Approximation of a Function in Hölder Class Using Double Karamata (Kλ,μ) Method

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EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 3, 2020, 567-578

ISSN 1307-5543 – www.ejpam.com Published by New York Business Global

Approximation of a function in H¨

older class using

double Karamata (

K

λ,µ

) method

H. K. Nigam1_{, Md Hadish}1,∗

1 _{Department of Mathematics, Cental University of South Bihar, Gaya, Bihar, India}

Abstract. In this paper, we establish a new theorem on the best approximation of a function of two variables belonging to H¨older class by double Karamata (Kλ,µ_{) means of its double Fourier} series.

2020 Mathematics Subject Classifications: 40C10, 40G05, 40G10, 42A10, 42A24, 40C05, 41A25, 42B05

Key Words and Phrases: H¨older class, double Karamata (Kλ,µ_{), double Fourier series, stirling} numbers, error approximation.

1. Introduction

Kλ-method was first introduced by Karamata [7]. Lotosky [10] re-introduced the spe-cial case λ = 1. Only after the study of Agnew [1], an intensive study of these and similar cases took place. Vuˆckoviˆc [19] applied this method for summability of Fourier series. Kathal[8] extended the result of Vuˆckoviˆc [19]. The approximation of a 2π-periodic function f(x) in different Lipschitz classes using Ces`aro, N¨orlund and Kλ summability methods of Fourier series and conjugate Fourier series has been studied by the researchers [2, 5, 6, 12, 14–16].

The approximation of a 2π-periodic functionf(x) in H¨older metric by different summa-bility transforms of Fourier series has been studied by the researchers like [4, 11, 13]. The approximation of a functionf(x, y) (2π-periodic with respect to the variablesx, y) of their Fourier series has been studied by [17, 18]. Lal [9] studied the approximation of a function in Lipschitz class by matrix means of its double Fourier series. But nothing seems to have done to obtain the best approximation of the function f(x, y), a 2π-periodic with respect to the variable x, y, of its double Fourier series. Thus, in this paper, we obtain the best approximation of the functionh(ζ,Θ) in H¨older class byKλ,µmethod of its double Fourier series.

∗

Corresponding author.

DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3663

Email addresses: [email protected](H. K. Nigam),[email protected](Md Hadish)

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2. Preliminaries

Under usual assumptions, Fourier series of h(t) is defined as

h(t)∼ 1

2a0+

∞

X

ρ=1

(aρcosρt+bρsinρt). (1)

Under usual assumptions, double Fourier series of h(ζ,Θ) is given by

h(ζ,Θ)∼ ∞

X

ν=0

∞

X

ρ=0

αν,ρ[aν,ρcosνζcosρΘ+bν,ρsinνζcosρΘ+cν,ρcosνζsinρΘ+dν,ρsinνζsinρΘ],

(2) where

αν,ρ =

  

 

1

4 for ν =ρ= 0 1

2 for ν >0, ρ= 0 and ν = 0, ρ >0 1 for ν >0, ρ >0.

(3)

and

aν,ρ =

1 π2

Z Z

S2

h(ζ,Θ) cosνζcosρΘdζ dΘ (4) with similar expressions for bν,ρ, cν,ρ and dν,ρ forν = 0,1,2, . . . and ρ= 0,1,2, . . ., where

S2 denotes the fundamental square (−π, π;−π, π). The partial sums of (2) can be denoted by

sν,ρ(h;ζ,Θ) = ν

X

i=0

ρ

X

j=0

[aijcosiζcosjΘ+bijsiniζcosjΘ+cijcosiζsinjΘ+dijsiniζsinjΘ].

We can also write

sν,ρ(ζ,Θ) =

1 π2

Z Z

S2

h(ζ+σ,Θ +τ)sin ν+ 1 2

σsin ρ+1₂τ 4 sin σ₂.sin τ₂ dσ dτ. The numberν_pfor 0≤p≤ν and ν ∈_N∪ {0} is defined by

ν−1 Y

l=0

(ζ+l) =ζ(ζ+ 1)· · ·(ζ+ν−1) =

ν

X

p=0

ν p

ζp= Γ(ζ+ν) Γζ .

Let us also define forρ∈_N∪ {0}, the number ρ_qfor 0≤q≤ρ, by

ρ−1 Y

l=0

(Θ +l) = Θ(Θ + 1)· · ·(Θ +ρ−1) =

ρ

X

q=0

ρ q

Θq= Γ(Θ +ρ) ΓΘ .

