A characterization of the k functional for the algebraic version of the trigonometric jackson integr

Result.Math. 54 (2009), 397–413c© 2009 Birkhauser Verlag Basel/Switzerland1422-6383/030397-17, published online July 14, 2009

DOI 10.1007/s00025-009-0362-4 Results in Mathematics

A Characterization of the K-Functional for theAlgebraic Version of the Trigonometric JacksonIntegrals Gs,n and the K-Functionals for Cao–Gonska Operators G∗

s,n and G+s,n

Teodora Dimova Zapryanova

Abstract. We construct moduli of functions which are computable character-istic equivalents of the error in approximation by algebraic versions of trigono-metric Jackson integrals and Cao–Gonska operators.

Mathematics Subject Classification (2000). Primary 41A25; Secondary 41A36.

Keywords. K-functional, moduli of functions, linear operators.

1. Introduction

Let X be a normed space. For a given differential operator D we set X∩D−1(X) ={g ∈ X : Dg ∈ X} . Given a subspace Y of X∩D−1(X), we define for every f ∈ Xand t > 0 the Peetre K-functional by

K(f, t;X,Y,D) := inf {‖f − g‖X + t ‖Dg‖X : g ∈ Y } . (1)

Let I be the identity operator and Lf denote the linear function interpolatingf at −1 and 1, i.e.,

L(f, x) := f(1)1 + x

2+ f(−1)

1 − x

2, −1 ≤ x ≤ 1 .

Let X be the space C[−1, 1]. We denote the norm in C[−1, 1] by ‖.‖ . Exam-ples of the operator D are

D1g := Hg , where H := (H1)2 , (H1g)(x) := (1 − x2)1/2 d

dxg(x) ,

D2g := H(I − L)g and D3g := (I − L)Hg .

398 T.D. Zapryanova Result.Math.

In these cases the K-functionals defined in (1) are

K(f, t;C[−1, 1], C2,H

):= inf{‖f − g‖ + t‖Hg‖ : g ∈ C2} , (2)

K∗(f, t;C[−1, 1], C2,H(I − L))

:= inf{‖f − g‖ + t‖H(I − L)g‖ : g ∈ C2} , (3)

K+(f, t;C[−1, 1], C2, (I − L)H

):= inf{‖f − g‖ + t‖(I − L)Hg‖ : g ∈ C2} . (4)

The K-functional (2) is equivalent to the approximation error of Jacksontype operators Gs,n in uniform norm, while the K-functional (3) is equivalentto the approximation error of operators G∗

s,n (such equivalence was establishedin [5]). The K-functional (4) is equivalent to the approximation error of operatorsG+

s,n (such equivalence was established in [6]).We recall that the operators Gs,n :C[−1, 1] → Πsn−s, G

∗s,n : C[−1, 1] → Πsn−s and G+

s,n : C[−1, 1] → Πsn−s aredefined by

Gs,n(f, x) =

π∫

−π

f(cos(arccos x + v)

)Ks,n(v)dv ,

G∗s,n(f, x) = Gs,n(f, x) + L(f, x) − Gs,n(Lf, x) ;

G+s,n(f, x) = Gs,n(f, x) + L(f, x) − L(Gs,nf, x) ,

where

Ks,n(v) = μs,n

(sin(nv/2)sin(v/2)

)2s

,

π∫

−π

Ks,n(v)dv = 1 .

Notation Φ(f, t) ∼ Ψ(f, t) means that there is a positive constant γ, inde-pendent of f and t, such that γ−1Ψ(f, t) ≤ Φ(f, t) ≤ γΨ(f, t).

By c we denote positive constants, independent of f and t, that may differat each occurrence.

For a natural number r number we putCr[a, b] = {f : f, f

′, . . . , f (r) ∈ C[a, b](continuous function in [a, b])}.

For r ∈ N and 1 ≤ p ≤ ∞ we denoteW r

p [a, b] = {f : f, f′, . . . , f (r−1) ∈ AC[a, b], f (r) ∈ Lp[a, b]}.

Here, in the case p = ∞ we mean only boundedness of the r-th derivative.Thus, W r

∞ = Cr.

2. Known results

The idea for the equivalence of the approximation errors of a given sequence ofoperators and the values of proper K-functionals was studied systematically in [1].Such equivalence was established for the algebraic version of trigonometric Jacksonintegrals Gs,n and K-functionals (2) and for the operators G∗

s,n and K-functionals(3) in uniform norm in [5] and later for the operators G+

s,n and K-functionals (4)in [6] (see Theorem A a), b), c) , respectively).

