Lip( ,p)
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667 U.K. India
** Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
Abstract
Mittal and Rhoades (1999-2001), Mittal et al. (2005) Mittal, and Rhoades and Mishra (2006) have initiated the studies of error estimates E (f ) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows.
In this paper, we continue the work in the direction. Here we extend two theorems of Leindler [L. Leindler, Trigonometric approximation in
L p
-norm, J. Math. Anal. Appl. 302 (2005) 129-136], where he has weakened the conditions on given by Chandra [P. Chandra, Trigonometric approximation of functions in norm, J. Math. Anal. Appl. 275 (2002) 13-26], to more general classes of triangular matrix methods.
Our Theorem also generalizes Theorem 4 partially of Mittal et al. [M.L. Mittal, et al., Using infinite

using trigonometric polynomials, J. Math. Anal.
Appl. 326 (2007) 667-676] by dropping the monotonicity on the elements of matrix rows which in turn generalize the results of Quade [E. S. Quade, Trigonometric approximation in the mean, Duke Math. J.
3 (1937) 529-542].
Key words: Signals,
Class Lip( , ) , Trigonometric-Fourier approximation, L -norm p
1.
Introduction
Let
 n
(f ) denote the nth term of the (C, 1) transform of the partial sums of the Fourier series of a 2

- periodic function (signal) f. In 1937, Quade [15] has proved that if f

Lip
  for 0
 α 
1, then f
  n
(f ) p

O (n

) for either (i) p

1 and
0
 α  f
  n
1
(f )
or (ii)

1
 p

1 and

0
 α 
1.
He also showed that, if p

1
 α,
then
Chandra[2] has extended the work of Quade[15] and proved three theorems. In [2] among others the following theorems, where N (f) and n
R (f ) denote the nth terms of the Nörlund and weighted mean transforms of the n sequences of partial sums, respectively were proved.
Theorem A[2]. Let f

Lip ( , p)

and let
  n
 
 p be positive such that n
 
. (1)
If either
(i) p

1 , 0
  
1 , and (ii)
  n
is monotonic, or
(i) p

1 , 0
  
1 and (ii)
 
is non-decreasing. Then f

N (f ) p
  α
O (n ).
(2)
Theorem B[2]. Let f

Lip ( ,1) , 0<
 
1, and let
  n be positive and non-decreasing with (1). Then
   f R (f ) O( n ).
n 1
(3)

NOTE:
1.
Some of the estimates, as mentioned by Chandra [2, p.15] himself, are sharper than the results proved by Quade[15], Mohapatra and Russell[14], and by himself earlier [1] and also new[2, p.253], hence are interesting (in view of Leindler [4, p.130] also).
2. From the point of view of applications, the sharper estimates of infinite matrices, as
Gil mentioned [3, p.176], are useful to get bounds for the lattice(occurs in solid state
Physics) norms of matrix valued functions, and enable us to investigate perturbations of matrix valued functions and compare them. Few more applications are mentioned in section 2.
3. Here we shall use all the notations of Mittal et al.[11].
A positive sequence c :

 
is called almost monotone decreasing (increasing) if there exists a constant K: K (c), depending on the sequence c only, such that for all n m, c n

K c m
(K c n
 c ).
m
Such sequences will be denoted by c

A M D S and c

A M IS, respectively. A sequence which is either AMDS or AMIS is called almost monotone and will be denoted by c

A M S .
Leindler [4] extended Theorems A and B of Chandra [2] without monotonicity on
  n
. He proved:
Theorem C [4].
Let f

Lip(

, p) and let
  n
be positive. If one of the conditions
(i) p

1, 0
(ii) p

1, 0
  
1
  
and
  
A M DS,
  
A M IS and (1) holds,
(iii) p 1, 1 and

(iv) 1,

(v) p 1, 0
 
1,
 k
 p k

O (P ), n
 p k

O (P / n ), n
and (1) holds,
 p k

O (P / n ), n maintains, then (2) holds.
Theorem D [4].
Let f

Lip ( ,1), 0 <

<1.
If the positive
  satisfies (1) and the condition
  p k

then (3) holds.
For a given signal f
 p p
  
let

 n
 a
0

2 n   a cos kx
 b sin kx
  n  denote the partial sum, a trigonometric polynomial of degree(or order) n, of the first
(n

1) terms of the Fourier series f.
Define
   n
(f ) n
(f ; x)
 n  a s (f ) , n , k k n 0.
The Fourier series of signal f is said to be T –summable to s, if
Throughout
 n
(f )
 s as n

.
T

(a n , k
) -a linear operator, will denote a lower triangular regular matrix with non-negative entries with row sums 1. Such a matrix T is said to have monotone rows if, for each n,
 
is either non-increasing or non-decreasing in k, 0
 k
 n.
The integral modulus of continuity of f is defined by
 p
(δ; f )
 sup
0
 h
 δ