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Let{sν,ρ}be the sequence of partial sums of double infinite seriesP_ν∞₌₀P∞_ρ₌₀aν,ρand

write

sλ,µ_ν,ρ = Γλ Γ(λ+ν)×

Γµ Γ(µ+ρ)

ν

X

p=0

ρ

X

q=0

ν p

ρ q

λpµqsp,q (5)

to denote (ν, ρ)th Kλ,µ-means of order (λ, µ)>0. If

sλ,µ_ν,ρ →sas (ν, ρ)→ ∞, (6)

then the sequence sν,ρ or the series P∞ν=0 P∞

ρ=0aν,ρ summable to s by double Karamata

(Kλ,µ) method of order (λ, µ)>0. Thus,

sλ,µ_ν,ρ →s(Kλ,µ) as (ν, ρ)→ ∞.

The methodKλ,µis regular for (λ, µ)>0. The regularity of theKλ,µmethod is supposed throughout the paper.

“Sinceh(ζ) is continuous and 2π-periodic function then the H¨older class forh(ζ) is defined as

Hα={h∈C2π :|h(ζ)−h(Θ)| ≤K|ζ−Θ|},

where K is a positive constant. It can be verified that Hα is a Banach space with the

norm k.k_α defined by

khkα=khkC + sup ζ6=Θ

∆αh(ζ,Θ), (7)

where

∆αh(ζ,Θ) = |h(ζ)−h(Θ)|

|ζ−Θ|α forζ 6= Θ.

By convention ∆0h(ζ,Θ) = 0 and khkC = supζ∈[−π,π]|h(ζ)|. The metric induced by the norm (7) on Hα is called the H¨older metric [13].”

Since h(ζ,Θ) is continuous and 2π-periodic function then the H¨older class for h(ζ,Θ) is defined as

Hα,β =

n

h:|h(ζ,Θ;z, w) :=|h(ζ,Θ)−h(z, w)| ≤C1

|ζ−z|α₊_|_Θ₋_w_|βo

for some α, β > 0 and for all ζ,Θ, z, w. In above class of function, C1 is some positive constant, which may depend onh, but not onζ,Θ, t. Hα,β class of function is identical to

Lip(α, β) class of function.

Hα,β is a Banach space, whose norm k.kα,β is defined by

khkα,β =khkC+ sup ζ6=z,

Θ6=w

∆α,βh(ζ,Θ;z, w) (8)

i.e.

khkα,β=khkC+ sup ζ6=z,

Θ6=w

|f(ζ,Θ)−f(z, w)|

(4)

where ∆α,βh(ζ,Θ;z, w) = _||h_ζ₋(ζ,_zΘ)_|α₊−_|h_Θ(₋z,w_w_|)β| (ζ 6=z,Θ6=w).

By convention ∆0,0h(ζ,Θ;z, w) = 0 and

khkC = sup

(ζ,Θ)∈S2

|h(ζ,Θ)|. (9)

“The η-order error of approximation of a functionh∈C2π is defined by

Eη(h) = inf tη

kh−tηk,

wheretη is a trigonometric polynomial of degreeη (Bernstein [3])”.

“If Eη(h) → 0 as η → ∞, then Eη(h) is said to be the best approximation of h ([20])”.

We write

φ(t) =h(ζ+t) +h(ζ−t)−2h(ζ). Φ(t) =

Z t

0

|φ(σ)|dσ. Φ(σ, τ) = Φ(ζ,Θ;σ, τ)

= 1

4[h(ζ+σ,Θ +τ) +h(ζ+σ,Θ−τ) +h(ζ−σ,Θ +τ) +h(ζ−σ,Θ−τ)−4h(ζ,Θ)]

where Ψ(σ, τ) := Ψ(z, w;σ, τ).

Since h(ζ,Θ)∈Hα,β,then|F(σ, τ)|=O(|ζ−z|α+|Θ−w|β).

K_νλ(σ) = Pν

p=0 ν

p

λpsin p+1₂ σ

Γ(λ+ν) sin σ₂ . K_ρµ(τ) =

Pρ

q=0 ρ

q

µqsin q+1₂τ Γ(µ+ρ) sin τ₂ .

