Approximation of functions in Besov space by

Singh and Mittal Journal of Inequalities and Applications (2016) 2016:118 DOI 10.1186/s13660-016-1060-x

RESEARCH

Open Access

Approximation of functions in Besov space by deferred Cesàro mean Mradul Veer Singh* and Madan Lal Mittal *

Correspondence: [email protected] Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand 247667, India

Abstract In this paper we study the degree of approximation of functions (signals) in a Besov space by trigonometric polynomials using deferred Cesàro mean. We also deduce a few corollaries of our main result and compare them with the existing results. MSC: 42A10; 42A25; 40C05 Keywords: Besov space; degree of approximation; deferred Cesàro mean

1 Introduction During the last few decades, various investigators such as Alexits [], Chandra [, ], Das et al. [, ], Leindler [, ], Mittal et al. [–], Mohapatra and Chandra [], Prössdorf [], Quade [], etc. have studied the approximation properties of functions in Lipschitz and Hölder spaces using different summability methods. Here it is difficult to mention all the relevant published research papers in this area. However, some of the well-known results regarding the Lipschitz and Hölder norms are presented in survey papers [–] in an elegant way. Besov spaces are a much more general tool in describing the smoothness properties of functions and contain a large number of fundamental spaces such as Sobolev spaces, Hölder spaces, Lipschitz spaces, etc. []. This has motivated us to work on the degree of approximation of functions in Besov spaces. We recall a few definitions and some notation from DeVore and Lorentz [] that are necessary before introducing our results. Let Cπ := C[, π] denote the Banach space of all π -periodic continuous functions (signals) f defined on [, π] under the supremum π norm, and Lp := Lp [, π] := {f : [, π] → R;  |f (x)|p dx < ∞}, p ≥ , be the space of all π -periodic integrable functions. The Lp -norm of a function f is defined by π  p /p ( π  |f (x)| dx) ,  ≤ p < ∞, f p := p = ∞. ess sup ,

  be given, and let k denote the smallest integer k > α, that is, k = [α] + . For f ∈ Lp , if ωk (f , t)p = O t α ,

t > ,

(.)

then the signal f belongs to the generalized Lipschitz space Lip∗ (α, p). Then the seminorm is |f |Lip∗ (α,p) = supt> (t –α ωk (f , t)p ). Thus, Lip(α, p) ⊆ Lip∗ (α, p). Hölder spaces For  < α ≤ , let Hα = {f ∈ Cπ : ω(f , t) = O(t α )}. It is well known that Hα is a Banach space with norm f α = f C + sup t –α ω(t) for  < α ≤  and f  = f C , t>

and Hα ⊆ Hβ ⊆ Cπ for  < β ≤ α ≤ . The metric induced by the norm · α on Hα is called the Hölder metric. For  < α ≤  and  < p ≤ ∞, let Hα,p := Hα,p [, π] = {f ∈ Lp : ω(f , t)p = O(t α )} with the norm · α,p defined as follows: f α,p = f p + sup t –α ω(f , t)p

for  < α ≤  and f ,p = f p .

t>

Then Hα,p is a Banach space for p ≥  and a complete p-normed space (Maddox [], p.) for  < p < . Also, Hα,p ⊆ Hβ,p ⊆ Lp for  < β ≤ α ≤ . Besov space Let α >  be given, and let k = [α] + . For  < p, q ≤ ∞, the Besov space Bαq (Lp ) is the collection of all the signals (π -periodic functions) f ∈ Lp such that π (  [t –α ωk (f , t)p ]q dtt )/q ,  < q < ∞, |f |Bαq (Lp ) := ωk (f , ·) α,q = q = ∞, supt> (t –α ωk (f , t)p ),

(.)

is finite (Wojtaszczyk [], p.). It is known that (.) is a seminorm if  ≤ p, q ≤ ∞ and a quasi-seminorm in other cases (DeVore and Lorentz [], p.). The (quasi-)norm for Bαq (Lp ) is f Bαq (Lp ) := f p + |f |Bαq (Lp ) = f p + ωk (f , ·)α,q . Note  (i) In particular, for q = ∞, Bα∞ (Lp ) = Lip∗ (α, p). (ii) When  < α < , the space Bα∞ (Lp ) reduces to the space Hα,p (Das et al. []). (iii) By taking p = ∞ = q and  < α < , the Besov space reduces to the space Hα (Prössdorf []). (iv) In this paper, we consider the cases where p ≥  and  < q ≤ ∞.

(.)

Singh and Mittal Journal of Inequalities and Applications (2016) 2016:118

Page 3 of 12

2 Preliminaries Deferred Cesàro mean (DCM) Let un be a given infinite series with the sequence of partial sums {sn }. The DCM of sequence {sn } is defined by [], p., D(an , bn ; sn ) =

san + + san + + · · · + sbn , bn – a n

(.)

where {an } and {bn } be sequences of nonnegative integers satisfying a n < bn

lim bn = ∞.

and

(.)

n→∞

In the notation of matrix transformation,

D(an , bn ; sn ) =

∞

an,k sk ,

where an,k =

k=

 , bn –an



a n < k ≤ bn , otherwise.

This method is regular [] under condition (.). If an = n –  and bn = n, then D(an , bn ; sn ) is the identity transformation, and if an =  and bn = n, then D(an , bn ; sn ) is the Cesàro transformation (of order ) of sn , that is, σn . It is known that [] (C, ) ⊂ D(an , bn )

if and only if

an = O(). bn – a n

Also, note that

n n D(n – , n + k – ; sn ) = σn,k =  + σn+k– – σn– , k k

(.)

which is called the delayed arithmetic mean (DAM) of sequence {sn } [], p.. Some of its interesting properties can also be found in [, ]. Putting k = n, n, n, . . . in (.) gives a variety of DAM. For k = n, σn,k is called the second-type DAM [], p.. For a given signal f ∈ Lp , let a sn (f ; x) ≡ + (ak cos kx + bk sin kx) = uk (f ; x)  n

n

k=

k=

(.)

denote the partial sums, called trigonometric polynomials of degree (or order) n, of the first (n + ) terms of the trigonometric Fourier series of f . Let Dn (f ) := D(an , bn , sn (f ; x)) denote DCM of sn (f ; x), again a trigonometric polynomial. Then by ordinary calculations [], p., using (.) we get  Dn (f ) = π



π

sin[((bn + an + )/)u] sin[((bn – an )/)u] f (x + u) + f (x – u) du.  (bn – an ) sin (u/)

Let bn = (j + )an + j, where j ∈ N [], p.. Then

Dn (f ) =

 j(an + )π



π

sin[(j + )(an + )u] sin[j(an + )u] f (x + u) + f (x – u) du.   sin (u/)


Page 4 of 12

Using the identity

 j(an + )π

π

sin[(j + )(an + )u] sin[j(an + )u] du = ,  sin (u/)



we get  π

ln (x) := Dn (f ) – f (x) =

π 

KnD (u)φx (u) du,

(.)

where  sin[(j + )(an + )u] sin[j(an + )u] , j(an + )  sin (u/)

KnD (u) =

φx (u) = f (x + u) + f (x – u) – f (x). We write φx+t (u) – φx (u), (x, t, u) = φx+t (u) + φx–t (u) – φx (u), ln (x + t) – ln (x), Ln (x, t) = ln (x + t) + ln (x – t) – ln (x),

 < α < ,  ≤ α < ,  < α < ,  ≤ α < .

By elementary computations we get  π

Ln (x, t) =

π



ωk (ln , t)p = Ln (·, t)p .

KnD (u)(x, t, u) du and

We need the following lemmas in the proof of our main result. Lemma  ([]) Let  ≤ p ≤ ∞ and  < α < . If f ∈ Lp , then for  < t, u ≤ π , (i) (·, t, u)p ≤ ωk (f , t)p , (ii) (·, t, u)p ≤ ωk (f , u)p , (iii) φ· (u)p ≤ ωk (f , u)p , where k = [α] + . In view of our observation [], p., we replace the ordinary kernel Kn (u) by the deferred kernel KnD (u) in Lemma . of []. Lemma  ([]) Let  ≤ β < α < . If f ∈ Bαq (Lp ), p ≥ ,  < q < ∞, then K D (u)

π

(i)

n





π

= O() 

π

(ii) 

q

(·, t, u)p dt t βq t

u

K D (u) n

π

u π

= O()

q

(·, t, u)p dt t βq t

/q du

–(/q) D q/(q–) du . Kn (u)

α–β+(/q)

u



du

–(/q) q/(q–) uα–β KnD (u) du ,

/q


Page 5 of 12

The proofs run similarly to that of Lemma  of [], p.. Lemma  ([]) Let  ≤ β < α < . If f ∈ Bαq (Lp ), p ≥ , q = ∞, then sup  , ⎨(an + ) , – ln (·) β = O() (an + )–α+β+q , α – β – q– < , Bq (Lp ) ⎪ ⎩ – –q– , α – β – q– = . (an + ) [log(an + )]

(.)

