# Degree of Approximation of function belonging to Lip(α, r) functions by Product Summability Method

DOI : 10.17577/IJERTV2IS90263

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#### Degree of Approximation of function belonging to Lip(α, r) functions by Product Summability Method

Lip(, r) functions by Product Summability Method

Ripendra Kumar, B.K. Singh & Aditya Kumar Raghuvanshi

Department of Mathematics

Abstract

In this paper author have been determined the degree of approxi- mation of certain functions belonging to Lip(, r) class by (C, 1)(E, q) means of its Fourier series.

Let f(t) be periodic functions with period 2 and integrable in the Lebesgue sense. The fourier series f(t) is given by

)

)

1

f(t) = a0

2

+

n=1

(an

• cos nt + bn

• sin nt) (1)

A function f Lip(, r) for 0 x 2, if

r

r

(r 2

0

0

|f(x + t) f(x)| dx

\1/r

= O(|t|

), 0 < 1, r 1, t > 0 (2)

0

0

The degree of approximation of a function f : R R by trigonometrical polynomial tn of order n is defined by Zygmund [1]

||tn f|| = sup{|tn(x) f(x)| : x R} (3)

1 )

1 )

n

n

n

If (E, q) = Eq

)

)

Â·

Â·

= qnk sk

(1 + q)n

k=0

s as n . Then an infinite

series uk with the partial sums sn is said to be summable (E, q) to the

k=0

definite number s. (Hardy [4]).

Ripendra Kumar, B.K. Singh & Aditya Kumar Raghuvanshi

The series uk

k=0

1 n

k

k

is said to be (C, 1) summable to s. If (C, 1) = s

(n + 1)

k=0

s as n . The (C, 1) transform of the (E, q) transform defines the (C, 1)(E, q)

transform of the partial sums sn of the series uk.

k=0

Thus if

(CE)q

n

1

1

= Eq s as n (4)

n (n + 1)

k

k=0

n

n

where Eq denotes the (E, q) transform of sn, then the series uk is said to be

k=0

summable (C, 1)(E, q) means or simply summable (C, 1)(E, q) to s. We shall

use following notation:

(t) = f(x + t) + f(x t) 2f(x)

1. In this paper we have generalized the theorem of S. Lal [12].

Theorem 2.1. If f : R R is 2 periodic, Lebesgue integrable on [ , ] and belonging to the Lipschitz (, r) class then the degree of approximation of f by the (C, 1)(E, q) product means of its Fourier series satisfies for n = 0, 1, 2, 3, …

q

q

n

n

||(CE) (x) f(x)||

= O / 1 \ for 0 < < 1 and r > 1

1

(n + 1)r

1

(n + 1)r

2. For proof of our theorem, we shall use the following lemmas [12].

Lemma 1. Let

then

Mn(t) =

n

1

1

2(n + 1)

k=0

1 k

(q + 1)k

r=0

k

r

r

qkr

1

1

sin(r + )t

sin(r + )t

2

2

2

sin t

Lemma 2.

Mn(t) = O(n + 1) for 0 < t < n + 1

n

n

t

t

n + 1

n + 1

M (t) = O 1 , for 1 < t <

Degree of Approximation of function belonging to Lip(, r) functions by …

3. 2

2

The nthpartial sum sn(x) of the series (1) at t = x is written as

r

r

1

0

0

sn(x) = f(x) + 2

(t) Â·

sin(n + 1 )t sin ( t ) dt

2

2

So that (E, q) means of the series (1) are

n

n

Eq (x) =

n

1

1

(q + 1)n

k=0

n

k

k

qnksk(x)

= f(x) +

2(q + 1)n

= f(x) +

2(q + 1)n

sin ( t )

2

sin ( t )

2

sin

sin

k +

2

k +

2

t

t

dt.

dt.

1 r (t) / n

0

0

n

k

k

1\

0

k=0

k

0

k=0

k

Therefore (C, 1)(E, q) means of the series (1) are

n

n

n

n

(n + 1)

(n + 1)

k

k

(CE)q (x) = 1 Eq (x) (n = 0, 1, 2, 3, …)

k=0

k=0

n

n

1 ( 1 r (t) / k

2

2

k

kr

1

\'l

= f(x) +

2(n + 1)

r

Â·

Â·

k=0

(q + 1)k

0 sin ( t )

r q

r=0

sin r +

2

t dt

where

= f(x) +

(t) Mn(t)dt (5)

0

Mn(t) =

so

n

1

1

2(n + 1)

k=0

1 k

(q + 1)k

r=0

k

r

r

qkr

1

sin(r + )t

sin(r + )t

2

sin(t/2)

n

n

(CE)q (x) f(x) =

r

r

Â·

Â·

(t) Mn(t)dt

/r

/r

0

1

n+1

0

0

r \

= +

1

n+1

(t) Â· Mn(t)dt

Now

\

\

I1 =

1

r

r

n+1

/r

/r

0

(t) Â· Mn(t)dt

= I1 + I2 (6)

1

n+1

1 1

r 1 s

n+1

r s

Â·

Â·

|I1|

[(t)] dt

0

[Mn(t)] dt

/r

/r

\

\

0

, using HÂ¨olders inequality

Ripendra Kumar, B.K. Singh & Aditya Kumar Raghuvanshi

( 1 \

!r

!r

\

\

1

1 s

n+1

s

|I1| O

(n + 1) Â·

(n + 1) dt

0

( 1 \

r(n + 1)s 1

|I1| O

(n + 1)

Â· n + 1

s

s

1

1

(n + 1)

(n + 1)

1s

(n + 1) s

1s

(n + 1) s

|I | O ( 1 \ Â· ! 1 \

1

1

1

(n + 1)+ 1s

1

(n + 1)+ 1s

|I | O ! 1 \

1

1

(n + 1)(1 1 )

s

(n + 1)(1 1 )

s

r

r

s

s

|I | O ! 1 \ 1 + 1 = 1

1

1

1

(n + 1)r

1

(n + 1)r

|I | O ! 1 \

Next

r

r

Â·

Â·

I2 = (t) Mn(t)dt

1

n+1

|I2| =

|I2| =

1

n+1

1

n+1

1

n+1

1

n+1

r

1

n+1

1

n+1

(t) Â· Mn(t)dt

1

n+1

1

n+1

r

s

r

s

!r

|I2|

|I2|

\1 !r \1

s(Mn(t)) dt

s(Mn(t)) dt

|I2|

r((t)) dt

s(Mn(t)) dt

|I2|

r((t)) dt

s(Mn(t)) dt

( \

( \

1

(n + 1)

1

(n + 1)

1

n+1

1

n+1

1

s dtt

1

s dtt

s

s

!r \1

|I2| O

|I2| O

|I2| O

|I2| O

s

s

( 1 \ r 1 1s

|I2| O

(n + 1) n + 1

2

2

1s

(n + 1)+ s

1s

(n + 1)+ s

|I | O ! 1 \

2

2

1

(n + 1)+ 1s

1

(n + 1)+ 1s

|I | O ! 1 \

2

2

1

(n + 1)r

1

(n + 1)r

|I | O ! 1 \

Then from (6) and the above inequalities we have

||tn

f||

= sup{|tn

(x) f(x) : x R = O 1 , 0 < < 1, r > 1.

1

1

| } / \

| } / \

(n + 1)r

This completes the Proof of the theorem.

for 0 < < 1

for 0 < < 1

If r then degree of approximation of a function f Lip is given by

q

n

which reduces to the theorem of S. Lal [12].

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4. Hardy, G.H.; Divergent Series, Oxford, at the Clarendon Press, 1949.

5. Alexits, G.; Convergence Problems of Orthogonal Series. Pergamon Press London (1961).

6. Qureshi, K.; On degree of approximation of function belonging to the Lip class, Indian Jour. of pure appl. Math., 13 (1982) 8, 898.

7. Qureshi, K. and Nema, H.K.; A class of function and their degree of approximation, Ganita, 41 (1990) 1, 37.

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Ripendra Kumar, B.K. Singh & Aditya Kumar Raghuvanshi

11. Lal, S. and Kushwaha, J.K.; Approximation of Conjugate of functions belonging to the generalized Lipschitz class by lower triangular matrix means, Int. Journal of Math. Analysis, 3(2009) 21, 1031-1041.

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13. Sarangi Sunita et al.; Degree of approximation of Fourier series by HousdÂ¨orff and NÂ¨orlund Product means, Journal of Computation and Modelling, Vol. 3, no. 1, 2013, 145-152.