3. Main Theorem

Theorem 1. The best approximation of a2π-periodic function h(ζ,Θ)of two variables ζ

andΘand Lebesgue integrable overS2(−π, π;−π, π) inHα,β,0< α, β≤1class by double Karamata (Kλ,µ) method of its double Fourier series is given by

ksλ,µ_ν,ρ(ζ,Θ)−h(ζ,Θ)kα,β

=O

M N ΓλΓµ (ν+ 1)(ρ+ 1)

1 (ν+ 1)α +

1

(ρ+ 1)β + 1

+O

M ΓλΓµ (ν+ 1)Γ(µ+ρ)

1 +lnπ(ρ+ 1)

(ν+ 1)α + lnπ(ρ+ 1)

(5)

+O

N ΓλΓµ (ρ+ 1)Γ(λ+ν)

1 +lnπ(ρ+ 1)

(ρ+ 1)β + lnπ(ν+ 1)

+O

ΓλΓµ

Γ(λ+ν)Γ(µ+ρ) ln (ν+ 1)(ρ+ 1)π 2

+{lnπ(ν+ 1)}{lnπ(ρ+ 1)}

where M =λln(ν+ 1) + 1 andN =µln(ρ+ 1) + 1.

4. Lemmas

Lemma 1. “([19]). Letλ >0 and0< t < π₂ then

ImΓ(λeit+ρ) Γ(λcost+ρ) sin ₂t =

|sin(λln(ρ+ 1).sint)|

sin t₂ +O(1)

as ρ→ ∞ uniformly in t”.

Lemma 2. “([12]). For0< σ < _ν₊₁1 ,

K_νλ(σ) =O[λln(ν+ 1)] +O(1)

and for 0< τ < _ρ₊₁1 ,

K_ρµ(τ) =O[µln(ρ+ 1)] +O(1)

”.

Lemma 3. For _ν₊₁1 ≤σ ≤π

K_νλ(σ) =O

1 σ Γλ

.

Proof. Using sinσ₂ ≥ σ_π and |sin(ρσ)| ≤1

|K_νλ(σ)|=

Pν

p=0 ν

p

λpsin p+1₂ σ

Γ(λ+ν) sin σ₂

≤ 1

Γ(λ+ν)

ν

X

p=0

ν p

λp 1 sin σ₂ =O

1 σ Γλ

.

Lemma 4. For _ρ₊₁1 ≤τ ≤π

K_ρµ(τ) =O

1 τ Γµ

.

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5. Proof of the Main Theorem

Let sν,ρ(ζ,Θ) denote the (ν, ρ)th partial sum of the series (2), we have

sν,ρ(ζ,Θ)−h(ζ,Θ) =

1 π2 Z π 0 Z π 0

Φ(σ, τ)sin ν+ 1 2

σ sin σ₂ .

sin ρ+1₂τ sin τ₂ dσ dτ. Denoting Kλ,µ means of{sν,ρ(ζ,Θ)} by sλ,µν,ρ(ζ,Θ), we get

sλ,µ_ν,ρ(ζ,Θ)−h(ζ,Θ) = Γλ

Γ(λ+ν). Γµ Γ(µ+ρ)

ν X p=0 ρ X q=0 ν p ρ q

λpµq(sp,q(ζ,Θ)−h(ζ,Θ))

= ΓλΓµ π2

Z π

0 Z π

0

Φ(σ, τ)

ν X p=0 ν p

_λp_sin _p₊1 2

σ Γ(λ+ν) sin σ₂

ρ X q=0 ρ q

_µq_sin _q₊1 2

τ Γ(µ+ρ) sin τ₂dσ dτ = ΓλΓµ

π2 Z π

0 Z π

0

Φ(σ, τ)K_νλ(σ)K_ρµ(τ)dσ dτ

=Iν,ρ(ζ,Θ) (say). (10)

Let us estimate

sup

ζ6=z,

Θ6=w

|Iν,ρ(ζ,Θ)−Iν,ρ(z, w)|

|ζ−z|α₊_|_Θ₋_w_|β =O(1). (11)

Now,

= ΓλΓµ π2 Z π 0 Z π 0

F(σ, τ)K_νλ(σ)K_ρµ(τ)dσ dτ

≤ ΓλΓµ

π2 Z 1 ν+1 0 Z 1 ρ+1 0 + Z 1 ν+1 0 Z π 1 ρ+1 + Z π 1 ν+1 Z 1 ρ+1 0 + Z π 1 ν+1 Z π 1 ρ+1 !

|F(σ, τ)K_νλ(σ)K_ρµ(τ)|dσ dτ

=O

ΓλΓµ

π2 (J1+J2+J3+J4)

. (12)

Using the fact that |F(σ, τ)|=O(|ζ−z|α₊_|_Θ₋_w_|β_{) and Lemma 4.2 for 0}_{< σ <} 1

