CLASS USING LOWER TRIANGULAR MATRIX SUMMABILITY METHOD

APPROXIMATION OF FUNCTIONS

BELONGING

TO GENERALIZED

Lip

 



,

r

CLASS USING

LOWER

TRIANGULAR MATRIX

SUMMABILITY METHOD

Shyam Lal

Department of Mathematics Faculty of Science Banaras Hindu University

Varanasi-221005, India

[email protected]

JITENDRA KUMAR KUSHWAHA

Department of Mathematics Faculty of Science Banaras Hindu University

Varanasi-221005, India

[email protected]

Abstract. In this paper, a new estimate for the approximation of a function

f



Lip

 



,

r

by lower triangular matrix means without monotonic rows has been determined.

Keywords and phrases: Generalized Lipschitz Class of functions, Fourier series,

Degree of approximation, Linear operator, generalized Minkowski's inequality.

Subject Classification (2011): 42B05, 42B08

1. Introduction and preliminaries. Qureshi ([7],[8]), Chandra [2], Sahney & Rao [9] and Khan [3] have determined the degree of approximation of functions belonging to Lip(α,r) class by Nörlund



N,p_n



and generalized Nörlund (N,p,q) methods. But till now no work seems to have been done for determining the degree of approximation of a function

f



Lip

(



,

r

)

class by lower triangular summability method. The purpose of this paper is to determine an estimate for the degree of approximation of a function belonging to generalized

Lip

 



,

r

class by lower triangular matrix method of its Fourier series. Class

Lip

 



,

r

includes Lipα and Lip(α, r) classes,

0







1

,

r



1

,

as particular cases. Lower triangular matrix summability means includes



N,p_n



and (N,p,q) means as particular cases. The lower triangular matrix method used in this paper is free from monotonocity conditions.

)

,

(

r

Lip

f





if



 



(

)

,

0

1

,

1

/ 1 2

0





























f

x

t

f

x

t

dx

O

t

r

r r



 

.

(def 5.38 of Mc Fadden [5])

Given a positive increasing function



( )

t

and an integer

1



r





,we find (Khan [4] ) that f belongs to generalized Lipschitz class i.e.

f x

 



Lip





 

t

,

r



if







 





 



1/ 2

0

.

r r

f x t

f t

dx

t





 

 



In case



(

t

)



t

 then

Lip





 

t

,

r



coincides to

Lip





,

r



. If

r

 

in

Lip





,

r



then it reduces to

Lip



.

Let f (x) be periodic with period 2π, integrable in the sense of Lebesgue and belonging to

)

,

(

r

Lip



class. The Fourier series of f(x) is given by

















1 0 2

1

_cos

_sin

)

(

n

n

n

nx

b

nx

a

a

x

f

(1)

We define norm by

1/ 2

0

( )

,

1.

r r r

f

f x

dx

r









_

_









The degree of approximation E_n

 

f

is given by (Zygmund [12], p.114 )

E

_n

(

f

)



min

t

_n



f

_r (2)

where t (x)_n =















1 0 2

1

_cos

_sin

n

n

n

mx

b

mx

a

a

is a trigonometric polynomial of degree n.

Let

T



(

a

n k_,

)

be an infinite triangular matrix satisfying Silverman-Töeplitz [11] conditions of

regularity i.e.











n

as

,

1

a

n

0 k

k ,

n ,

a

n,k



0

,

for

k



n

and

a

M

n

0 k

k ,

n







, a finite constant.

Let





0

n n

u

be an infinite series with sequence of th

k

partial sum

 

s

_k where

.

0







k

k

u

s

 

The sequence-to-sequence transformation _k

n

k k n

n

a

s

t



 

_

0

, n k

n

k

k n n

s

a

_

 





0

, (3)

defines the sequence

 

t

_n of lower triangular matrix means of the sequence

 

s

_n , generated by the sequence

of coefficients

 

a

n,k . The series





0

n n

u

is said to summable to the sum s by lower triangular matrix method if

n n

t

lim

exists and is equal to s (Zygmund [12], p. 74) and we write

t

_n



s

(

T

),

as

n





. Important particular cases of lower triangular matrix means are

(i) Harmonic mean, when

n

k

n

a

_{n k}

log

)

1

(

1

,



_

_

.

