# Approximation of Signals (Functions) by Trigonometric Polynomials inLp-Norm

Authors: M. L. Mittal; Mradul Veer Singh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

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## Abstract

Mittal and Rhoades (1999, 2000) and Mittal et al. (2011) have initiated a study of error estimatesEn(f) through
trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper, the first author continues the work in the direction for T to be a Np-matrix. We extend two theorems on summability matrix Np of Deger et al. (2012) where they have extended two theorems of Chandra (2002) using Cλ-method obtained by deleting a set of rows from Cesàro matrix C1. Our theorems also generalize two theorems of Leindler (2005) to Np-matrix which in turn generalize the result of Chandra (2002) and Quade (1937).


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## Body

## 1. Introduction

Letf be a 2π periodic signal (function) and let f∈Lp:=Lp[0,2π],p≥1. Let
(1)sn(f)≔sn(f;x)=a02+∑k=1n‍(akcos⁡kx+bksinkx)=∑k=0n‍uk(f;x)
denote the partial sums, called trigonometric polynomials of degree (or order) n, of the first (n+1) terms of the Fourier series of f at a point x.The integral modulus of continuity off is defined by
(2)ωp(δ;f):=sup⁡0<|h|≤δ{12π∫02π‍|f(x+h)-f(x)|pdx}1/p.If, forα>0,
(3)ωp(δ;f)=O(δα),
then f∈Lip⁡(α,p)(p≥1). Throughout ∥·∥p will denote the Lp-norm, defined by
(4)∥f∥p≔{12π∫02π‍|f(x)|pdx}1/p,f∈Lp(p≥1).A positive sequencec:={cn} 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,
(5)cn≤Kcm(Kcn≥cm).Such sequences will be denoted byc∈AMDS and c∈AMIS, respectively. A sequence which is either AMDS or AMIS is called almost monotone sequence and will be denoted by c∈AMS. Let 𝔽 be an infinite subset of ℕ and 𝔽 as the range of strictly increasing sequence of positive integers; say 𝔽={λ(n)}n=1∞. The Cesàro submethod Cλ is defined as
(6)(Cλx)n=1λ(n)∑k=1λ(n)‍xk,(n=1,2,3,…),
where {xk} is a sequence of real or complex numbers. Therefore, the Cλ-method yields a subsequence of the Cesàro method C1, and hence it is regular for any λ. Cλ is obtained by deleting a set of rows from Cesàro matrix. The basic properties of Cλ-method can be found in [1, 2]. In the present paper, we will consider approximation of f∈Lp by trigonometric polynomials Nnλ(f;x) and Rnλ(f;x) of degree (or order) n, where
(7)Nnλ(f;x)=1Pλ(n)∑m=0λ(n)‍pλ(n)-msm(f;x),Rnλ(f;x)=1Pλ(n)∑m=0λ(n)‍pmsm(f;x),sn(f;x)=1π∫02π‍f(x+t)Dn(t)dt,Dn(t)=(sin(n+1/2)t)2sin(t/2),Pλ(n)=p0+p1+⋯+pλ(n)≠0(n≥0),
and by convention p-1=0=P-1.The casepn=1 for all n(≥0) of either Nnλ(f;x) or Rnλ(f;x) yields
(8)σnλ(f;x)=1λ(n)+1∑m=0λ(n)‍sm(f;x).We also use(9)Δan=an-an+1,Δmg(n,m)=g(n,m)-g(n,m+1).Mittal and Rhoades [3, 4] have initiated the study of error estimates En(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper, the first author continues the work in the direction for T to be a Np-matrix. Recently, Chandra [5] has proved three theorems on the trigonometric approximation using Np-matrix. Some of them give sharper estimates than the results proved by Quade [6], Mohapatra and Russell [7], and himself earlier [8]. These results of Chandra [5] are improved in different directions by different investigators such as Leindler [9] who dropped the monotonicity on generating sequence {pn} and Mittal et al. [10, 11] who used more general matrix while very recently Deger et al. [12] used more general Cλ-method in view of Armitage and Maddox [1].