ν+1, we obtain

J1= Z _ν₊₁1

0

Z _ρ₊₁1

0

= [O{λln(ν+ 1)}+O(1)] [O{µln(ρ+ 1)}+O(1)] Z _ν₊₁1

0

Z _ρ₊₁1

0

|F(σ, τ)|dσ dτ

=O(1)[{λln(ν+ 1)}+ 1][{µln(ρ+ 1)}+ 1][(|ζ−z|α₊_|_Θ₋_w_|β_)]

Z _ν₊₁1

0

Z _ρ₊₁1

0

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=O(1)

[{λln(ν+ 1)}+ 1]

ν+ 1 ×

[{µln(ρ+ 1)}+ 1] ρ+ 1

(|ζ−z|α+|Θ−w|β). (13)

For 0< α, β ≤1, by using Lemmas 4.2 for 0< σ < _ν₊₁1 , 4.4 and the fact that |F(σ, τ)|= O(|ζ−z|α₊_|_Θ₋_w_|β_{), we get}

J2= Z _ν₊₁1

0

Z π 1

ρ+1

=O{λlog(ν+ 1) + 1}

Z 1

ν+1

0

Z π 1

ρ+1

|F(σ, τ)||K_ρµ(τ)|dσ dτ

=O(1){λlog(ν+ 1) + 1} ν+ 1

Z π 1

ρ+1

|F(σ, τ)||K_ρµ(τ)|dτ

=O _{

λln(ν+ 1)}+ 1 ν+ 1

(|ζ−z|α+|Θ−w|β) Z π

1

ρ+1 1 τΓµdτ

=O _{

λln(ν+ 1)}+ 1 (ν+ 1)Γµ

(|ζ−z|α₊_|_Θ₋_w_|β_{) ln}_π₍_ρ_{+ 1)}_. ₍₁₄₎

Similarly by changing the order of integration inJ3and using Lemmas 4.2 for 0< τ < _ρ₊₁1 and 4.3, we obtain

J3 =O

{µln(ρ+ 1)}+ 1 (ρ+ 1)Γλ

(|ζ−z|α+|Θ−w|β) ln((ν+ 1)π). (15)

Now, using Lemmas 4.3 and 4.4, we get

J4 = Z π

1

ν+1 Z π

1

ρ+1

= Z π

1

ν+1

K_νλ(σ) Z π

1

ρ+1 1 τΓµdτ

!

|F(σ, τ)|dσ

=O

1 Γµ

Z π

1

ν+1

K_νλ(σ) ln((ρ+ 1)π)|F(σ, τ)|dσ

=O

ln((ρ+ 1)π).ln((ν+ 1)π)

ΓλΓµ (|ζ−z|

α₊_|_Θ₋_w_|β₎

. (16)

Combining (12) to (16), we obtain

(|ζ−z|α₊_|_Θ₋_w_|β₎

=O

[{λln(ν+ 1)}+ 1][{µ ln(ρ+ 1)}+ 1]ΓλΓµ

(ν+ 1)(ρ+ 1) +

[{λln(ν+ 1)}+ 1]Γλ lnπ(ρ+ 1) ν+ 1

(8)

+O

[{µln(ρ+ 1)}+ 1]Γµlnπ(ν+ 1)

ρ+ 1 +

ln(ρ+ 1)π) ln((ν+ 1)π) ΓλΓµ

. (17)

Now, from (10), we have

|Iν,ρ(ζ,Θ)|

= ΓλΓµ π2 Z π 0 Z π 0

Φ(σ, τ)K_νλ(σ)K_ρµ(τ)dσ dτ

≤ΓλΓµ

π2 " Z 1 ν+1 0 Z 1 ρ+1 0 +

Z ν+1

0 Z π 1 ρ+1 + Z π 1 ν+1 Z 1 ρ+1 0 + Z π 1 ν+1 Z π 1 ρ+1

|Φ(σ, τ)||K_νλ(σ)||K_ρµ(τ)|dσ dτ #

= ΓλΓµ

4π2 (I1+I2+I3+I4). (18)

Now, I1 = Z 1 ν+1 0 Z 1 ρ+1 0

|Φ(σ, τ)||K_νλ(σ)||K_ρµ(τ)|dσ dτ

I1 =O "

(λlog(ν+ 1) + 1)(µlog(ρ+ 1) + 1) Z 1 ν+1 0 Z 1 ρ+1 0

(σα+τβ)dσ dτ # =O " M N Z 1 ν+1 0 σα ( Z 1 ρ+1 0 dτ ) dσ # +O " M N Z 1 ρ+1 0 τβ ( Z 1 m+1 0 dσ ) dτ #

where M ={λln(ν+ 1)}+ 1 and N ={µln(ρ+ 1)}+ 1 =

" M N ρ+ 1

Z 1

ν+1

0

σαdσ #

+ "