(ii) Nörlund means [6] when

n k n k n

P

p

a

_,



 where







n

k k

n

p

P

0

.

(iii) Generalized Nörlund means [1] when

n k k n k n

R

q

p

a

_,



 where

.

0



 



n

k

k n k

n

p

q

R

Throughout this paper



 

x t

,



f x t



 



f x t



 



2

f x

 

,

A

a

,

n

n k

k , n ,

n



   







 

1_t



The

greatest

int

eger

not

greater

than

(

1

/

t

)

,

 

1 ,





2 0

sin

1/ 2

sin( / 2)

n n k n

k

a

k

t

K

t

t

 







_{k n, k}a  a_{n, k} a_{n, k 1}



0



k



n



1

.

Sahney and Rao [9] proved the following theorem:

Theorem 1. f(x) is periodic and belongs to the class Lip(α,p), for 0  1. Let

 

p_n be a negative, non-increasing generating sequence for the

(

N

,

p

_n

)

method such that













n

k k

n

P

n

p

P

0

)

(

as

n





and if





1/ q

n _q

q 2 q 1/ q 1

1

p(y) P(n)

dy O ,

y   n 

  _{ }

  _{  }

 

   







then *_{n n p}

Tn

E (f ) min f N  =













 p

n

O

_

1

_1/

where N (x)_n is the



N,p_n



mean of (1), and in particular,T (x) N (x)_n  _n . Khan [3] generalized the above result in the following form.

Theorem. Let f(x) be a periodic function and belongs to the class Lip(α, p), for 0  1. Let

 

p_n and

 

q_n be two non-negative, non-increasing sequences such that

n 0 1 n

P p p  ... p  , as n ,

n 0 1 n

Q q q  ... q

n 0 n 1 n 1 n 0

R p q p q _  ... p q  , as n  and R(y)

y is non-decreasing

then *_n p,q_n

p

E (f ) min f t  =O 1_{1/ p} n

 

 

 .

2. Main Theorem. In this paper, a new estimate for the approximation of a function f belonging to generalized

 

,

Lip



r

class by lower triangular matrix means

t

_nof its Fourier series has been obtained in the following form.

Theorem: Let

T



(

a

_{n k,}

)

be an infinite regular triangular matrix with non-negative elements

 

a_{n, k} such that

n

k n, k k 0

1

a O

n 1



 

 _{  }



 



. (2.1)

If a function f :[0,2 ] R is 2π periodic, Lebesgue integrable and belonging to generalized

Lip

 



,

r

class,







r

1

, then its degree of approximation by lower triangular matrix summability operator

k n

k k n

n

a

s

t



 

_

0

( )_{n n}

r

1

E (f ) t f O log(n 1)

n 1

 _  _{ } _  _ 

 _{ } _ 

 , n=0,1,2,3…, (2.2)

provided (t) is a positive increasing function of t such that













t

t

)

(



is monotonic decreasing. (2.3)

3. Lemmas. For the proof of our theorem, the following Lemmas are required.

Lemma 1. Under the condition of our theorem on

(

a

_{n k,}

)

,

K t

_n

( )



O n

(



1),

for

0

 

t

(

n



1)

1.

Proof.

For

0

 

t

(

n



1)

1,

sin

nt



nt

,sin( / 2) ( / )

t



t



 

1  1/ 2 ,

2 ( / )

0

n

k t

n n k t

k

K

t

_

a

_







_,

0

(2

1)

n n k k

n

a









= O(n+1).

Lemma 2.

For n

(



1)

1

 

t



,

sin( / 2) ( / )

t



t



, _n

 



 

_{( }1₁₎ 2

.

n t

K

t

O

Proof.

For n

(



1)

1

 

t



,

sin( / 2) ( / )

t



t



,

 

1 ,





2 0

sin

1/ 2

sin( / 2)









n n k

n

k

a

k

t

K

t

t



1





, 2

0

sin

1/ 2





_t



n _{n k}



k

a

k

t

=













1 1

, , 1 ,

2

0 0 0

sin

1/ 2

sin

1/ 2





  



_

_

_

_

















n k n

n k n k n n

t

k r k

a

a

k

t

a

k

t

by Abel’s lemma

=

2 2

1 1

, ,

2 0

sin

/ 2

sin

/ 2

sin / 2

sin / 2























n

n k n n

t k

kt

kt

a

a

t

t

_























 nn

n

k

k n t

n

t

a

a

K

_,

1

0 , 2 2

)

(



2 _,

2 0







n



_{n k}

t k

a



=

 

2

1 (n1)t

O

, by (2.1)

Lemma 3.