## 2. Known Results

Leindler [9] proved the following.Theorem 1 (see [9]).
Iff∈Lip⁡(α,p) and {pn} be positive. If one of the conditions(i)
p>1,0<α<1, and {pn}∈AMDS,(ii)
p>1,0<α<1, and {pn}∈AMIS,
(10)(n+1)pn=O(Pn)holds,(iii)
p>1,α=1, and ∑k=1n-1‍k|Δpk|=O(Pn),(iv)
p>1,α=1,∑k=0n-1‍|Δpk|=O(Pn/n), and (10) holds,(v)
p=1,0<α<1, and ∑k=-1n-1‍|Δpk|=O(Pn/n)
maintains, then(11)∥f-Nn(f)∥p=O(n-α).Theorem 2 (see [9]).
Letf∈Lip⁡(α,1),0<α<1. If the positive {pn} satisfies conditions (10) and ∑k=0n-1‍|Δpk|=O(Pn/n) hold, then
(12)∥f-Rn(f)∥1=O(n-α).Deger et al. [12] proved.Theorem 3 (see [12]).
Letf∈Lip⁡(α,p) and let {pn} be positive such that
(13)(λ(n)+1)pλ(n)=O(Pλ(n)).
If either (i)p>1,0<α≤1, and {pn} is monotonic or (ii)  p=1,0<α<1, and {pn} is nondecreasing, then
(14)∥f-Nnλ(f)∥p=O(n-α).Theorem 4 (see [12]).
Letf∈Lip⁡(α,1),0<α<1. If the positive {pn} satisfies condition (13) and is nondecreasing, then
(15)∥f-Rnλ(f)∥1=O(n-α).


## 3. Main Results

In this paper we generalize Theorems 3 and 4 of Deger et al. [12], by dropping monotonicity on the elements of the matrix rows which in turn generalize Theorems 1 and 2, respectively, of Leindler [9] to a more general Cλ-method. We prove the following.Theorem 5.
Iff∈Lip⁡(α,p) and {pn}is positive and if one of the following conditions(i)
p>1,0<α<1, and {pn}∈AMDS,(ii)
p>1,0<α<1, {pn}∈AMIS, and (13) holds,(iii)
p>1,α=1, and ∑k=1λ(n)-1‍k|Δpk|=O(Pλ(n)),(iv)
p>1,α=1,∑k=0λ(n)-1‍|Δpk|=O(Pλ(n)/λ(n)), and (13) holds,(v)
p=1,0<α<1, and ∑k=-1λ(n)-1‍|Δpk|=O(Pλ(n)/λ(n))maintains, then
(16)∥f-Nnλ(f)∥p=O((λ(n))-α).Theorem 6.
Letf∈Lip⁡(α,1),0<α<1. If the positive {pn} satisfies (13) and the condition ∑k=0λ(n)-1‍|Δpk|=O(Pλ(n)/λ(n)) holds, then
(17)∥f-Rnλ(f)∥1=O((λ(n))-α).Remarks. (1) If λ(n)=n, then our Theorems 5 and 6 reduce to Theorems 1 and 2, respectively.(
2
)
Deger et al. [12] have used monotone sequences {pn} in Theorems 3 and 4, while our Theorems 5 and 6 claim less than the requirements of their theorems. For example, the condition of the sum in (iii) of Theorem 5 is always satisfied if the sequence {pn} is nonincreasing; that is,
(18)∑k=1λ(n)-1‍k|Δpk|=∑k=1λ(n)-1‍k(pk-pk+1)=Pλ(n)-1-(λ(n)-1)pλ(n)=O(Pλ(n)),
while if sequence {pn} is nondecreasing and condition (13) holds, then the condition in (iv) of Theorem 5 is also satisfied; that is,
(19)∑k=0λ(n)-1‍|Δpk|=∑k=0λ(n)-1‍(pk+1-pk)=pλ(n)-p0≤pλ(n)=O(Pλ(n)λ(n)).Thus our theorems generalize the two theorems of Deger et al. [12] under weaker assumptions and give sharper estimates because all the estimates of Deger et al. [12] are in terms of n, while our estimates are in terms of λ(n) and (λ(n))-α≤n-α for 0<α≤1.


## 4. Lemmas

We will use the following lemmas in the proof of our theorems.Lemma 1 (see [6]).
Iff∈Lip⁡(α,p), for 0<α≤1 and p>1, then
(20)∥f-sn(f)∥p=O(n-α).Lemma 2 (see [6]).
Iff∈Lip⁡(1,p), for p>1, then
(21)∥σn(f)-sn(f)∥p=O(n-1).Lemma 3 (see [6]).
Iff∈Lip⁡(α,1),0<α<1, then
(22)∥f-σn(f)∥1=O(n-α).Lemma 4.
Let{pn}∈AMDS or let {pn}∈AMIS and satisfy (13). Then, for 0<α<1,
(23)∑m=1λ(n)‍m-αpλ(n)-m=O((λ(n))-αPλ(n))
holds.Proof.
Letr denote the integral part of (λ(n)/2). Then, if {pn}∈AMDS,
(24)∑m=1λ(n)‍m-αpλ(n)-m=∑m=1r‍m-αpλ(n)-m+∑m=r+1λ(n)‍m-αpλ(n)-m≤Kpλ(n)-r∑m=1r‍m-α+(r+1)-α∑m=r+1λ(n)‍pλ(n)-m≤Kpλ(n)-r∑m=1λ(n)‍m-α+(r+1)-α∑m=0λ(n)‍pλ(n)-m=Kpλ(n)-r(λ(n))1-α+O((λ(n))-αPλ(n))=O((λ(n))-αPλ(n)).
If{pn}∈AMIS and (13) holds, then
(25)∑m=1λ(n)‍m-αpλ(n)-m=∑m=1r‍m-αpλ(n)-m+∑m=r+1λ(n)‍m-αpλ(n)-m≤Kpλ(n)∑m=1r‍m-α+(r+1)-α∑m=r+1λ(n)‍pλ(n)-m=O(Pλ(n)λ(n)+1)∑m=1λ(n)‍m-α+O((λ(n))-αPλ(n))=O((λ(n))-αPλ(n)).
This completes the proof of Lemma4.