Vol. 54 (2009) A Characterization of K-functionals 399

Theorem A.a) For every f ∈ C[−1, 1] and s, n ∈ N, s ≥ 3 we have

‖f − Gs,nf‖ ∼ K

(f,

1n2

;C[−1, 1], C2,H

).

b) For every f ∈ C[−1, 1] and s, n ∈ N,n ≥ n0, s ≥ 3 we have∥∥f − G∗

s,nf∥∥ ∼ K∗

(f,

1n2

;C[−1, 1], C2,H(I − L))

.

c) For every f ∈ C[−1, 1] and s, n ∈ N, s ≥ 3 we have∥∥f − G+

s,nf∥∥ ∼ K+

(f,

1n2

;C[−1, 1], C2, (I − L)H)

.

On the other hand the idea for the equivalence of K- functionals and moduli ofsmoothness can be traced back to the 1960,s in the works of Peetre (see [2, Chap. 6,Theorem 2.4]).

Theorem B. For every 0 < t ≤ 1, f ∈ Lp[0, 1], 1 ≤ p ≤ ∞, r ∈ N and Dg = g(r)

we haveK

(f, tr;Lp[0, 1],W r

p ,D)∼ ωr(f, t)p .

The classical moduli of smoothness are defined by

ωr(f, t)p := sup0<h≤t

‖Δrhf(.)‖p ,

and the finite difference with a fixed step h is given by

Δrhf(x) =

{r∑k=0 (−1)r−k

(rk

)f(x + kh) , if x, x + rh ∈ [a, b] ,

0 , otherwise .

Similar characterizations of some weighted Peetre K-functionals in terms of weight-ed moduli were established in [3].

3. New results

The aim of this paper is to introduce moduli that are equivalent to the K-functionals (2), (3) and (4), respectively.

We define the modulus Ω2(f, t) for characterizing the K-functional K(f, t;C[−1, 1], C2,H) which describes the rate of approximation by operators Gs,n.

For a function f on [−1, 1] we consider

Δ2h

(f ◦ θ, θ−1(x)

):= (f ◦ θ)(y + h) + (f ◦ θ)(y − h) − 2(f ◦ θ)(y)

= f(cos(y + h)

)+ f

(cos(y − h)

)− 2f(cos y)

= f(cos(arccos x + h)

)+ f

(cos(arccos x − h)

)− 2f(x) ,

where θ(x) = cos x, y = θ−1(x).


Definition 1. For given f ∈ C[−1, 1] and t > 0 we set

Ω2(f, t) := sup0<h≤t

sup−1≤x≤1

∣∣Δ2

h

(f ◦ θ, θ−1(x)

)∣∣ .

Obviously, Ω2(f, t) inherits the properties of the classical modulus ω2(f, t) as(f, g ∈ C[−1, 1]):

• Ω2(f + g, t) ≤ Ω2(f, t) + Ω2(g, t);• Ω2(λf, t) = |λ|Ω2(f, t), λ ∈ R;• Ω2(f, t) ≤ 4 ‖f‖ .

We point out that at the end of the interval [−1, 1] the second differenceΔ2

h(f ◦ θ, θ−1(1)) = 2(f(cos h) − f(1)) and hence for F1(x) = arccos x, Δ2h(F1 ◦

θ, θ−1(1)) = 2h.We show that the new modulus defined above can be used for a characteri-

zation of the K-functional (2). We prove

Theorem 1. For every f ∈ C[−1, 1] and natural number n we have

K

(f,

1n2

;C[−1, 1], C2,H

)∼ Ω2

(f,

1n

).

From Theorem 1 and Theorem A a) we get.

Corollary 1. For every f ∈ C[−1, 1] and natural numbers n, s, s ≥ 3 fixed, we have

‖f − Gs,nf‖ ∼ Ω2

(f,

1n

).

We define the modulus Ω∗2(f, t) for characterizing the K-functional

K∗(f, t;C[−1, 1], C2,H(I − L))

which describes the rate of approximation by operators G∗s,n.

We consider

Δ2h

((I − L)f ◦ θ, θ−1(x)

):= Δ2

h

(f ◦ θ, θ−1(x)

)− Δ2

h

((Lf) ◦ θ, θ−1(x)

)

= Δ2h

(f ◦ θ, θ−1(x)

)

− Δ2h

(f(1)

1 + θ

2+ f(−1)

1 − θ

2, θ−1(x)

)

= Δ2h

(f ◦ θ, θ−1(x)

)− f(1)

2Δ2

h

(θ, θ−1(x)

)

+f(−1)

2Δ2

h

(θ, θ−1(x)

)


)

+ f(cos(arccos x − h)

)− 2f(x)

− f(1) − f(−1)2

(cos(arccos x + h)

+ cos(arccos x − h) − 2x)



)+ f


)

− 2f(x) + [f(1) − f(−1)] (1 − cos h)x ,

where θ(x) = cos x.

Definition 2. For given f ∈ C[−1, 1]and t > 0 we set

Ω∗2(f, t) := sup

0<h≤tsup

−1≤x≤1

∣∣Δ2

h

((I − L)f ◦ θ, θ−1(x)

)∣∣ .