1/ r 2

r

0

(x, t) dx O (t) .



 

 _  _{ }

 

 







Proof.

1/ r 2

r

0

(x, t) dx



 

 _ 

 

 

=

1/ r 2

r

0

f (x t) f (x t) 2f (x) dx



 

 _{   } 

 

 







=



 



1/ r 2

r

0

f (x t) f (x) f (x t) f (x) dx



 

 _{    } 

 

 







1/ r 1/ r

2 2

r r

0 0

f (x t) f (x) dx f (x t) f (x) dx

 

   

   

_   _ _   _

   





 





,

by Minkowski’s inequality =O (t) O (t)   =O (t) .

4. Proof of the Theorem: Following Titchmarsh([10], p.402), the

n

thpartial sum of the Fourier series (1) is given by

0

1

sin(

1/ 2)

( ; )

( )

( , )

2

sin / 2







_

n

n

t

s

f x

f x

x t dt

t







.

By taking lower triangular matrix means, we have





x

t

dt

t

t

n

x

f

x

f

s

a

n

k n

k

k k

n





 







_



0 0

0 2

1

,

_sin

_/

₂

(

,

)

)

2

/

1

sin(

)

(

)

;

(

.

(

)

(

)

(

,

)

(

)

.

0

dt

t

K

t

x

x

f

x

t

_n







_n





Next,

1/ r 2

r

n _r n

0

t f t (x) f (x) dx



_{ }  _ 

 

 







1/ r r 2

n 0 0

(x, t) K (t)dt dx

 

 

 

_  _

 



 



1/ r 2

r n 0 0

(x, t) K (t) dx dt,

 _  _

 

 _  _

 

 

 

by generalized Minkowski’s inequality

1/ r 2

r n

0 0

K (t) (x, t) dx dt,

 _  _

 

 _  _

 

 





_n

0

O K (t) (t)dt ,



 

 

 

 

 







by lemma 3

1/(n 1)

n n

0 1/(n 1)

O K (t) (t)dt O K (t) (t)dt

 



 

 

 

 

 _  _ _  _

 

 

=

O I

( )

1



O I

( )

2 . (4.1)

Now,

I

K

t

t

dt

n

n

(

)

(

)

) 1 /( 1

0

1









_















1/(

_

1)

0

)

1

(

)

(

n

dt

n

t

O



, by lemma 1

_















1/(

_

1)

0

)

(

)

1

(

n

dt

t

n

O



=

1/(n 1)

1

O (n 1) dt

n 1





 _{ }   _{ } _

 



, where

1 n 1

0   _

by first mean value theorem

O 1 n 1

    __{ }



 

 . (4.2)

Lastly, ₂

1/( 1)

( ) ( )







 











n n



I

O

K t

t dt





= ₂

1/( 1)

( )

(

1)





















n



t

O

dt

n

t



_

, by Lemma 2



 























) 1 /( 1 1 1

)

1

/(

1

)

(

)

1

(

1

n n

t

dt

n

n

O

, by (2.3)

=

1

log(

1)

1







_









_

_

_







O

n

n





. (4.3)

Combining (4.1)-(4.3),

1

log(

1)

1



_

_{ }







_









_

_

_







n _r

t

f

n

e

n





=

_











_















1

log(

1

)

1

n

n

O



,

log(

n



1

)



e



O

(

log(

n



1

)).

This completes the proof of the theorem.

5. Corollaries. Following corollaries can be derived from our theorem:

Cor. 1.The degree of approximation of a function f Lip( ,r)  ,

1



r





by

t

_n-means of its Fourier series is given by

( )

1

, 0

1

(

1)

( )

log(

1)

,

1.

(

1)



 



_{ }

 

_



 







_{ }









_







_

_

_



n n _r

O

n

E

f

t

f

n

O

n

 





Proof of this corollary can be developed parallel to the main theorem by taking(t) t .