1
2π
If, for α 
0 ,
2π
0
  


1/ p
.
 p
(δ; f )
 O(δ ), then f

Lip ( , p) (p

1).
The L p
-norm of f is defined by f p



1
2π
2 π

0 p f (x) dx
 1/ p


 f
 p


We write s (f) n
 π
where
-1

2 π
 n
D (t) n
0

   
, A n, k
 n  a n, r
, t n
 n  a n, k

A n ,0
1, n 0,
  
, the Dirichlet Kernel of degree n. b n, k
 k

1

A n, k

A n, 0

   n ,
Δ a k n, k
 a n, k
 a ,
B n
  n  b , k n, k
[x]- the greatest integer contained in x,
Here, a signal (function) f is approximated by trigonometric polynomials

of order n
(or degree) n and the degree of approximation E ( f ) is given by n
E (f ) n

Min f (x) n
  n
(f ; x) p
, in terms of n. This method of approximation is called trigonometric Fourier
Approximation.

Recently Mittal et al. [11] have generalized two theorems A and B of Chandra ([2,
Theorem 1 and a part of Theorem 2]) to more general classes of triangular matrix methods. They prove:
Theorem E [11].
Let f

Lip ( , p) and let T have monotone rows and satisfy t n
 
O(n
 
).
(4)
(i) If p 1, 0

1, and T also satisfies

 n ,0
, a n, r


O (1), (5)
where r: [n / 2] , then f
  n
(f ) p
= O (n
 
).
(6)
(ii) If p > 1,

=1, then (6) is satisfied.
(iii) If p 1, 0

1, and T also satisfies

 n, 0
, a n, n


O (1), (7) then (6) is satisfied.
2. Mittal, Rhoades ([5]-[8]), Mittal et al. [9] and Mittal, Rhoades, Mishra [10] have obtained many results on tfa (these approximations have assumed important new dimensions due to their wide applications in signal analysis [12], in general and in digital signal processing [13] in particular, in view of the classical Shannon sampling theorem), using summability methods without monotone rows of the matrix T: a digital filter. In this paper, we extend two theorems C and D of
Leindler [4, Theorems 1 and 2] to more general classes of triangular matrix methods. Our Theorem1 also generalize partially Theorem E of Mittal et al. [11], respectively by dropping monotonicity on the elements of matrix rows (that is weakening the conditions on the filter, we improve the quality of the filter). We prove:
Theorem 1.
Let f

Lip
 p ), and let T

( a ) be an infinite regular triangular matrix.
(i) If p 1, 0

1,
 


AMS in k and satisfies
 n ,0
, a n, r


O (1), (8)
where r: [n / 2] , then (6) is satisfied.
(ii) If p

1,
 
1 and n k

1 

0
( n k )
 a k ,
(iii) If p

1,
 
1 , k

1 n 

0
 k a
,


O (1), or (9)
( n ,0
) , or (10)
(iv) If p

1, 0
  
1 , k n 

0
 a k , and also

( n , 0
) , (11) n ,0

O (1) , (12) hold then (6) is satisfied.

We note that:
(i). In case of Nörlund  p
) or weighted (R ) p
-matrix, condition (8)[or(12)] reduces to
(1)[11, p.674], while conditions (9), (10) and (11) reduce to conditions (iii), (iv) and
(v) of theorem C respectively. Thus our Theorem1 generalizes theorem C.
(ii). Further, it is easy to examine that the conditions of Theorem 1 claim less than the requirements of Theorem E for A n , 0
1 t in (9) is always satisfied if the sequence a n n , k
. For example, the condition on the sum
is non-decreasing in k, then using (12), we get
   a k n , k
   n , k
 a
   a ) n , k
If

A n ,0
  n ,0

O (1).
is non-increasing in k and (12) holds then
  a k n , k
  a n , k
 a
 
(a n , k
 a )
 a n ,0
 a n , n
 a n,0
 
1
O (n ) is also true.
3. Lemmas
In order to prove our Theorem 1, we require the following lemmas:
Lemma 1 [15].
If f

Lip (1, p), p

1, then
σ (f ) n
 s ( f ) n p

Lemma 2 [15].
Let, for 0
  
1 and p f
 s ( f ) n

1, f

O (n

).
p

Lip(

, p).
Then
Lemma 3 [11]. Let T have monotone rows and satisfy (8). Then, for 0
  
1, n  a n , k
     α
Note: Since every monotone sequence is also an almost monotone, the proof of
Lemma 3 is valid for the almost monotone (i.e. AMDS or AMIS) sequences.
4. Proof of Theorem 1.
Cases I.
If p 1, 0
 
1.
Dropping the second term as t n
  n ,0
,
 n, the proof runs similar to the case (I) of Theorem E[11]. Let
Lemmas 2 and 3, we get be AMS in k. Thus, using
 n
(f ) f n  a s (f) f n, k k
  n  a n, k
 k
 
(13) hence
 n
(f) f p
 n  a n, k s (f) f k
 p
 n  a n, k
O (k
  
).