We prove

Theorem 2. For every f ∈ C[−1, 1] and natural numbers n ≥ n0, we have

K∗(

f,1n2

;C[−1, 1], C2,H(I − L))

∼ Ω∗2

(f,

1n

).

From Theorem 2 and Theorem A b) we get

Corollary 2. For every f ∈ C[−1, 1] and natural numbers n, s, n ≥ n0, s ≥ 3 fixed,we have

‖f − G∗s,nf‖ ∼ Ω∗

2

(f,

1n

).

Now we introduce the modulus Ω+2 for the characterization of the functional

K+. We consider

(I − L)Δ2h

(f ◦ θ, θ−1(x)

):= Δ2

h

(f ◦ θ, θ−1(x)

)− LΔ2

h

(f ◦ θ, θ−1(x)

)

= Δ2h

(f ◦ θ, θ−1(x)

)

−[Δ2

h

(f ◦ θ, θ−1(1)

)1 + x

2

+Δ2h

(f ◦ θ, θ−1(−1)

)1 − x

2

]


)+ f


)− 2f(x)

−[(

f(cos(arccos 1 + h)

)

+ f(cos(arccos 1 − h)

)− 2f(1)

)1 + x

2

+(

f(

cos(arccos(−1) + h

))

+f(

cos(arccos(−1) − h

))− 2f(−1)

)1 − x

2

]


)+ f


)− 2f(x)

−[(f(cos h) + f

(cos(−h)

)− 2f(1)

)1 + x

2

+(f(cos(π + h)

)+ f

(cos(π − h)

)− 2f(−1)

)1 − x

2

]



)+ f


)− 2f(x)

+[f(1) − f(cos h)

](1 + x)

+[f(−1) − f(− cos h)

](1 − x) ,

where θ(x) = cos x.

Definition 3. For given f ∈ C[−1, 1] and t > 0 we set

Ω+2 (f, t) := sup

0<h≤tsup

−1≤x≤1

∣∣(I − L)Δ2

h

(f ◦ θ, θ−1(x)

)∣∣ .

We prove

Theorem 3. For every f ∈ C[−1, 1] and natural number n we have

K+

(f,

1n2

;C[−1, 1], C2, (I − L)H)

∼ Ω+2

(f,

1n

).

From Theorem 3 and Theorem A c) we get

Corollary 3. For every f ∈ C[−1, 1] and natural numbers n, s, s ≥ 3 fixed, we have

‖f − G+s,nf‖ ∼ Ω+

2

(f,

1n

).

4. Proofs of the theorems

Definition 4. Set

Y ={g ∈ C[−1, 1] : H1g ∈ C[−1, 1],Hg ∈ C[−1, 1],H1g(±1) = 0

}.

Lemma 1. Let Y be the space from Definition 4, g ∈ Y and g(σ) := g(cos σ).Theng ∈ C2(R) and g′′(σ) = Hg(cos σ) for σ ∈ R.

Proof. To prove the lemma we observe that

g(σ) = g(cos σ) = g(cos arccos cos σ) = g(arccos cos σ) .

We have

arccos cos σ ={

σ − 2kπ , if σ ∈(2kπ, (2k + 1)π

), k = 0 ,±1 ,±2 . . .

−σ + 2kπ , if σ ∈((2k − 1)π, 2kπ

), k = 0 ,±1 ,±2 . . .

.

Hence

(arccos cos σ)′=

{1 , if σ ∈

(2kπ, (2k + 1)π

), k = 0 ,±1 ,±2 . . .

−1 , if σ ∈((2k − 1)π, 2kπ

), k = 0 ,±1 ,±2 . . .

,

dg(σ)dσ

= g′(arccos cos σ)(arccos cos σ)′ .

From g ∈ Y and H1g(cos σ) = −dg(σ)dσ we get limσ→kπ±0

dg(σ)dσ = 0 for k =

0,±1,±2 . . .. Hence the left and the right derivative of the function g(σ) at pointsσ = kπ are equal to zero, dg(σ)

dσ = 0 at σ = kπ , k = 0,±1,±2 . . . and thereforeg(σ) ∈ C1(−∞,∞).


For σ ∈ (2kπ, (2k + 1)π), k = 0,±1,±2 . . . and g ∈ Y we have

dg(σ)dσ

= g′(arccos cos σ)(arccos cos σ)′ = g′(σ − 2kπ) = g′(σ) .

d2g(σ)dσ2

= g′′(σ) = g′′(σ − 2kπ) .

From Hg ∈ C[−1, 1] (g ∈ Y ) and d2g(σ)dσ2 = Hg(cos σ) we get

limσ→2kπ+0

d2g(σ)dσ2

= limσ→0+0

g′′(σ) = limx→1−0

Hg(x) = Hg(1) .

For σ ∈ ((2k − 1)π, 2kπ), k = 0,±1,±2 . . . and g ∈ Y we have

dg(σ)dσ

= g′(arccos cos σ)(arccos cos σ)′ = g′(−σ + 2kπ)(−1) = −g′(−σ) .

d2g(σ)dσ2

= g′′(−σ) = g′′(−σ + 2kπ) .

