ϕ t t α r d t 1 r 1 n + 1 π t δ + α t s d t 1 s = O 1 Γ λ + n 1 n + 1 - δ 1 n + 1 π t δ + α - 1 s d t 1 s = O 1 Γ λ + n 1 n + 1 - δ t δ + α - 1 s + 1 δ + α - 1 s + 1 1 n + 1 π 1 s = O 1 Γ λ + n 1 n + 1 - δ 1 n + 1 δ + α - 1 s + 1 1 s = O 1 Γ λ + n 1 n + 1 - δ 1 n + 1 δ + α - 1 + 1 s = O 1 Γ λ + n 1 n + 1 α - 1 + 1 s = O 1 Γ λ + n 1 n + 1 α - 1 r .
(5.4)
Combining (5.1), (5.2) and (5.4),
S m - f x = O log n + 1 e n + 1 1 n + 1 α - 1 r + O 1 Γ λ + n 1 n + 1 α - 1 r = O 1 n + 1 α - 1 r log n + 1 e n + 1 + 1 Γ λ + n .

This completes the proof of Theorem 1.

5.2 Proof of Theorem 2

Following the proof of Theorem 1,
S m x - f x = Γ λ 2 π 0 1 n + 1 + 1 n + 1 π ϕ t K n t d t = O I 2 . 1 + O I 2 . 2 say .
(5.5)
We have
ϕ x + t - ϕ x f u + x + t - f u + x + f u - x - t - f u - x .
Hence, by Minkowiski's inequality,
0 2 π ϕ x + t - ϕ x sin β x r d x 1 r 0 2 π f u + x + t - f u + x s i n β x r d x 1 r + 0 2 π f u - x - t - f u - x sin β x r d x 1 r = O ξ t .

Then f W (L r (t)) φ W(L r , ξ (t)).

Now we consider,
I 2 . 1 0 1 n + 1 ϕ t K n t d t .
Using Lemma 2,
I 2 . 1 = O λ log n + 1 + O 1 0 1 n + 1 ϕ t d t .
Using Hölder's inequality and the fact that φ (t) W (L r , ξ (t)),
I 2.1 = O [ { λ log ( n + 1 ) } + 1 ] [ 0 1 n + 1 { t | ϕ ( t ) | sin β ( t ) ξ ( t ) } r d t ] 1 r [ 0 1 n + 1 { ξ ( t ) t sin β t } s d t ] 1 s = O { λ log ( n + 1 ) e } ( 1 n + 1 ) [ 0 1 n + 1 { ξ ( t ) t sin β t } s d t ] 1 s by ( 3.4 )
Since sin t2t/π,
I 2 . 1 = O log n + 1 e n + 1 0 1 n + 1 ξ t t 1 + β s d t 1 s .
Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,
I 2 . 1 = O log n + 1 e n + 1 ξ 1 n + 1 1 n + 1 1 t 1 + β s d t 1 s for some 0 < < 1 n + 1 = O log n + 1 e n + 1 ξ 1 n + 1 t - 1 + β s + 1 - 1 + β s + 1 1 n + 1 1 s = O log n + 1 e n + 1 ξ 1 n + 1 n + 1 1 + β - 1 s = O log n + 1 e n + 1 ξ 1 n + 1 n + 1 β + 1 r since 1 r + 1 s = 1 .
(5.6)
Next we consider,
I 2 . 2 1 n + 1 π ϕ t K n t d t .
Using Hölder's inequality, |sin t| ≤ 1,sin t ≥ 2t/π, (5.3), conditions (3.3), (3.5) and second mean value theorem for integrals,
I 2 . 2 = O 1 n + 1 π 1 Γ λ + n t ϕ t d t = O 1 Γ λ + n 1 n + 1 π t - δ ϕ t sin β t ξ t r d t 1 r 1 n + 1 π ξ t t - δ sin β t t s d t 1 s = O 1 Γ λ + n 1 n + 1 π t - δ ϕ t ξ t r d t 1 r 1 n + 1 π ξ t t - δ + β + 1 s d t 1 s = O 1 Γ λ + n n + 1 δ 1 n + 1 π ξ t t - δ + β + 1 s d t 1 s .
Putting t = 1 y
I 2 . 2 = O n + 1 δ Γ λ + n π n + 1 ξ 1 y y δ - β - 1 s d y y 2 1 s = O n + 1 δ Γ λ + n ξ 1 n + 1 η n + 1 d y d s δ - β - 1 + 2 d t 1 s for some 1 π η n + 1 = O n + 1 δ Γ λ + n ξ 1 n + 1 1 n + 1 d y y s δ - β - 1 + 2 d t 1 s for some 1 π 1 n + 1 = O n + 1 δ Γ λ + n ξ 1 n + 1 y s β + 1 - δ - 1 s β + 1 - δ - 1 1 n + 1 1 s = O n + 1 δ Γ λ + n ξ 1 n + 1 n + 1 1 + β - δ - 1 s = O ξ 1 n + 1 Γ λ + n n + 1 β + 1 r since 1 r + 1 s = 1 .
(5.7)
Now combining (5.5)-(5.7),
s m x - f x = O log n + 1 e n + 1 ξ 1 n + 1 n + 1 β + 1 r + O 1 Γ λ + n ξ 1 n + 1 n + 1 β + 1 r = O n + 1 β + 1 r ξ 1 n + 1 log n + 1 e n + 1 + 1 Γ λ + n .
Now using L r -norm, we get
S m ( x ) f ( x ) = { 0 2 π | S m ( x ) f ( x ) | r d x } 1 r = O [ 0 2 π { ( n + 1 ) β + 1 r ξ ( 1 n + 1 ) } { log ( n + 1 ) e ( n + 1 ) + 1 Γ ( λ + n ) } d x ] 1 r = [ { ( n + 1 ) β + 1 r ξ ( 1 n + 1 ) } { log ( n + 1 ) e ( n + 1 ) + 1 Γ ( λ + n ) } ] [ { 0 2 π d x } 1 r ] = O { ( n + 1 ) β + 1 r ξ ( 1 n + 1 ) } [ log ( n + 1 ) e ( n + 1 ) + 1 Γ ( λ + n ) ] .