M N ν+ 1

Z 1

ρ+1

0

τβdτ #

=O(1) M N (ν+ 1)(ρ+ 1)

1 (ν+ 1)α +

1 (ρ+ 1)β

. (19) I2 = Z 1 ν+1 0 Z π 1 ρ+1

=O "

{λlog(ν+ 1) + 1}

Z 1 ν+1 0 Z π 1 ρ+1 1 τΓµ(σ

α₊_τβ₎_{dσ dτ}

# =O " Γµ Z 1 ν+1 0 Z π 1 ρ+1 1 τσ

α_{dσ dτ}

# +O " M Γµ Z 1 ν+1 0 Z π 1 ρ+1 1 ττ

β_{dσ dτ}

#

=O(1) M (ν+ 1)Γµ

ln(ρ+ 1)π (ν+ 1)α + 1

. (20)

Similarly,

I3=O(1) N (ρ+ 1)Γλ

ln(ν+ 1)π (ρ+ 1)β + 1

(9)

and I4 = Z π 1 ν+1 Z π 1 ρ+1

=O " Z π 1 ν+1 Z π 1 ρ+1 1 στΓλΓµ[σ

α₊_τβ_]_{dσ dτ}

#

=O "

1 ΓλΓµ

Z π 1 ν+1 σα σ ( Z π 1 ρ+1 dτ τ ) dσ # +O " 1 ΓλΓµ

Z π 1 ν+1 1 σ ( Z π 1 ρ+1 τβ τ dτ ) dσ # . =O 1

ΓλΓµln(ρ+ 1)π

+O

1

ΓλΓµln(ν+ 1)π

=O

1

ΓλΓµln(ν+ 1)(ρ+ 1)π 2

(22)

Combining (18) to (22), we get

Iν,ρ =O

ΓλΓµ

π2

M N (ν+ 1)(ρ+ 1)

1 (ν+ 1)α +

1 (ρ+ 1)β

+ M

(ν+ 1)Γµ

1 +ln(ρ+ 1)π (ν+ 1)α

+O

ΓλΓµ π2

N (ν+ 1)Γλ

1 +ln(ν+ 1)π (ρ+ 1)β

+ 1

ΓλΓµlog(ν+ 1)(ρ+ 1)π 2

. (23)

By (17) and (23), we obtain

kIν,ρkα,β=ksλ,µν,ρ(ζ,Θ)−h(ζ,Θ)kα,β=O

M NΓλΓµ (ν+ 1)(ρ+ 1)

1 (ν+ 1)α +

1

(ρ+ 1)β + 1

+O

MΓλΓµ (ν+ 1)Γ(µ+ρ)

1 +log((ρ+ 1)π)

(ν+ 1)α + log((ρ+ 1)π)

+

NΓλΓµ (ν+ 1)Γ(λ+ν)

1 +log((ρ+ 1)π)

(ρ+ 1)β + log((ν+ 1)π)

+O

ΓλΓµ

Γ(λ+ν)Γ(µ+ρ) log (ν+ 1)(ρ+ 1)π 2

+{log((ν+ 1)π)}{log((ρ+ 1)π)}

whereM ={λln(ν+ 1)}+ 1 and N ={µln(ρ+ 1)}+ 1.

6. Verification

We calculate error by putting some values of ν, ρ, α, β, λ, µ.

6.1. ν =ρ= 10, α=β = 0.5, λ= 2, µ = 3

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6.2. ν =ρ= 20, α=β = 0.5, λ= 2, µ = 3

error E2∼0.46798

6.3. ν =ρ= 50, α=β = 0.5, λ= 2, µ = 3

error E3∼0.111632

6.4. ν =ρ= 500, α=β= 0.5, λ= 2, µ= 3

error E4∼0.0011456

6.5. ν =ρ= 1000, α=β = 0.5, λ= 2, µ= 3

error E5∼6.83193×10−4.

6.6. ν =ρ= 10000, α=β = 0.5, λ= 2, µ= 3

error E6∼1.13411×10−5.

6.7. ν =ρ= 100000, α=β = 0.5, λ= 2, µ = 3

error E7∼1.1191×10−7.

6.8. ν =ρ= 1010, α =β = 0.5, λ= 2, µ= 3

error E8∼5.36301×10−15.

7. Conclusion

From above verification, we observed that error estimation approaches to zero rapidly asν, ρincrease infinitely. Thus, we arrive at the best approximation of the function.

Acknowledgements

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