Now we deduce a few corollaries of Theorem  for DAM of second type. If j =  and an = n – , then D(an , bn ; sn ) reduces to σn,n , and we obtain the following: Corollary  If  ≤ β < α <  and f ∈ Bαq (Lp ), p ≥ ,  < q ≤ ∞, then ⎧ – ⎪ α – β – q– > , ⎨n , – –α+β+q σn,n (f ; ·) – f (·) β = O() n , α – β – q– < , Bq (Lp ) ⎪ – ⎩ – n [log(n)]–q , α – β – q– = .


Page 6 of 12

We note that the estimates in Corollary  are similar to that of [], p., for the ordinary Cesàro mean. Now in view of Note , we get the following: Corollary  If  ≤ β < α <  and f ∈ Lip∗ (α, p), p ≥ , then ⎧ – ⎪ α – β > , ⎨n , –α+β = O() , α – β < , n ∞ (Lp ) ⎪ ⎩ – n log n, α – β = .

σn,n (f ; ·) – f (·) β B

We further deduce the following results from Corollary . Corollary  ([], p.) If  ≤ β < α <  and f ∈ Hα,p , p ≥ , then σn,n (f ; ·) – f (·) = O n–α+β . β,p Taking p = ∞ in Corollary , we have the following: Corollary  If  ≤ β < α <  and f ∈ Hα , then σn,n (f ; ·) – f (·) = O n–α+β . β This result can be compared with that of Prössdorf []. For β = , we get the following: Corollary  If  < α <  and f ∈ Lip(α, p), p ≥ , then σn,n (f ; ·) – f (·) = O n–α . p Corollary  If α = p = , that is, f ∈ Lip(, ), then σn,n (f ; ·) – f (·) = O n– log n .  We note that the estimates in Corollaries  and  are analogous to the results of Quade [].

4 Proof of main result The proof of Theorem  is divided into two sections. 4.1 The proof for 1 < q < ∞, p ≥ 1, 0 ≤ β < α < 2 Replacing α by β in (.), we have ln (·) β = ln (·)p + ωk (ln , ·)β,q . B (L ) q

(.)

p

Using the generalized Minkowski inequality [], p., and Lemma (iii), from (.) we have ln (·) ≤  p π

π 

φ· (u) K D (u) du ≤  p n π



π

ωk (f , u)p KnD (u) du.

(.)


Page 7 of 12

Using Hölder’s inequality and definition (.) of the Besov space, we get

–q– π q– ωk (f , u)p q K D (u)uα+q– q/(q–) du du n uα+q–   π –q– D α+q– q/(q–) Kn (u) u = O() du ,

ln (·) ≤  p π

π



ln (·) = O() p

π /(an +)

π

+ π /(an +)



:= O()[I + J],

–q– D α+q– q/(q–) K (u)u du n

say.

(.)

By the first part of Lemma ,

–q– K D (u)uα+q– q/(q–) du n

π /(an +)

I= 

π /(an +)

= O(an + )

q

u q–

(α+q– )

–q– du



π /(an +)

= O(an + )

u

q q– (α+)–

–q– du = O (an + )–α .

(.)



Now using the second part of Lemma , we have

–q– D α+q– q/(q–) K (u)u du

π

J=

n

π /(an +)

= O (an + )

–

π

u

q – q– (α+q –)

–q– du

π /(an +)

= O (an + )

–

π

u

q q– (α–)–

–q– du

π /(an +)

⎧ – ⎪ α > , ⎨(an + ) , –α = O() (an + ) , α < , ⎪ – ⎩ (an + )– [log(an + )]–q , α = .

(.)

Thus, combining (.)-(.), we have ⎧ – ⎪ α > , ⎨(an + ) , ln (·) = O() (an + )–α , α < , p ⎪ ⎩ – –q– , α = . (an + ) [log(an + )]

(.)