This completes the proof of Theorem 2.

5.3 Proof of Theorem 3

Following Lal [7], the m th partial sum S ̃ m x of series (2.5) at t = x
S ̃ m x - - 1 2 π 0 π ψ t cot t 2 d t = 1 2 π 0 π ψ t cos m + 1 2 t sin t 2 d t .
Therefore,
Γ ( λ ) Γ ( λ + n ) m = 0 n [ n m ] λ m { S ˜ ( x ) ( 1 2 π 0 π ψ ( t ) cot ( t 2 ) d t ) } = 1 2 π 0 π ψ ( t ) Γ ( λ ) Γ ( λ + n ) m = 0 n [ n m ] λ m cos ( m + 1 2 ) t sin t 2 d t , S ˜ m ( x ) f ˜ ( x ) = Γ ( λ ) 2 π 0 π ψ ( t ) K ˜ n ( t ) d t = Γ ( λ ) 2 π [ 0 1 n + 1 + 1 n + 1 π ] | ψ ( t ) | | K ˜ n ( t ) | d t = O ( I 3.1 ) + O ( I 3.2 ) .
(5.8)
We consider,
I 3 . 1 = 0 1 n + 1 ψ t K ̃ n t d t .
Using Lemma 3,
I 3 . 1 = O 0 1 n + 1 e - λ 2 t 2 log n + 1 t ψ t d t + O λ log n + 1 0 1 n + 1 sin λ log n + 1 . sin t ψ t d t + O 0 1 n + 1 e - λ 2 t 2 log n + 1 sin t 2 ψ t d t = I 3 . 1 . 1 + I 3 . 1 . 2 + I 3 . 1 . 3 say .
(5.9)
Now consider,
I 3 . 1 . 1 = O 0 1 n + 1 e - λ 2 t 2 log n + 1 t ψ t d t = O 0 1 n + 1 t ψ t t α r d t 1 r 0 1 n + 1 t α - 2 e - λ 2 t 2 log n + 1 s d t 1 s .
Using second mean value theorem for integrals,
I 3 . 1 . 1 = O e - λ 2 log n + 1 n + 1 2 n + 1 1 n + 1 t α - 2 s d t 1 s for 0 < < 1 n + 1 = O e - λ 2 log n + 1 n + 1 2 n + 1 t s α - 2 s + 1 s α - 2 s + 1 1 n + 1 1 s = O 1 n + 1 1 n + 1 α s - 2 s + 1 1 s = O 1 n + 1 1 n + 1 α - 2 + 1 s = O 1 n + 1 1 n + 1 α - 1 - 1 r since 1 r + 1 s = 1 = O 1 n + 1 α - 1 r .
(5.10)
Now we consider,
I 3 . 1 . 2 = O λ log n + 1 0 1 n + 1 sin λ log n + 1 sin t ψ t d t .
Since, for 0 < t < 1 n + 1 , sin n t n t ,
I 3 . 1 . 2 = O λ log n + 1 0 1 n + 1 t ψ t d t .
Using Hölder's inequality and Lemma 4,