## 5. Proof of the Main Results

Proof of Theorem5.
We prove cases (i) and (ii) together. Since(26)Nnλ(f;x)-f(x)=1Pλ(n)∑m=0λ(n)‍pλ(n)-m{sm(f;x)-f(x)},
thus in view of Lemmas 1 and 4 and condition (13), we have
(27)∥f-Nnλ(f)∥p≤1Pλ(n)∑m=0λ(n)‍pλ(n)-m∥f-sm(f)∥p=1Pλ(n)∑m=1λ(n)‍pλ(n)-m∥f-sm(f)∥p+pλ(n)Pλ(n)∥s0-f∥p=1Pλ(n)∑m=1λ(n)‍pλ(n)-mO(m-α)+O(pλ(n)Pλ(n))=O((λ(n))-α).
Next we consider case (iv).
Letp>1 and α=1. By Abel’s transformation, we get
(28)Nnλ(f;x)=1Pλ(n)∑m=0λ(n)‍Pλ(n)-mum(f;x),
and thus
(29)snλ(f;x)-Nnλ(f;x)=1Pλ(n)∑m=1λ(n)‍(Pλ(n)-Pλ(n)-m)um(f;x).
Hence again by Abel’s transformation, we get(30)snλ(f;x)-Nnλ(f;x)=1Pλ(n)∑m=1λ(n)‍Δm(m-1(Pλ(n)-Pλ(n)-m))×∑k=1m‍kuk(f;x)+1(λ(n)+1)×∑k=1λ(n)‍kuk(f;x).
Thus(31)∥snλ(f)-Nnλ(f)∥p≤1Pλ(n)∑m=1λ(n)‍|Δm(m-1(Pλ(n)-Pλ(n)-m))|·∥∑k=1m‍kuk(f)∥p+1(λ(n)+1)∥∑k=1λ(n)‍kuk(f)∥p.
Since(32)sn(f;x)-σn(f;x)=1n+1∑k=1n‍kuk(f;x),
thus by Lemma 2(33)∥∑k=1n‍kuk∥p=(n+1)∥σn(f)-sn(f)∥p=O(1).
In view of (31) and (33), we obtain
(34)∥snλ(f)-Nnλ(f)∥p=O(1Pλ(n))∑m=1λ(n)‍|Δm(m-1(Pλ(n)-Pλ(n)-m))|+O((λ(n))-1).
Now(35)Δm(Pλ(n)-Pλ(n)-mm)=1mΔm(Pλ(n)-Pλ(n)-m)+Pλ(n)-Pλ(n)-m-1m(m+1)=Pλ(n)-m-1-Pλ(n)-mm+Pλ(n)-Pλ(n)-m-1m(m+1)=Pλ(n)-Pλ(n)-m-1m(m+1)-pλ(n)-mm=1m(m+1)[Pλ(n)-Pλ(n)-m-1hhhhhhhhh-(m+1)pλ(n)-m],(36)Δm(Pλ(n)-Pλ(n)-mm)=1m(m+1)×[∑k=λ(n)-mλ(n)‍pk-(m+1)pλ(n)-m].
Next we will verify by the induction that(37)|∑k=λ(n)-mλ(n)‍pk-(m+1)pλ(n)-m|≤∑k=1m‍k|pλ(n)-k+1-pλ(n)-k|.
Ifm=1,
(38)|∑k=λ(n)-1λ(n)‍pk-2pλ(n)-1|=|pλ(n)-pλ(n)-1|.
Thus (37) holds for m=1. Now let us assume that (37) is true for m=j and we verify m=j+1(≤λ(n)). Since
(39)|∑k=λ(n)-(j+1)λ(n)‍pk-(j+2)pλ(n)-(j+1)|=|∑k=λ(n)-jλ(n)‍pk-(j+1)pλ(n)-(j+1)|=|∑k=λ(n)-jλ(n)‍pk-(j+1)pλ(n)-jhh+(j+1)pλ(n)-j-(j+1)pλ(n)-(j+1)∑k=λ(n)-jλ(n)‍|≤|∑k=λ(n)-jλ(n)‍pk-(j+1)pλ(n)-j|+|(j+1)pλ(n)-j-(j+1)pλ(n)-(j+1)|≤∑k=1j‍k|pλ(n)-k+1-pλ(n)-k|+(j+1)|pλ(n)-j-pλ(n)-(j+1)|=∑k=1j+1‍k|pλ(n)-k+1-pλ(n)-k|,
thus (37) is proved for m=j+1; that is, (37) is true for any 1≤m≤λ(n). Using (36) and (37) and interchanging the order of summation, we get
(40)∑m=1λ(n)‍|Δm(Pλ(n)-Pλ(n)-mm)|≤∑m=1λ(n)‍1m(m+1)×∑k=1m‍k|pλ(n)-k+1-pλ(n)-k|≤∑k=1λ(n)‍k|pλ(n)-k+1-pλ(n)-k|×∑m=k∞‍1m(m+1)=∑k=0λ(n)-1‍|Δpk|.