Cor. 2. Let



N,p_n



be a regular Nörlund method generated by a non-negative, monotonic decreasing sequence

 

p_n such that

n

n k

k 0

P

p O .

n 1



 

 _{  }



 

approximation of f Lip( ,r)  by Nörlund summability means

n

N 1

n _Pn n k k

k 0

t p _ s







of its

Fourier series (1) is given by























































.

1

,

)

1

(

)

1

log(

1

0

,

)

1

(

1







n

n

O

n

O

f

t

r N n

Proof: By taking _n,k pn k

Pn

a   and (t) t  in our theorem, proof can be obtained.

Cor. 3. Let (N,p,q) be a regular generalized Nörlund method generated by two non- negative, non-increasing sequences

 

p_n &

 

q_n such that

.

1

0





















 

n

R

O

q

p

n n k k n k

If f Lip( ,r)  then its degree of approximation by generalized Nörlund means

n p,q 1

n _Rn n k k k

k 0

t p _ q s







of

its Fourier series (1) is given by























































.

1

,

)

1

(

)

1

log(

1

0

,

)

1

(

1

,







n

n

O

n

O

f

t

r q p n

Proof: Proof of this corollary can be obtained by taking

n k k n k n

R

q

p

a

_,



 and



(

t

)



t

 in our theorem. Remarks.

(1) The proof given by Sahney and rao([9] p. 15 line 3) in particular

1/ p

/ n _p

0 (t) dt O(1) t    _{  }   _{ }  _  _{ }    





is not true. This step is not derived from

1/ p 2

p

0

f (x t) f (x) dx O(t )

     _{ }  _     



 .

The correct form of this step should be

1/ p

/ n _p

1 1/ p n 0 (t) dt O t    _{  }     _{ }  _{  }  _{ }  _{ }   



 .

(2) The proof given by Khan ([3], p. 135, line 3) in particular,

 

1/ p

/ n _p

1 n 0 t (t) dt O t    _{  }   _{ }  _  _{ }    





is not true. The author has taken (t) t 1/ p which is not satisfied if f Lip( ,p)  .

The modified and correct form of this step is

1/ p

/ n _p

(3) By taking





₄1

,

p



2

and other values in sahney & Rao [9] and Khan [3], the degree of approximation tends to infinity. Thus, in their theorems, it must be mentioned that the degree of approximation

.

1

,

1

)

(

1

/ 1

*

_

_

_





 p p n

n

f

E

(4) Since

1

1

_1/p

,

n

n





 for p>1, therefore our estimate is new, sharper and better than all

previously known results in the area of approximation theory.

(5) Cor. 2 & 3 are corrected and generalized forms of Sahney &Rao [9] and Khan [3] respectively.

References

[1] Borwein, D., (1958): On product of sequence, Jour. London Math. Soc., 33, 352-357.

[2] Chandra, Prem, (2002): Trigonometric approximation of functions in Lp-norm, J. Math Anal. Appl. 275, no. 1, 13--26.

[3] Khan, Huzoor H., (1974): On the degree of approximation of functions belonging to class Lip(α, p). Indian J. Pure Appl. Math. 5, no. 2, 132-136.

[4] Khan, Huzoor H., (1974): Approximation by class of functions, Thesis, Aligarh Muslim University, Aligarh, India.

[5] McFadden, Leonard, (1942): Absolute Nörlund summability. Duke Math. J. 9, 168- 207.

[6] Nörlund, N. E., (1919): Surene application des fonctions permutables, Lund. Universitiets Arsskrift, 16, 1-10.

[7] Qureshi, K., (1982): On the degree of approximation of a function belonging to the class

Lip

(



,

p

)

. Indian J. Pure. appl. 13 no.4, 466-470.

[8] Qureshi, K., (1982): On the degree of approximation of a function belonging to the Lipschitz class,Indian J. Pure.appl. 13 no.8, 898-903.

[9] Sahney, B.N.; Rao, V. Venu Gopal, (1972): Error bounds in the approximation of functions, Bull. Austral.Math Soc. 6, 11-18. [10] Titchmarsh, E. C., (1939): Theory of functions, Second Edition, Oxford University Press.

[11] Töeplitz, (1913): Uberallagemein lineara Mittle bil dunger P. M. F. 22,113-119.