I 3 . 1 . 2 = O λ log n + 1 0 1 n + 1 t ψ t t α r d t 1 r 0 1 n + 1 t α s d t 1 s = O λ log n + 1 1 n + 1 t α s + 1 α s + 1 0 1 n + 1 1 s = O λ log n + 1 1 n + 1 1 n + 1 α s + 1 1 s = O λ log n + 1 1 n + 1 1 n + 1 α + 1 s = O log n + 1 1 n + 1 1 n + 1 α + 1 - 1 - 1 s = O log n + 1 1 n + 1 1 n + 1 α + 1 - 1 r since 1 r + 1 s = 1 = O log n + 1 n + 1 2 1 n + 1 α - 1 r .
(5.11)
Next we consider,
I 3 . 1 . 3 = O 0 1 n + 1 e - λ 2 t 2 log n + 1 sin t 2 ψ t d t = O 0 1 n + 1 t ψ t d t = O 0 1 n + 1 t ψ t t α r d t 1 r 0 1 n + 1 t α s d t 1 s = O 1 n + 1 t α s + 1 α s + 1 0 1 n + 1 1 s = O 1 n + 1 1 n + 1 α s + 1 1 s = O 1 n + 1 1 n + 1 α + 1 s = O 1 n + 1 1 n + 1 α + 1 - 1 - 1 s = O 1 n + 1 2 1 n + 1 α - 1 r since 1 r + 1 s = 1 .
(5.12)
Combining (5.9)-(5.12),
I 3 . 1 = O 1 n + 1 α - 1 r + O log n + 1 n + 1 2 1 n + 1 α - 1 r + O 1 n + 1 2 1 n + 1 α - 1 r = O log n + 1 e n + 1 2 n + 1 α - 1 r + O 1 n + 1 α - 1 r .
(5.13)
Since, for 1 n + 1 < t < π , sin t 2 t π
K ̃ n t = O 1 Γ λ + n sin t 2 = O 1 Γ λ + n t .
(5.14)
Next we consider,
I 3 . 2 1 n + 1 π ψ t K ̃ n t d t .
Using Hölder's inequality, (5.14) and Lemma 4,
I 3 . 2 = O 1 Γ λ + n 1 n + 1 π t - δ ψ t t α r d t 1 r 1 n + 1 π t δ + α t s d t 1 s = O 1 Γ λ + n 1 n + 1 - δ 1 n + 1 π t δ + α - 1 s d t 1 s = O 1 Γ λ + n 1 n + 1 - δ t δ + α - 1 s + 1 δ + α - 1 s + 1 1 n + 1 π 1 s = O 1 Γ λ + n 1 n + 1 - δ 1 n + 1 δ + α - 1 s + 1 1 s = O 1 Γ λ + n 1 n + 1 - δ 1 n + 1 δ + α - 1 + 1 s = O 1 Γ λ + n 1 n + 1 α - 1 + 1 s = O 1 Γ λ + n 1 n + 1 α - 1 r since 1 r + 1 s = 1 .
(5.15)
Collecting (5.8), (5.13) and (5.15),
S ̃ m - f ̃ x = O log n + 1 e n + 1 2 n + 1 α - 1 r + 1 n + 1 α - 1 r + O 1 Γ λ + n 1 n + 1 α - 1 r = O 1 n + 1 α - 1 r log n + 1 e n + 1 + 1 Γ λ + n + 1 .

This completes the proof of Theorem 3.