Now combining this with the assumption∑k=0λ(n)-1‍|Δpk|=O(Pλ(n)/λ(n)), we get from (34)
(41)∥snλ(f)-Nnλ(f)∥p=O((λ(n))-1).
This and Lemma1 with α=1 yield
(42)∥f-Nnλ(f)∥p=O((λ(n))-1).
In the proof of case (iii), we first verify that the condition∑k=1λ(n)-1‍k|Δpk|=O(Pλ(n)) implies that
(43)∑m=1λ(n)‍Δm(Pλ(n)-Pλ(n)-mm)=O(Pλ(n)λ(n)).
In view of (36) and (37)
(44)∑m=1λ(n)‍Δm(Pλ(n)-Pλ(n)-mm)≤∑m=1λ(n)‍1m(m+1)∑k=1m‍k|Δkpλ(n)-k|=∑1r‍+∑r+1λ(n)‍:=B1+B2,say.
Denoting again byr the integral part of (λ(n)/2), then, by Abel’s transformation, we have
(45)B1=∑m=1r‍1m(m+1)∑k=1m‍k|Δkpλ(n)-k|≤∑k=1r‍|Δkpλ(n)-k|≤∑i=r-2λ(n)-1‍|Δpi|=O(Pλ(n)λ(n)),
at the last step; we have used the condition ∑k=1λ(n)-1‍k|Δpk|=O(Pλ(n)). Consider the following:
(46)B2=∑m=r+1λ(n)‍1m(m+1)∑k=1m‍k|Δpλ(n)-k|=∑m=r+1λ(n)‍1m(m+1)×[∑k=1r‍k|Δpλ(n)-k|+∑k=rm‍k|Δpλ(n)-k|]:=B21+B22.
Furthermore, using again our assumption, we get(47)B21≤∑m=rλ(n)‍1m+1∑i=r-2λ(n)-1‍|Δpi|=O(Pλ(n)λ(n)),B22≤∑m=rλ(n)‍1m+1∑k=rm‍|Δpλ(n)-k|B22=O(1λ(n))[|Δp0|+2|Δp1|+⋯+(r+1)|Δpr+1|]B22=O(Pλ(n)λ(n)).
Summing up our partial results, we verified (43). Thus (34) and Lemma 1 again yield
(48)∥f-Nnλ(f)∥p=O((λ(n))-1).
Now, we prove case (v), by using (26), p-1=0, and Abel’s transformation
(49)Nnλ(f;x)-f(x)=1Pλ(n)∑m=0λ(n)‍pλ(n)-m{sm(f;x)-f(x)}=1Pλ(n)∑m=0λ(n)‍(Δmpλ(n)-m)∑k=0m‍{sk(f;x)-f(x)}=1Pλ(n)∑m=0λ(n)‍(m+1)(Δmpλ(n)-m)hhhhhhh×{σm(f;x)-f(x)}.
Hence in view of Lemma3(50)∥Nnλ(f)-f∥1≤1Pλ(n)∑m=0λ(n)‍(m+1)|Δmpλ(n)-m|∥f-σm(f)∥1=O(1Pλ(n))∑m=0λ(n)‍(m+1)1-α|Δmpλ(n)-m|=O(λ(n)1-αPλ(n))∑m=-1λ(n)-1‍|Δpm|=O(λ(n)1-αPλ(n))·O(Pλ(n)λ(n))=O((λ(n))-α).
Herewith case (v) is also verified and thus the proof of Theorem5 is complete.Proof of Theorem6.
SinceRnλ(f;x)=(1/Pλ(n))∑m=0λ(n)‍pmsm(f;x), so, in view of the assumptions of Theorem 6, we get
(51)∥f-Rnλ(f)∥1=1Pλ(n)∥∑m=0λ(n)‍pm{f-sm(f)}∥1≤1Pλ(n)∑m=0λ(n)-1‍(m+1)|Δpm|·∥f-σm(f)∥1+(λ(n)+1)pλ(n)Pλ(n)-1∥f-σnλ(f)∥1=O((λ(n))1-αPλ(n)-1)∑m=0λ(n)-1‍|Δpm|+O((λ(n))-α)=O((λ(n))-α).
This proves Theorem6.



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