If p

1, α 
1.
Using Lemma 2, we get
  n
(f ) f p
  n
(f ) s (f) n p
 s (f ) f n

  n
 n p
 
1
O (n ).
So, it remains to show that
 n
(f ) s (f ) n p
(14) p
 
1
O (n ).
(15)
We may write
 n
(f )
 n  a s (f ) n , k k
 n  a s (f )
 n 
A u (f ) , n , k k and thus, as A n ,0

1, we have
 n
(f ) s (f ) n
 n 


A n , k k

A n , 0

 k u (f ) k
 n  b k u (f ).
n ,k k
By Abel’s transformation, we get
 n
   
 k b n , k
 

 k  ju (f )



 b n ,n n 
Thus by triangle inequality, we find
 n
 p
   k b n ,k k 
But by direct computations, we have s (f ) n
 n
(f )
 n  u (f ) m

1

1 n  ju (f ; x) j n  s (f ) m
 p

1 b n ,n n  n  p
(16)
   m
 m
Therefore by Lemma 1, we get n  ju (f ; x) j
   n
 n p
  p

1 
(17)
We note that b n,n


A n , n

A n, 0

/ n


A n ,0

A n, n

/ n

A n, 0
/ n

O (1/ n).
Thus b n , n n   b n , n n  p p
As in case (II) of proof of Theorem E[11, p.672], we write
 
O (n ).
(18)

 b k n , k

1
Next we claim that k  a n ,r
 


 n, k
 k  a n, r

 .
(19) n ,k
   n ,r
 a , (20) holds, k N . We verify it by mathematical induction.
For k 1, then inequality (20) reduces to equality i.e.
1  a n , r

2 a n , 1
 a n ,0
 a .
n ,1 holds. Now let us suppose that (20) holds for k m i.e. m  a n ,r
  n ,m
   n ,r
 a , (21) and we will show that (20) is true for k m 1.
So, let k=m+1. Using (21), we get
 a n ,r
   m  a n , r
  n , m

(m 1) a n ,m

(m 1) a
   n , r
 a

(m 1) a n , m
 a
   n , r
 a .
Thus (20) holds for k m 1.
Consequently (20) holds k N.
Using (10), (12), (19) and (20), we find n   k b n , k
 n  1 n , k
 k  a n , m
 n  1 k  a n , m n , k
 n 
 n  m
1 
 m a n , m 1

 n , m
 a
1
 n  1 k  m a
   a k n , k

O (a n ,0
 
1
 a n , m
Combining (16), (17), (18) and (22) yields (15). Thus, from (15) and (14), we get
 n
 p
 
1
O (n ).
If p 1, 1.
For this, we first prove that the condition
(2 2)
   a k n , k

O (1)

B n
 
1
O (n ).
(23)

As in case (III), using (20) and taking r:=[n/2] throughout in this case, we have
B n
 n  
1 k (k 1)

1  n , k
 k  a n , r
 n 

1

1
 n , m
 a



 r n   k 1





1

1 k  m
 m a n , m 1

B
1

B , say.
2
Interchanging the order of summation and using (9), we get
B
1
 r  
1

1 k  m
 m a n , m 1
 r  m
 m a n , m 1

 1
 r   m a n , m 1
 n   n m a

 n  m r 1

1
 n m a


 n  m
 n m a
 m



1    a k n , k

1

1
(24)
Now
B
2
 n  k
 r
1 k  m
 m a n , m 1
 n  n , 2
, say.
Using arguments as in B and (9), we obtain
1
B n ,1
 n 

1

1 r  m
 m a n , m 1
1
 



 r
 k   m 1



 m
 m a n , m 1


(26)
 r
1 n 
 r

1 n 

1 n 

1
 m
 n m a

  a k n , k

O(r ) n 
(k 1)

1 
O (n ), (27)
Again interchanging the order of summation and using (9), we have
B n , 2
 n 


1
1

1 n  m
 r


1 k  m
 m a n , m 1
 n  k
 r
 m a n , m 1 n 
1

1

1 k   m a n , m 1 n  m
 r
(n m 1)
 m a n , m 1
  a k n , k

1

1
(28)

From (25)-(28), we get (23). Thus (14), (17) and Lemma 2 again yield (6).
If p 1, 0
 
1.
Using (13), conditions (11), (12), convention a

0 ,
Abel’s transformation and the result of Quade [15] cited in the introduction, we obtain
 n
(f ) f
1
 
(
 a k n , k
)


 k   s (f ) f r
 




(a n , n
 a ) n   s (f ) f r
 
1

 n n 
(
 a k n , k
)



 k 
  k a n , k
 s (f ) f r
 



1
 n 
(
 a k n , k
   k

 k
(f ) f
1

O


 n   1
  k a n , k




1

O (n
1

) n   a k n , k

O (n
1
 a n ,0
)

O (n

).
This completes the proof of case (V) and consequently the proof of Theorem 1 is complete.
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