5.4 Proof of Theorem 4

Following the calculations of Theorem 3,
S ̃ m x - f ̃ x = Γ λ 2 π 0 1 n + 1 + 1 n + 1 π ψ t K ̃ n t d t = O I 4 . 1 + O I 4 . 2 .
(5.16)
Now,
I 4 . 1 = O 0 1 n + 1 ψ t K n t d t .
Using Lemma 3,
I 4 . 1 = O 0 1 n + 1 e - λ 2 t 2 log n + 1 t ψ t d t + O λ log n + 1 0 1 n + 1 sin λ log n + 1 sin t ψ t d t + O 0 1 n + 1 e - λ 2 t 2 log n + 1 sin t 2 ψ t d t = I 4 . 1 . 1 + I 4 . 1 . 2 + I 4 . 1 . 3 say .
(5.17)
Using Minkowiski's inequality, we have a fact that f W (L r , ξ (t)) ψ W (L r , ξ (t)). Now we consider,
I 4 . 1 . 1 = O 0 1 n + 1 e - λ 2 t 2 log n + 1 t ψ t d t = O 0 1 n + 1 t ψ t sin β t ξ t r d t 1 r 0 1 n + 1 ξ t e - λ 2 t 2 log n + 1 sin β t s d t 1 s = O e - λ 2 log n + 1 n + 1 2 O 1 n + 1 0 1 n + 1 ξ t sin β t s d t 1 s by ( 3.4 ) = O 1 n + 1 0 1 n + 1 ξ t sin β t s d t 1 s = O 1 n + 1 0 1 n + 1 ξ t t β s d t 1 s since sin t 2 t π .
Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,
I 4 . 1 . 1 = O 1 n + 1 ξ 1 n + 1 1 n + 1 1 t β s d t 1 s for some 0 < < 1 n + 1 = O 1 n + 1 ξ 1 n + 1 t - β s + 1 - β s + 1 1 n + 1 1 s = O 1 n + 1 ξ 1 n + 1 n + 1 β - 1 + 1 - 1 s = O 1 n + 1 2 ξ 1 n + 1 n + 1 β + 1 r since 1 r + 1 s = 1 .
(5.18)
Now,
I 4 . 1 . 2 = O λ log n + 1 0 1 n + 1 sin λ log n + 1 sin t ψ t d t .
Since for 0 < t < 1 n + 1 , sin n t n t ,
I 4 . 1 . 2 = O λ log n + 1 0 1 n + 1 t ψ t d t .
Hölder's inequality and the fact that ψ (t) W (L r , ξ (t)),
I 4 . 1 . 2 = O λ log n + 1 0 1 n + 1 t ψ t sin β t ξ t r d t 1 r 0 1 n + 1 ξ t sin β t s d t 1 s = O log n + 1 O 1 n + 1 0 1 n + 1 ξ t sin β t s d t 1 s by ( 3.4 ) = O log n + 1 n + 1 0 1 n + 1 ξ t t β s d t 1 s since sin t 2 t π .
Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,
I 4 . 1 . 2 = O log n + 1 n + 1 ξ 1 n + 1 1 n + 1 1 t β s d t 1 s for some < < 1 n + 1 = O log n + 1 n + 1 ξ 1 n + 1 t - β s + 1 - β s + 1 1 n + 1 1 s = O log n + 1 n + 1 ξ 1 n + 1 n + 1 β - 1 + 1 - 1 s = O log n + 1 n + 1 2 ξ 1 n + 1 n + 1 β + 1 r since 1 r + 1 s = 1 .
(5.19)
Next we consider,
I 4 . 1 . 3 = O 0 1 n + 1 e - λ 2 t 2 log n + 1 sin t 2 ϕ t d t = O 0 1 n + 1 t ψ t d t .
Using Hölder's inequality and the fact that ψ (t) W (L r (t)),
I 4 . 1 . 3 = O 0 1 n + 1 t ψ t sin β t ξ t r d t 1 r 0 1 n + 1 ξ t sin β t s d t 1 s = O 1 n + 1 0 1 n + 1 ξ t sin β t s d t 1 s by ( 3.4 )
Since, sin t ≥ 2t/π,
I 4 . 1 . 3 = O 1 n + 1 0 1 n + 1 ξ t t β s d t 1 s .
Since ξ (t) is a positive increasing function and using second mean value theorem for integrals,
I 4 . 1 . 3 = O