By repeated application of the generalized Minkowski inequality as in [], p., and Lemma  for the second term on the right-hand side of (.) we have – π –

Ln (·, t)p q dt q dt q = t tβ t   q– u π q D (·, t, u)p dt  K (u) du ≤ n π  t βq t 

ωk (ln , ·) = β,q

π

ωk (ln , t)p tβ

q


Page 8 of 12

– q (·, t, u)p dt q t βq t  u π –(/q) α–β D q/(q–) u Kn (u) = O() du

 + π

K D (u) du n

π



π

+ O() 

π

–(/q) q/(q–) uα–β+(/q) KnD (u) du

:= O()(I + J ),

say,

(.)

since (x + y)r ≤ xr + yr for positive x, y and  < r ≤  (for r =  – q– < ). Now

π

I =

–q– D q/(q–) Kn (u) du

α–β

u



–q– q/(q–) uα–β KnD (u) du +

π /(an +)

≤ 

:= I + I ,

π π /(an +)

–q– α–β D q/(q–) u Kn (u) du

say.

(.)

Using the first part of Lemma , we have

π /(an +)

I = O(an + )

q

u q–

(α–β)

–q– du



π /(an +)

= O(an + )

u

q q– (α–β+–(/q))–

–q– du



= O (an + )–α+β+(/q) .

(.)

By the second part of Lemma  we have I = O (an + )–

π

q

u q–

(α–β–)

–q– du

π /(an +)

= O (an + )–

π

u

q q– (α–β–(/q)–)–

–q– du

π /(an +)

⎧ – ⎪ α – β – q– > , ⎨(an + ) , – –α+β+q = O() (an + ) , α – β – q– < , ⎪ – ⎩ (an + )– [log(an + )]–q , α – β – q– = .

(.)

Now collecting (.)-(.) and using a similar argument as in (.), we have ⎧ – ⎪ α – β – q– > , ⎨(an + ) , – I = O() (an + )–α+β+q , α – β – q– < , ⎪ ⎩ – –q– , α – β – q– = . (an + ) [log(an + )] Using the earlier argument as in (.), we have J = 

π

–(/q) q/(q–) uα–β+(/q) KnD (u) du ,

(.)


J ≤

π /(an +)

–q– D q/(q–) Kn (u) du

α–β+(/q)

u



π

+

Page 9 of 12

π /(an +)

–q– q/(q–) uα–β+(/q) KnD (u) du

:= J + J ,

say.

(.)

By the first part of Lemma  we get

π /(n+)

J = O(an + )

u

q q– (α–β+(/q))

–q– du



π /(an +)

= O(an + )

q

u q–

(α–β+)–

–q– du



 . =O (an + )α–β

(.)

Using Lemma  and computing similarly as in I , we have

J = O (an + )

–

π

u

q q– (α–β+(/q)–)

–q– du

π /(an +)

= O (an + )

–

π

u

q q– (α–β–)–

–q– du

π /(an +)

⎧ – ⎪ α – β > , ⎨(an + ) , –α+β = O() (an + ) , α – β < , ⎪ – ⎩ (an + )– [log(an + )]–q , α – β = .

(.)

Now collecting (.)-(.) and using the earlier argument as in I , we have ⎧ – ⎪ α – β > , ⎨(an + ) , J = O() (an + )–α+β , α – β < , ⎪ – ⎩ (an + )– [log(an + )]–q , α – β = .

(.)

Combining (.), (.), and (.), we get ⎧ – ⎪ α – β – q– > , ⎨(an + ) , – –α+β+q ωk (ln , ·) = O() (an + ) , α – β – q– < , β,q ⎪ – ⎩ (an + )– [log(an + )]–q , α – β – q– = .

(.)

From (.), (.), and (.) we have ⎧ – ⎪ α – β – q– > , ⎨(an + ) , – –α+β+q ln (·) β = O() (an + ) , α – β – q– < , Bq (Lp ) ⎪ – ⎩ (an + )– [log(an + )]–q , α – β – q– = . This completes the proof of our Theorem  for p ≥ ,  < q < ∞, and  ≤ β < α < .

(.)


Page 10 of 12

4.2 The proof for q = ∞ and 0 ≤ β < α < 2 ln (·) β = ln (·)p + ωk (ln , ·)β,∞ . B (Lp )

(.)

∞

Using condition (.) in (.), we have ln (·) ≤  p π

π 

ωk (f , u)p KnD (u) du

π /(an +)

= O() 

:= O()[I + J ],

uα KnD (u) du +

π π /(an +)

uα KnD (u) du

say.

(.)

Using Lemma , we get

uα KnD (u) du = O(an + )

π /(an +)

I = 

π

J =

u π /(n+)

α

KnD (u) du = O

(an + )

–

π /(an +)

uα du = O (an + )–α ,

(.)



π

uα– du π /(n+)

⎧ – ⎪ α > , ⎨(an + ) , = O() (an + )–α , α < , ⎪ – ⎩ (an + )– [log(an + )]–q , α = .

(.)

Combining (.)-(.), we get ⎧ – ⎪ α > , ⎨(an + ) , ln (·) = O() (an + )–α , α < , p ⎪ – ⎩ (an + )– [log(an + )]–q , α = .

(.)

Using the generalized Minkowski inequality and Lemma , we have ωk (ln , ·) = sup t –β ωk (ln , t)p = sup t –β Ln (·, t)p β,∞ t>

t>

p /p π π   D dx = sup t –β K (u)(x, t, u) du n π  π  t> /p –β /p π π D p  t K (u) (x, t, u)p dx du ≤ sup n π π t>   –β π t (·, t, u) K D (u) du = sup p n π  t> π  = sup t –β (·, t, u)p KnD (u) du π  t> π uα–β Kn (u) du = O() 

π /(an +)

= O() 

:= O()[I + J ],

uα–β KnD (u) du + say.

π

π /(an +)

uα–β KnD (u) du (.)


Page 11 of 12

Using Lemma , we get

π /(an +)

I = 

π

J = π /(an +)

uα–β KnD (u) du = O (an + )β–α ,

uα–β KnD (u) du = O (an + )–

(.)

π

uα–β– du π /(an +)

⎧ – ⎪ α – β > , ⎨(an + ) , = O() (an + )–α+β , α – β < , ⎪ – ⎩ (an + )– [log(an + )]–q , α – β = .

(.)

Combining (.)-(.), we have ⎧ – ⎪ α – β > , ⎨(an + ) , –α+β ωk (ln , ·) = O() + ) , α – β < , (a n β,∞ ⎪ ⎩ – –q– , α – β = . (an + ) [log(an + )]

(.)

From (.), (.), and (.) we have ⎧ – ⎪ α – β – q– > , ⎨(an + ) , – –α+β+q = O() (an + ) , α – β – q– < , ∞ (Lp ) ⎪ – ⎩ (an + )– [log(an + )]–q , α – β – q– = .

ln (·) β B

(.)

This completes the proof of Theorem  for q = ∞. Combining Sections . and . completes the proof of Theorem .

5 Conclusions It is known that Besov spaces serve as generalizations of more elementary function spaces and are effective at measuring the smoothness properties of functions. As mentioned by DeVore and Popov [], p., “There are two definitions of Besov spaces that are currently in use. One uses the Fourier transform, and the second uses the modulus of smoothness of a function f . These two definitions are equivalent only under certain restrictions on the parameters. The Besov spaces defined by the modulus of smoothness occur more naturally in many areas of analysis including approximation theory.” In this paper we compute the error estimates of a function f in a Besov space by DCM of partial sums of the trigonometric Fourier series of f . We also deduce a few corollaries of our main result for the second-type DAM in a Besov space and other function spaces such as Lipschiz and Hölder spaces as particular cases and compare these results with earlier known results. As in [], p., we have used more general trigonometric polynomials (i.e., the secondtype DAM σn,n ) in Corollaries -; however, we can obtain similar estimates using other types of DAM such as D(n – , (j + )n – ) (or σn,jn ). Remark  Recently, Deˇger and Küçükaslan [] generalized the concept of DCM and studied approximation of a function using deferred Nörlund mean/deferred Riesz mean in Hölder metric, which may be the future interest of investigators in this direction.


Page 12 of 12

Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript. Acknowledgements The first author is thankful to Ministry of Human Resource Development, India, for financial support to carry out this research work. Received: 30 October 2015 Accepted: 7 April 2016 References 1. Alexits, G: Convergence problems of orthogonal series. Translated from the German by I. Földer. International Series of Monographs in Pure and Applied Mathematics, vol. 20. Pergamon, New York (1961) 2. Chandra, P: On the generalized Fejér means in the metric of Hölder space. Math. Nachr. 109, 39-45 (1982) 3. Chandra, P: Trigonometric approximation of functions in Lp -norm. J. Math. Anal. Appl. 275, 13-26 (2002) 4. Das, G, Ghosh, T, Ray, BK: Degree of approximation by their Fourier series in generalized Hölder metric. Proc. Indian Acad. Sci. Math. Sci. 106, 139-153 (1996) 5. Das, G, Nath, A, Ray, BK: An estimate of the rate of convergence of Fourier series in the generalized Hölder metric. In: Analysis and Applications, pp. 43-60. Narosa, New Delhi (2002) 6. Leindler, L: Trigonometric approximation in Lp -norm. J. Math. Anal. Appl. 302, 129-136 (2005) 7. Leindler, L: A relaxed estimate of the degree of approximation by Fourier series in generalized Hölder metric. Anal. Math. 35(1), 51-60 (2009) 8. Mittal, ML, Rhoades, BE, Mishra, VN, Singh, U: Using infinite matrices to approximate functions of class Lip(α , p) using trigonometric polynomials. J. Math. Anal. Appl. 326(1), 667-676 (2007) 9. Mittal, ML, Rhoades, BE, Sonker, S, Singh, U: Approximation of signals of class Lip(α , p) by linear operators. Appl. Math. Comput. 217(9), 4483-4489 (2011) 10. Mittal, ML, Singh, MV: Approximation of signals (functions) by trigonometric polynomials in Lp -norm. Int. J. Math. Math. Sci. 2014, Article ID 267383 (2014) 11. Mohapatra, RN, Chandra, P: Degree of approximation of functions in the Hölder metric. Acta Math. Hung. 41(1-2), 67-76 (1983) 12. Prössdorf, S: Zur Konvergenz der Fourierreihen Hölder stetiger Funktionen. Math. Nachr. 69, 7-14 (1975) 13. Quade, ES: Trigonometric approximation in the mean. Duke Math. J. 3(3), 529-543 (1937) 14. Bustamante, J, Jiménez, MA: Trends in Hölder approximation. In: Approximation, Optimization and Mathematical Economics (Pointe-ã-Pitre, 1999), pp. 81-95. Springer, Berlin (2001) 15. Chandra, P: Degree of approximation of functions in the Hölder metric. In: Fourier Analysis, Approximation Theory and Applications, Aligarh, 1993, pp. 37-51. New Age International Publishers, New Delhi (1997) 16. Holland, ASB: A survey of degree of approximation of continuous functions. SIAM Rev. 23(3), 344-379 (1981) 17. Triebel, H: Theory of Function Spaces. Reprint of 1983 edition. Birkhäuser/Springer, Basel (2010) 18. DeVore, RA, Lorentz, GG: Constructive Approximation. Springer, New York (1993) 19. Maddox, IJ: Elements of Functional Analysis. Cambridge University Press, Cambridge (1970) 20. Wojtaszczyk, P: A Mathematical Introduction to Wavelets. London Mathematical Society Student Texts, vol. 37. Cambridge University Press, New York (1997) 21. Agnew, RP: On deferred Cesàro means. Ann. Math. (2) 33(3), 413-421 (1932) 22. Zygmund, A: Trigonometric Series. Vols. I, II. Reprint of the 1979 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2002) 23. Stein, EM, Shakarchi, R: Fourier Analysis. Princeton Lectures in Analysis, vol. 1. Princeton University Press, Princeton (2003) 24. Nayak, L, Das, G, Ray, BK: An estimate of the rate of convergence of Fourier series in the generalized Hölder metric by deferred Cesàro mean. J. Math. Anal. Appl. 420(1), 563-575 (2014) 25. Mohanty, H, Das, G, Ray, BK: Degree of approximation of Fourier series of functions in Besov space by (N, pn ) mean. J. Orissa Math. Soc. 30(2), 13-34 (2011) 26. Mittal, ML, Singh, MV: Degree of approximation of signals (functions) in Besov space using linear operators. Asian-Eur. J. Math. 9(1), Article ID 1650009 (2016). doi:10.1142/S1793557116500091 27. Proakis, JG: Digital Communications. McGraw-Hill, New York (1995) 28. Psarakis, EZ, Moustakides, GV: An L2 -based method for the design of 1-D zero phase FIR digital filters. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 44(7), 591-601 (1997) 29. Bachman, G, Narici, L, Beckenstein, E: Fourier and Wavelet Analysis. Springer, New York (2000) 30. Nayak, L, Das, G, Ray, BK: An estimate of the rate of convergence of the Fourier series in the generalized Hölder metric by delayed arithmetic mean. Int. J. Anal. 2014, Article ID 171675 (2014) 31. Mohanty, H: Some aspects of measure of approximation. Ph.D. Thesis, Utkal University, Odisha, India (2012) 32. DeVore, RA, Popov, VA: Interpolation of Besov spaces. Trans. Am. Math. Soc. 305(1), 397-414 (1988) ˇ U, Küçükaslan, M: A generalization of deferred Cesáro means and some of their applications. J. Inequal. Appl. 33. Deger, 2015, Article ID 14 (2015)