From Hg ∈ C[−1, 1] (g ∈ Y ) and d2g(σ)dσ2 = Hg(x) we get

limσ→2kπ−0

d2g(σ)dσ2

= limσ→0−0

g′′(−σ) = limσ→0+0

g′′(σ) = limx→1−0

Hg(x) = Hg(1) .

Therefore the left and the right second derivative of the function g(σ) at pointsσ = 2kπ are equal. Similarly, using 2π periodisity of g′′(σ) we get that the left andthe right second derivative of the function g(σ) at points σ = (2k + 1)π are equal.This proves the lemma. �

Lemma 2. Let Y be the space from Definition 4. Then for every f ∈ C[−1, 1] andt > 0, we have

K(f, t;C[−1, 1], Y,H

)= K

(f, t;C[−1, 1], C2,H

),

K+(f, t;C[−1, 1], Y, (I − L)H

)= K+

(f, t;C[−1, 1], C2, (I − L)H

).

Proof. From C2 ⊂ Y we see that K(f, t;C[−1, 1], Y,H) ≤ K(f, t;C[−1, 1], C2,H).In order to prove K(f, t;C[−1, 1], C2,H) ≤ K(f, t;C[−1, 1], Y,H) it is sufficientto show (see Lemma 2 in [4, p. 116]) that for every g ∈ Y and ε > 0 there existsG ∈ C2 such that ‖G − g‖ ≤ ε and ‖HG‖ ≤ ‖Hg‖ + ε. Let g(x) ∈ Y. We putx = cos σ and consider g(σ) := g(cos σ). Since g(x) ∈ Y, g(σ) ∈ C2 (see Lemma 1).We use the Jackson integrals of the following type

Jn(g, σ) :=

π∫

−π

g(σ + v)K1s,n(v)dv =

π∫

−π

g(v)K1s,n(σ − v)dv, (5)

where K1s,n(v) := λs,n( sin mv/2sin v/2 )2s,

π∫−π

K1s,n(v)dv = 1, n, s > 0,m := [n/s] + 1.


Since 12m ( sin mv/2

sin v/2 )2 = 12+

m−1∑k=0 (1 − k/m) cos kv it follows that K1s,n is an

even non-negative trigonometric polynomial of degree ≤ n. Jn(g, σ) is a trigono-metric polynomial of degree ≤ n, which is even as g is even. From Jackson’stheorem (see [3, Chap. 7, Theorem 2.2]) we get

‖g − Jn(g)‖ ≤ cω2

(g,

1n

)≤ O

(1n2

). (6)

By the substitution σ = arccos x in Jn(∼g, σ) we obtain an algebraic polynomial,

which is the desired function from C2. We set

G(x) = Jn(g, arccos x) . (7)

From ‖g − G‖C[−1,1] = ‖g − Jn(g)‖C[0,π] and (6) we get ‖g − G‖ ≤ O( 1n2 ). From

(5) we get

d2

dσ2Jn(g, σ) =

π∫

−π

g′′(σ + v)K1s,n(v)dv =

π∫

−π

g′′(v)K1s,n(σ − v)dv = Jn(g′′, σ) .

Using the Jackson theorem we get

‖g′′ − Jn(g′′)‖C[0,π] ≤ cω2

(g′′,

1n

). (8)

Since (Hg)(x) = d2

dσ2 g(σ) and (HG)(x) = d2

dσ2 Jn(g, σ) (8) implies

‖Hg − HG‖C[−1,1] ≤ cω2

(g′′,

1n

). (9)

For given ε > 0 we choose n such that cω2(g′′, 1n ) < ε and obtain ‖HG‖ ≤ ‖Hg‖+ε.

To prove the second statement in Lemma 2 it is sufficient to show thatfor every g ∈ Y and ε > 0 there exists G ∈ C2 such that ‖G − g‖ ≤ ε and‖(I − L)HG‖ ≤ ‖(I − L)Hg‖ + ε. Let G(x) be defined by (7). From (9) we get‖(I − L)H(g − G)‖ ≤ 2 ‖Hg − HG‖ ≤ cω2(g′′, 1

n ). For given ε > 0 we choose n

such that cω2(g′′, 1n ) < ε and obtain‖(I − L)HG‖ ≤ ‖(I − L)Hg‖ + ε. This com-

pletes the proof of the lemma. �

Proof of Theorem 1. The theorem will be proved if we show that

K

(f,

1n2

, C[−1, 1], Y,H

)∼ Ω2

(f,

1n

)

(see Lemma 2). First we show that Ω2(f, 1n ) ≤ cK(f, 1

n2 , C[−1, 1], Y,H). For g ∈ Y

we have Ω2(f, 1n ) ≤ Ω2(f − g, 1

n ) + Ω2(g, 1n ). But

Ω2

(f − g,

1n

)≤ 4 ‖f − g‖ ,


and

Ω2

(g,

1n

)= sup

0<h≤ 1n

−1≤x≤1

∣∣g

(cos(arccos x + h)

)+ g


)− 2g(x)

∣∣

= sup0<h≤ 1

n

0≤z≤π

∣∣g

(cos(z + h)

)+ g

(cos(z − h)

)− 2g(cos z)

∣∣ (z = arccos x)

= sup0<h≤ 1

n

0≤z≤π

|g(z + h) + g(z − h) − 2g(z)|(g(z) := g(cos z)

)

= sup0<h≤ 1

n

0≤z≤π

∣∣∣∣∣∣∣

h2∫

−h2

h2∫

−h2

g′′(z + u1 + u2)du1du2

∣∣∣∣∣∣∣

≤ 1n2

‖g′′‖C[0,π] =1n2

‖Hg‖C[−1,1] .

We used that for g ∈ Y we have (see Lemma 1) g ∈ C2(−∞,∞). Therefore

Ω2

(f,

1n

)≤ 4 ‖f − g‖ +

1n2

‖Hg‖ ≤ cK

(f,

1n2

, C[−1, 1], Y,H

).

In order to prove K(f, 1n2 , C[−1, 1], Y,H) ≤ cΩ2(f, 1

n ) it is enough to showthat for every function f ∈ C[−1, 1] there exists function g ∈ Y such that:

a) ‖f − g‖ ≤ cΩ2(f, 1n );

b) 1n2 ‖Hg‖ ≤ cΩ2(f, 1

n ).

We define the following functions

g1,t(x) :=1t

t2∫

− t2

f(cos(arccos x + u)

)du =

1t

t2∫

− t2

f(z + u)du ≡ g1,t(z) .

g2,t(x) :=1t2

t2∫

− t2

t2∫

− t2

f(cos(arccos x + u1 + u2)

)du1du2

=1t2

t2∫

− t2

t2∫

− t2

f(z + u1 + u2)du1du2

=1t

t2∫

− t2

g1,t(z + u)du ≡ g2,t(z) ,


where we have put z = arccos x, f(z) := f(cos z). We have

g2,kt(x) =1

k2t2

kt2∫

− kt2

kt2∫

− kt2

f(z + u1 + u2)du1du2

(ui = kvi,−

t

2≤ vi ≤

t

2

)

=1t2

t2∫

− t2

t2∫

− t2

f(z + k(v1 + v2)

)dv1dv2 .

We define

G2,t(x) :=12g2, t

2(x) +

12g2,− t

2(x) . (10)

We show later that G2,t ∈ Y. This function will serve for g in the above inequalitiesa) and b). We have

(G2,t−f)(x) =1

2t2

t2∫

−t2

t2∫

−t2

(f

(z+

12(u1+u2)

)+ f

(z − 1

2(u1+u2)

))du1du2−f(x)

=1

2t2

t2∫

− t2

t2∫

− t2

(f

(z +

12(u1 + u2)

)+ f

(z − 1

2(u1 + u2)

)

− 2f(z))

du1du2 .

Then

|(f − G2,t)(x)| ≤ 12t2

t2 sup0<u≤t

sup−1≤x≤1

∣∣∣f

(cos

(arccos x +

u

2

))

+f(cos

(arccos x − u

2

))− 2f(x)

∣∣∣ ,

and hence

‖f − G2,t‖ ≤ 12Ω2(f, t) .

We take t = 1n and obtain a).

Now we will prove b) with g = G2,t.We have

d

dzg2,t(z) =

1t

(g1,t

(z +

t

2

)− g1,t

(z − t

2

)), (11)

d2

dz2g2,t(z) =

1t

[1t

(f(z + t) − f(z)

)− 1

t

(f(z) − f(z − t)

)]

=1t2

(f(z + t) + f(z − t) − 2f(z)

),


and henced2

dz2g2,kt(z) =

1k2t2

(f(z + kt) + f(z − kt) − 2f(z)

). (12)

If we set G2,t(z) = 12 g2, t

2(z) + 1

2 g2,− t2(z) and use that d2

dz2 G2,t(z) = HG2,t(x) weget

HG2,t(x) =12

d2

dz2g2, t

2(z) +

12

d2

dz2g2,− t

2(z)

=12

1t2

4

(f

(cos

(arccos x +

t

2

))+ f

(cos

(arccos x − t

2

))− 2f(x)

)

+12

1t2

4

(f

(cos

(arccos x− t

2

))+f

(cos

(arccos x+

t

2

))−2f(x)

).

Hence t2 ‖HG2,t‖ ≤ 4Ω2(f, t). This completes the proof of b) .Finally, we observe that G2,t(x) ∈ Y. (10) implies that G2,t(x) ∈ C[−1, 1]. (11)

implies that ddz G2,t(z) ∈ C[0, π] and d

dz G2,t(z) |z=0,π= 0. From ddz G2,t(z) =

−H1G2,t(x) we get H1G2,t(x) ∈ C[−1, 1] and H1G2,t(±1) = 0. (12) implies thatd2

dz2 G2,t(z) ∈ C[0, π]. From d2

dz2 G2,t(z) = HG2,t(x) we get HG2,t(x) ∈ C[−1, 1].This completes the proof of the equivalence

K

(f,

1n2

, C[−1, 1], Y,H

)∼ Ω2

(f,

1n

).

Applying the first statement of Lemma 2 we prove the theorem. �

Proof of Theorem 2. According to Theorem A b) we have K∗(f, 1n2 ;C[−1, 1], C2,

H(I − L)) ∼∥∥G∗

s,nf − f∥∥ . On the other hand G∗

s,nf − f = (Gs,n − I)(I − L)f .Corollary 1 implies

∥∥G∗

s,nf − f∥∥ = ‖(Gs,n − I)(I − L)f‖ ∼ Ω2

((I − L)f,

1n

)= Ω∗

2

(f,

1n

).

This proves the theorem. �

Lemma 3. Let Y be the space from Definition 4. Then for f ∈ C[−1, 1] and naturalnumber n we have

K+

(f,

1n2

;C[−1, 1], Y, (I − L)H)

= infg∈Y

{‖ f − g‖ +

1n2

‖(I − L)Hg‖}

∼ infg∈Y

{‖(I−L)( f−g)‖ +

1n2

‖(I−L)Hg‖}

.

Proof. Obviously for every function g ∈ Y we have

‖(I − L)(f − g)‖ +1n2

‖(I − L)Hg‖ ≤ 2 ‖f − g‖ +1n2

‖(I − L)Hg‖ .


Therefore

infg∈Y

{‖(I − L)(f − g)‖ +

1n2

‖(I − L)Hg‖}

≤ 2K+

(f,

1n2

;C[−1, 1], Y, (I − L)H)

.

We will prove the inequality in the opposite direction. Let

infg∈Y

{‖(I − L)(f − g)‖ +

1n2

‖(I − L)Hg‖}

≥ ‖(I − L)(f − g0)‖ +1n2

‖(I − L)Hg0‖ − ε ,

i.e., the infimum is almost achieved for g = g0 (ε is a small positive arbitrarynumber). Set g1 = g0 + L(f − g0). We have g1 ∈ Y and

‖f − g1‖ +1n2

‖(I − L)Hg1‖

= ‖f − g0 − L(f − g0)‖ +1n2

∥∥(I − L)H

(g0 + L(f − g0)

)∥∥

= ‖(I − L)(f − g0)‖ +1n2

‖(I − L)Hg0‖ , as (I − L)HL(f − g0) ≡ 0 .

Hence

K+

(f,

1n2

;C[−1, 1], Y, (I − L)H)

≤ ‖f − g1‖ +1n2

‖(I − L)Hg1‖

≤ ‖(I − L)(f − g0)‖ +1n2

‖(I − L)Hg0‖

≤ inf{‖(I−L)(f−g)‖+

1n2

‖(I−L)Hg‖}

+ε .

The last inequality shows that

K+

(f,

1n2

;C[−1, 1], Y, (I−L)H)

≤ infg∈Y

{‖(I − L)(f−g)‖+

1n2

‖(I−L)Hg‖}

.

This proves the lemma. �

Proof of Theorem 3. The theorem will be proved if we show that

K+

(f,

1n2

, C[−1, 1], Y, (I − L)H)

∼ Ω+2

(f,

1n

)


(see Lemma 2). First we show that Ω+2 (f, 1

n ) ≤ cK+(f, 1n2 ;C[−1, 1], Y, (I −L)H).

For g ∈ Y we have Ω+2 (f, 1

n ) = Ω+2 (f − g + g, 1

n ) ≤ Ω+2 (f − g, 1

n ) + Ω+2 (g, 1

n ). But

Ω+2

(f − g,

1n

)≤ 8 ‖f − g‖ , as ‖I − L‖ = 2 and

Ω+2

(g,

1n

)= sup

0<h≤ 1n

−1≤x≤1

∣∣∣(I − L)

[g(cos(arccos x + h)

)

+ g(cos(arccos x − h)

)− 2g(x)

]∣∣∣

= sup0<h≤ 1

n

0≤z≤π

∣∣∣(I − L)

[g(cos(z + h)

)+ g

(cos(z − h)

)− 2g(cos z)

]∣∣∣

= sup0<h≤ 1

n

0≤z≤π

∣∣(I − L)

[g(z + h) + g(z − h) − 2g(z)

]∣∣

= sup0<h≤ 1

n

0≤z≤π

∣∣∣∣∣∣∣(I − L)

h2∫

−h2

h2∫

−h2

g′′(z + u1 + u2)du1du2

∣∣∣∣∣∣∣

= sup0<h≤ 1

n

−1≤x≤1

∣∣∣∣∣∣∣(I − L)

h2∫

−h2

h2∫

−h2

Hg(cos(arccos x + u1 + u2)

)du1du2

∣∣∣∣∣∣∣,

where we have put z = arccos x , g(z) := g(cos z) and

L(g, z) := g(0)1 + cos z

2+ g(π)

1 − cos z

2, 0 ≤ z ≤ π .

We used that for g ∈ Y we have (see Lemma 1) g ∈ C2(−∞,∞).Set cos(arccos x + u1 + u2) = y, u = u1 + u2. Next we have

Hg(cos(arccos x + u)

)= g′′

(cos(arccos x + u)

)sin2(arccos x + u)

− g′(cos(arccos x + u)

)cos(arccos x + u)

and

Hg(y) = (1 − y2)g′′(y) − yg′(y) .

Thus

Hg(y) = Hg(cos(arccos x + u1 + u2)

).

Further we have

LHg(y) =(Hg)(1) − (Hg)(−1)

2y +

(Hg)(1) + (Hg)(−1)2

.


Thenh2∫

−h2

h2∫

−h2

LHg(y)du1du2

=

h2∫

−h2

h2∫

−h2

((Hg)(1) − H(g)(−1)

2y +

(Hg)(1) + H(g)(−1)2

)du1du2

=

h2∫

−h2

h2∫

−h2

(c1 cos(arccos x + u1 + u2) + c2

)du1du2 .

We calculateh2∫

−h2

h2∫

−h2

cos(arccos x + u1 + u2)du1du2

=

h2∫

−h2

h2∫

−h2

[x cos(u1 + u2) −

√1 − x2 sin(u1 + u2)

]du1du2

= x

h2∫

−h2

h2∫

−h2

cos(u1 + u2)du1du2 −√

1 − x2

h2∫

−h2

h2∫

−h2

sin(u1 + u2)du1du2 = c(h)x ,

because the second term is zero. Hence

(I − L)

h2∫

−h2

h2∫

−h2

LHg(y)du1du2 = (I − L)(c1(h)x + c2(h)

)≡ 0 .

Further, using that Hg(cos(arccos x + u)) = Hg(y) and the above relation we get

(I − L)

h2∫

−h2

h2∫

−h2


)du1du2

= (I − L)

h2∫

−h2

h2∫

−h2

(I − L)Hg(cos(arccos x + u1 + u2)

)du1du2 .


Hence

Ω+2

(g,

1n

)= sup

0<h≤ 1n

−1≤x≤1

∣∣∣∣∣∣∣(I − L)

h2∫

−h2

h2∫

−h2


)du1du2

∣∣∣∣∣∣∣

= sup0<h≤ 1

n

−1≤x≤1

∣∣∣∣∣∣∣(I − L)

h2∫

−h2

h2∫

−h2

(I − L)Hg(cos(arccos x + u1 + u2)

)du1du2

∣∣∣∣∣∣∣

≤ 2 sup0<h≤ 1

n

−1≤x≤1

∣∣∣∣∣∣∣

h2∫

−h2

h2∫

−h2

(I − L)Hg(y)du1du2

∣∣∣∣∣∣∣

≤ 21n2

sup−1≤y≤1

|(I − L)Hg(y)| =2n2

‖(I − L)Hg‖ .

Therefore

Ω+2

(f,

1n

)≤ 8 ‖f − g‖+

2n2

‖(I − L)Hg‖ ≤ 8K+

(f,

1n2

;C[−1, 1], Y, (I − L)H)

.

Using Lemma 3 to prove K+(f, 1n2 ;C[−1, 1], Y, (I − L)H) ≤ cΩ+

2 (f, 1n ), it is

enough to show that for every f ∈ C[−1, 1] there exists g ∈ Y such that:

a) ‖(I − L)(f − g)‖ ≤ cΩ+2 (f, 1

n );b) 1

n2 ‖(I − L)Hg‖ ≤ cΩ+2 (f, 1

n ).

We define G2,t by (10). This function will serve for g in the above inequali-ties a) and b). In the proof of Theorem 1 we observed that G2,t ∈ Y . We have

(I − L)(G2,t − f)(x)

=1

2t2(I − L)

t2∫

− t2

t2∫

− t2

(I − L)(

f

(cos

(arccos x +

12(u1 + u2)

))

+ f

(cos

(arccos x − 1

2(u1 + u2)

))− 2f(x)

)du1du2

+1

2t2(I − L)

t2∫

− t2

t2∫

− t2

L

(f

(cos

(arccos x +

12(u1 + u2)

))

+ f

(cos

(arccos x − 1

2(u1 + u2)

))− 2f(x)

)du1du2 .


As the last term is zero we get

|(I − L)(G2,t − f)(x)| ≤ 22t2

t2 sup−t≤u≤t

−1≤x≤1

∣∣∣(I − L)

[f

(cos

(arccos x +

u

2

))

+f(cos

(arccos x − u

2

))− 2f(x)

]∣∣∣ .

Hence‖(I − L)(f − G2,t)‖ ≤ Ω+(f, t) .

We will show that t2 ‖(I − L)Hg‖ ≤ cΩ+(f, t) for g = G2,t. As in the proofof Theorem 1 we get

(I − L)HG2,t(x)

= (I − L)

[12

1t2

4

(f

(cos

(arccos x +

t

2

))+ f

(cos

(arccos x − t

2

))− 2f(x)

)]

+ (I−L)

[12

1t2

4

(f

(cos

(arccos x− t

2

))+f

(cos

(arccos x+

t

2

))−2f(x)

)]

.

Hence t2 ‖(I − L)HG2,t‖ ≤ 4Ω+2 (f, t). This completes the proof of b) .

Thus

K+

(f,

1n2

;C[−1, 1], Y, (I − L)H)

∼ Ω+2

(f,

1n

).

Applying the second statement of Lemma 2 we prove the theorem. �

5. Examples

In this section we give the order of magnitude of Ditzian–Totik modulus of smooth-ness ω2√

1−x2(f, t)∞ and moduli Ω2(f, t),Ω∗2(f, t),Ω+

2 (f, t) for several functions. Werecall

ω2√1−x2(f, t)∞ := sup

−1≤x≤1,0<h≤t

∣∣∣Δ

2

h√

1−x2f(x)∣∣∣ ,

where the second difference is given by

Δ2

t f(x) ={

f(x + t) + f(x − t) − 2f(x) , if x, x ± t ∈ [−1, 1] ,0 , otherwise .

Example 5.1. Let f(x) = (1 − x)α |log(1 − x)|β , x ∈ (−1, 1), α ∈ (0, 2], β ∈ R orα = 0, β < 0. Then

ω2√1−x2(f, t)∞∼

⎧⎪⎪⎨

⎪⎪⎩

t2α |log t|β for α ∈ (0, 1) , β ∈ R or α = 0 , β < 0 ;t2 |log t|β−1 for α = 1 , β > 1 ;t2 for α∈(1, 2] , β∈R or α=1 , β∈(−∞, 0) ∪ (0, 1] ;0 for α = 1 , β = 0 .


Ω2(f, t) ∼ Ω∗2(f, t) ∼ Ω+

2 (f, t) ∼ t2α |log t|β for α ∈ (0, 1), β ∈ R or α = 1, β > 0 orα = 0, β < 0.

Ω2(f, t) ∼ t2 , Ω∗2(f, t) = Ω+

2 (f, t) = 0 for α = 1 , β = 0 .

Ω2(f, t) ∼ Ω∗2(f, t) ∼ Ω+

2 (f, t) ∼ t2

for α ∈ (1, 2] , β ∈ R or α = 1 , β < 0 .

Comparing the moduli Ω2(f, t),Ω∗2(f, t) and Ω+

2 (f, t) with the modulusω2√

1−x2(f, t)∞ for the function f(x) = (1 − x)α |log(1 − x)|β we see the differ-ence of the order of magnitude for α = 1, β ≥ 0. Therefore we cannot use themodulus ω2√

1−x2(f, t)∞ for a characterization of K-functionals (2), (3) and (4).

References

[1] Z. Ditzian and K.Ivanov, Strong converse inequalities, J. Anal. Math. 61 (1993),61–111.

[2] Z. Ditzian and V. Totik, Moduli of Smoothness. Berlin: Springer Verlag 1987.

[3] R. A. DeVore, G. G. Lorentz, Constructive Approximation. Berlin: Springer Verlag1993.

[4] K. Ivanov, A Characterization Theorem for the K-functional for Kantorovich andDurrmeyer Operators, Approximation Theory: A volume dedicated to Borislav Bo-janov, Marin Drinov Academic Publishing House, Sofia, (2004), 112–125.

[5] T. Zapryanova, Approximation by Operators of Cao–Gonska type Gs,n and G∗s,n.

Direct and Converse Theorem, Thirty Third Spring Conference of the Union ofBulgarian Mathematicians, Borovets (2004), pp. 189–194.

[6] T. Zapryanova, Approximation by operators of Cao–Gonska type G+s,n. Direct and

converse theorem. Thirty Seventh Spring Conference of the Union of Bulgarian Math-ematicians, Borovets (2008), pp. 169–175.

Teodora Dimova ZapryanovaVarna University of EconomicsBlvd. Kniaz Boris I, 77BG-9002 VarnaBulgariae-mail: [email protected]

Received: June 23, 2008.

Revised: September 28, 2008.

A characterization of the k functional for the algebraic version of the trigonometric jackson integr

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notation f

independent of f

f g thg

f xand t

f r lpa

f r ca

operators g s

f g thi lg