Skip to main content

A study on degree of approximation by Karamata summability method

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 (Kλ) 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.

1 Introduction

The method Kλwas 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 (Kλ) 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) = tαthen 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-cost< 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-cost< 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 tπ,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 dt.

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 dt.

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 dt.

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 ,sinntnt,

I 3 . 1 . 2 =O λ log n + 1 0 1 n + 1 tψ t dt.

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 dt.

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 dt.

Since for 0<t< 1 n + 1 ,sinntnt,

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 1 n + 1 ξ 1 n + 1 1 n + 1 1 t β s d t 1 s = 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 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 .
(5.20)

Combining from (5.17) to (5.20),

I 4 . 1 = O 1 n + 1 2 ξ 1 n + 1 n + 1 β + 1 r + O log n + 1 n + 1 2 ξ 1 n + 1 n + 1 β + 1 r + O 1 n + 1 2 ξ 1 n + 1 n + 1 β + 1 r .
(5.21)

Using Hölder's inequality, |sin t| ≤ 1, sin t ≥ 2t/π, conditions (3.3), (3.5) and second mean value theorem for integrals and the fact ψ (t) W (Lr,ξ (t)),

I 4 . 2 = O 1 n + 1 π 1 Γ λ + n t ψ t d t = O 1 Γ λ + n 1 n + 1 π t - δ ψ t sin β t ξ t r d t 1 r 1 n + 1 π ξ t t 1 - δ sin β t s d t 1 s = O 1 Γ λ + n 1 n + 1 π t - δ ψ t ξ t r d t 1 r 1 n + 1 π ξ t t 1 - δ sin β t s d t 1 s = O 1 Γ λ + n n + 1 δ 1 n + 1 π ξ t t 1 - δ sin β t 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 4 . 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 y 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 .
(5.22)

Combining from (5.16), (5.21) and (5.22)

S m x - f x = O 1 n + 1 2 ξ 1 n + 1 n + 1 β + 1 r + O log n + 1 n + 1 2 ξ 1 n + 1 n + 1 β + 1 r + O 1 n + 1 2 ξ 1 n + 1 n + 1 β + 1 r + O ξ 1 n + 1 Γ λ + n n + 1 β + 1 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 .

Now using Lr-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 2 n + 1 2 + log n + 1 n + 1 2 + 1 Γ λ + n d x 1 r = n + 1 β + 1 r ξ 1 n + 1 2 n + 1 2 + log n + 1 n + 1 2 + 1 Γ λ + n 0 2 π d x 1 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 = O n + 1 β + 1 r ξ 1 n + 1 1 + log n + 1 e n + 1 2 + 1 Γ λ + n .

This completes the proof of Theorem 4.

References

  1. Karamata J: Theorems surla sommabilite exponentielle etd autres Sommabilities sattachant. Math (Cliy) 1935, 9: 164.

    Google Scholar 

  2. Lotosky AV: On a linear transformation of sequences (in Russian). Ivanov Gos Red Inst Uchen Zap 1963, 4: 61.

    Google Scholar 

  3. Agnew RP: The Lotosky method for evaluation of series. Michigan Math J 1957, 4: 105.

    Article  MathSciNet  Google Scholar 

  4. Vuĉkoviĉ V: The summability of Fourier series by Karamata method. Maths Zeitchr 1965, 89: 192. 10.1007/BF02116860

    Article  Google Scholar 

  5. Kathal PD: A new criteria for Karamata summability of Fourier series. Riv Math Univ Parma Italy 1969, 10: 33–38.

    MathSciNet  Google Scholar 

  6. Ojha AK Ph.D. Thesis, B.H.U; 1982:120–126.

  7. Tripathi LM, Lal S: Kλ-summability of Fourier series. Jour Sci Res 1984, 34: 69–74.

    Google Scholar 

  8. Alexits G: Convergence problems of orthogonal series. Translated from German by I Folder. International series of Monograms in Pure and Applied Mathematics 1961., 20:

    Google Scholar 

  9. Sahney BN, Goel DS: On the degree of continuous functions. Ranchi Univ Math J 1973, 4: 50–53.

    MathSciNet  Google Scholar 

  10. Chandra P: Trigonometric approximation of functions in Lpnorm. J Math Anal Appl 2002,275(1):13–26. 10.1016/S0022-247X(02)00211-1

    Article  MathSciNet  Google Scholar 

  11. Qureshi K: On the degree of approximation of a periodic function f by almost Nörlund means. Tamkang J Math 1981,12(1):35–38.

    MathSciNet  Google Scholar 

  12. Qureshi K, Neha HK: A class of functions and their degree of approximation. Ganita 1990,41(1):37–42.

    MathSciNet  Google Scholar 

  13. Rhoades BE: On the degree of approximation of functions belonging to Lipschitz class by Hausdorff means of its Fourier series. Tamkang J Math 2003,34(3):245–247.

    MathSciNet  Google Scholar 

  14. Zygmund A: Trigonometric Series. Cambridge University Press, Cambridge; 1939.

    Google Scholar 

  15. McFadden L: Absolute Nörlund summability. Duke Math J 1942, 9: 168–207. 10.1215/S0012-7094-42-00913-X

    Article  MathSciNet  Google Scholar 

  16. Titchmarsh EC: The Theory of Functions. Oxford University Press, Oxford; 1939.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kusum Sharma.

Additional information

6 Competing interests

The authors declare that they have no competing interests.

7 Authors' contributions

HK framed the problems. HK and KS carried out the results and wrote the manuscripts. All the authors read and approved the final manuscripts.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Nigam, H.K., Sharma, K. A study on degree of approximation by Karamata summability method. J Inequal Appl 2011, 85 (2011). https://doi.org/10.1186/1029-242X-2011-85

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords