Approximation by Jackson-type operator on the ... - Semantic Scholar

MATHEMATICAL COMMUNICATIONS Math. Commun., Vol. 15, No. 2, pp. 331-346 (2010)

331

Approximation by Jackson-type operator on the sphere∗ Feilong Cao1,† and Xiaofei Guo1 1

Department of Information and Mathematics Sciences, China Jiliang University, Hangzhou-310 018, P. R. China

Received November 7, 2009; accepted March 20, 2010

Abstract. This paper discusses the approximation by a Jackson-type operator on the sphere. By using a spherical translation operator, a modulus of smoothness of high order, which is used to bound the rate of approximation of the Jackson-type operator, is introduced. Furthermore, the method of multipliers is applied to characterize the saturation order and saturation class of the operator. In particular, the function of saturation class is expressed by an apparent formula. The results obtained in this paper contain the corresponding ones of the Jackson operator. AMS subject classifications: 41A17, 41A05, 41A63 Key words: approximation, Jackson-type operator, saturation order, saturation class, sphere

1. Introduction A volume of work has been carried out for studying Jackson and Bernstein type theorems on the sphere during the past decades by Lizorkin and Nikol’skiˇı [13], Ditzian [8], Dai [7], Dai and Ditzian [6], Li and Yang [12], Rustamov [16] and many other mathematicians. In 1983, to prove direct and inverse theorems on a more special Banach space Hpr (Sn−1 ), Lizorkin and Nikol’skiˇı [13] constructed an iterated Boolean sum of Jackson operator. They introduced a Jackson-type operator (Qdk,s f ) (which will be defined in the next section), which is a linear combination of an iterated Jackson operator and can be reduced to the known Jackson operator when d = 1. Here we are interested in the approximation properties of (Qdk,s f ). After introducing a modulus of smoothness of high order by using the translation operator on the sphere, we will give an estimation of the rate of approximation for the operator (Qdk,s f ). We also use the method of multipliers to obtain saturation theorems of (Qdk,s f ), and the expression of the function of saturation class. The obtained results show that the operator (Qdk,s f ) has a higher degree of approximation and order of saturation. When d = 1, our results coincide with those of the known Jackson operator. ∗ The

research was supported by the National Natural Science Foundation of China, No. 60873206. author. Email addresses: [email protected] (F. Cao), [email protected] (X. Guo)

† Corresponding

http://www.mathos.hr/mc

c °2010 Department of Mathematics, University of Osijek

332

F. Cao and X. Guo

2. Definitions and auxiliary notations A function on the sphere Sn−1 is in Lp (Sn−1 ), 1 ≤ p ≤ ∞, if ½Z ¾1/p kf kp := kf kLp (Sn−1 ) := |f (x)|p dσ < ∞,

1≤p β + n − 1, 0 < θ ≤ π, and n ≥ 3, we have Z π θβ Dk,s (θ) sin2λ θdθ ≤ Cβ,s k −β

(11)

0

where λ =

n−2 2 ,

and s, n, k are positive integers.

The proof of Lemma 1 is simple (see also [13]). The following Lemma 2 gives the description of Qdk,s f (µ) by multiplies. Lemma 2. For f ∈ Lp (Sn−1 ), 1 ≤ p ≤ ∞, there holds (Qdk,s f )(µ) =

∞ X

d (j)Yj (f ; µ) ξk,s

(12)

j=0

where Z d ξk,s (j)

Ã

π

=1−

Dk,s (θ) 1 −

Gλj (cos θ)

!d

Gλj (1)

0

sinn−2 θdθ, j = 0, 1, 2 . . .

and the convergence of the series is meant in a weak sense. P∞ Proof. We know f (µ) = j=0 Yj (f ; µ). Then from Definition 1, (Qdk,s f )(µ) =

∞ X

Z Yj (f ; µ) −

π

Dk,s (θ)(I − Sθ )d f (µ) sinn−2 θdθ.

0

j=0

From [13], it follows that Sθ (f ) =

∞ X Gλj (cos θ) j=0

Gλj (1)

Yj (f ; µ).

(13)

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Approximation by Jackson-type operator on the sphere

So (Qdk,s f )(µ) =

∞ X j=0

=

∞ X



Z

Ã

π

1 −

Dk,s (θ) 1 − 0

Gλj (cos θ) Gλj (1)

!d

 sinn−2 θdθ Yj (f ; µ)

d ξk,s (j)Yj (f ; µ).

j=0

This finishes the proof of Lemma 2. The next lemma is useful for determining the saturation order. It can be deduced by the methods in [1] and [4]. Lemma 3. Suppose that {Iρ }ρ>0 is a sequence of operators on Lp (Sn−1 ), and there ∞ exists series {λρ (j)}j=1 with respect to ρ, such that Iρ (f )(x) =

∞ X

λρ (j)Yj (f )(x)

j=0

for every f ∈ Lp (Sn−1 ). If there exists ϕ(ρ) → 0 + (ρ → ∞) such that for any j = 0, 1, 2, . . . , 1 − λρ (j) lim = τj 6= 0, ρ→ρ0 ϕ(ρ) then {Iρ }ρ>0 is saturated on Lp (Sn−1 ) with the order O(ϕ(ρ)) and the collection of all constants is the invariant class for {Iρ }ρ>0 on Lp (Sn−1 ). In [9] Dunkl introduced the spherical convolution with respect to the measure and zonal function. Normally Mλ is denoted by the set of Borel measures µ, and it satisfies Z Ωn−2 π kµkMλ := |dµ(θ)| < ∞. Ωn−1 0 Now, let M (Sn−1 ) be the space of all the limited regular Borel measures on Sn−1 endowed with the norm Z 1 kµkM := |dµ(x)|. Ωn−1 Sn−1 Then it is a Banach space and every µ ∈ M (Sn−1 ) can be expanded as the following spherical series: ∞ X S(dµ; x) ∼ Yj (dµ; x), j=0

where

Γ(λ)(j + λ) Yj (dµ; x) = 2π λ+1

Z Sn−1

Gλj (x · y)dµ(y), j = 0, 1 . . . .

From the above we get Lemma 4 (see [9]).

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Lemma 4. Let f ∈ Lpλ , 1 ≤ p ≤ ∞ or Cλ , µ ∈ M (Sn−1 ). Then Z 1 (f ∗ dµ)(x) = f (x · y)dµ(y) Ωn−1 Sn−1

(14)

is in Lp (Sn−1 ) or C(Sn−1 ), kf ∗ dµkp ≤ kf kp,λ kµkM

(15)

and λ ˆ f (j)Yj (dµ; x). j+λ

Yj (f ∗ dµ; x) =

(16)

Moreover, if µ is a zonal function, so is f ∗ µ. To get the saturation class, we need to discuss spherical convolution and a singular integral together (see [1]). Let f ∈ X, Bρ ∈ L1λ , and also Z π c(0, ρ) Bρ (cos θ) sin2λ θdθ = 1, (17) 0 Z π c(0, ρ) |Bρ (cos θ)| sin2λ θdθ ≤ M, (18) 0

(for ρ > 0, it holds uniformly and also M ≥ 1) lim sup

ρ→∞ δ≤θ≤π

|Bρ (cos θ)| = 0.

(19)

Then Hρ (f ; x) =

1

Z

Ωn−1

Sn−1

f (y)Bρ (x · y)dy

(20)

is a singular integral with kernel Bρ (ρ > 0), where c(0, λ) = Ωn−2 /Ωn−1 . It is easy to know that the operator (Qdk,s f )(µ) is a singular integral with classical Jackson kernel Dk,s . Now we give the fifth lemma (see [1]). Lemma 5. Let f ∈ X. Suppose that the kernel Bρ of singular integral (20) satisfies (17),(18) and lim

ρ→∞

λ ˆ j+λ Bρ (j)

ϕ( ρ)

−1

= ψ(j),

(21)

where the meaning of ϕ(ρ) is the same as that of Lemma 3. Then the following statements hold: a) If there exists g ∈ X, such that ° ° ° 1 ° ° lim ° {Hρ f − f } − g ° (22) ° = 0, ρ→∞ ϕ(ρ) X

339

Approximation by Jackson-type operator on the sphere

then ψj Yj (f ; x) = Yj (g; x),

j = 0, 1, 2, . . . .

Especially, if g = 0, then f is constant. b) If kHρ f − f kX = O(ϕ(ρ)), ρ → ∞, then f ∈ H(X, ψ(j)). To represent the functions in the saturation class (see Theorem 3), we define the following kernel:  · ¸ Z 1 1 − r2λ 1 − r2 1   − 1 dr, d = 1;  r (1 − 2rt + r2 )λ+1 kd (1, t) := 2λ Z0 1 2λ  1 1−r   kd−1 (r, t)dr, d = 2, 3, . . . 2λ 0 r2λ+1 where |r| < 1,|t| ≤ 1, k1 (r, t) = and kd (r, t) =

1 2λ

Z

1 2λ

r

·

0

Z

r 0

¸ 2λ r − r12λ 1 − r12 − 1 dr1 2 λ+1 (1 − 2r1 t + r1 ) r1

r2λ − r12λ kd−1 (r1 , t)dr1 , d = 2, 3 . . . . r12λ+1

Lemma 6 gives some important properties of kd . Lemma 6. Let t = cos θ. For λ > 12 , 1 ≤ q < (2λ + 1)/(2λ − 1), we have kkd (1, cos(·))kq,λ < ∞.

(23)

Furthermore, (23) also holds for λ = 21 , 1 ≤ q < ∞. Proof. From (3) it follows that ∞

Xj+λ 1 − r2 = rj Gλj (t), (1 − 2rt + r2 )λ+1 λ j=0

(24)

where −1 ≤ t ≤ 1, 0 ≤ r < 1 and λ > − 12 , λ 6= 0. By (24) we have ∞ X j=1

j r1j+β

+λ λ Gj (t) = λ

·

¸ 1 − r12 − 1 r1β , (1 − 2r1 t + r12 )λ+1

0 ≤ r1 < 1, β ≥ −1.

Since the left series converges uniformly, by termwise integration with respect to r1 on [0, r] we have ¸ Z r· ∞ X 1 − r12 rj+β+1 j + λ λ β Gj (t) = 2 )λ+1 − 1 r1 dr1 . j + β + 1 λ (1 − 2r t + r 1 0 1 j=1

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We first take β = −1 and multiply r2λ , and take β = 2λ − 1 in the above formula, respectively. Then their difference is given by ∞ µ X 1 j=1

1 − j j + 2λ

¶ r

j+2λ j

+λ λ Gj (t) = λ

Z

r 0

·

¸ 2λ 1 − r12 r − r12λ − 1 dr1 . (1 − 2r1 t + r12 )λ+1 r1

For 0 ≤ r < 1 and −1 ≤ t ≤ 1, we have 1 k1 (r, t) = 2λ ∞ X = j=1

Z

r

0

·

¸ 2λ 1 − r12 r − r12λ dr1 − 1 (1 − 2r1 t + r12 )λ+1 r1

1 j+λ λ rj+2λ Gj (t). j(j + 2λ) λ

Similarly, k2 (r, t) = =

1 2λ ∞ X j=1

Z

r

r2λ − r12λ k1 (r1 , t)dr1 r12λ+1

0

j+λ λ 1 rj+2λ Gj (t). 2 (j(j + 2λ)) λ

It follows from induction that Z r 2λ 1 r − r12λ kd−1 (r1 , t)dr1 2λ 0 r12λ+1 ∞ X j+λ λ 1 rj+2λ Gj (t). = d (j(j + 2λ)) λ j=1

kd (r, t) =

(25) (26)

For 0 ≤ r ≤ 1, −1 ≤ t < 1, 1 − 2rt + r2 > 0 holds. So from the integral expression of k1 (r, t) we can attain that for any t ∈ [−1, 1), lim k1 (r, t) exists and r→1−

k1 (1, t) =

lim k1 (r, t) is continuous for t. Furthermore, kd (1, t) = lim kd (r, t)

r→1−

r→1−

also holds for t ∈ [−1, 1). Therefore, 1 kd (1, t) = 2λ

Z 0

1

1 − r12λ kd−1 (r1 , t)dr1 . r12λ+1

Next we prove the boundary of kd (1, t). From (25), kd (1, t) can be written as kd (1, t) =

1 2λ

Z

1 2

0

:= I1 + I2 .

1 − r12λ 1 k (r , t)dr1 + 2λ+1 d−1 1 2λ r1

Z

1 1 2

1 − r12λ kd−1 (r1 , t)dr1 r12λ+1 (27)

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Approximation by Jackson-type operator on the sphere

Applying (7) and (26), we have Z 1 ∞ 1 X 1 j + λ ¯¯ λ ¯¯ 2 1 − r12λ j+2λ G (t) r1 dr1 j 2λ j=1 (j(j + 2λ))d−1 λ r12λ+1 0 µ ¶µ ¶ ∞ 1 X 1 1 j + λ j + 2λ − 1 1 ≤ − 2λ j=1 (j(j + 2λ))d−1 λ 2j j (j + 2λ)2j+2λ j   ∞ ∞ 2λ 2λ X X j j 1 1  − ≤ C1 (λ)  d−1 2j j d−1 (j + 2λ) 2j+2λ (j(j + 2λ)) (j(j + 2λ)) j=1 j=1

|I1 | =

= C2 (λ) < ∞,

(28)

where Ci (λ)(i = 1, 2, . . . ) are constants depending on λ. Thus I1 is uniformly bounded for t. ¢ £ Next, we discuss I2 . For r1 ∈ 12 , 1 , ¸ 2λ r1 − r22λ 1 − r22 − 1 dr2 2 λ+1 (1 − 2r2 t + r2 ) r2 0 ¸ 2λ Z 12 · 1 1 − r22 r1 − r22λ = − 1 dr2 2λ 0 (1 − 2r2 t + r22 )λ+1 r2 ¸ 2λ Z r1 · r1 − r22λ 1 1 − r22 − 1 + dr2 2 λ+1 2λ 12 (1 − 2r2 t + r2 ) r2

k1 (r1 , t) =

1 2λ

Z

r1

·

:= B1 (t) + B2 (t). From the above discussion, B1 is uniformly bounded for t. Also ¯ Z r1 · ¯ ¸ 2λ ¯ 1 ¯ 1 − r22 r1 − r22λ ¯ ¯ |B2 (t)| = ¯ − 1 dr 2 2 ¯ 2λ 12 (1 − 2r2 t + r2 )λ+1 r2 Z r1 2λ Z r1 r12λ − r22λ 1 r1 − r22λ 1 − r22 1 dr2 + dr2 . ≤ 2 λ+1 2λ 12 r2 2λ 12 (1 − 2r2 t + r2 ) r2 It is easy to verify that the first integral of the right–hand side is uniformly bounded. Hence we consider the second integral on the right–hand side. Let Z r1 1 1 − r22 r12λ − r22λ I(t) = dr2 . 2λ 12 (1 − 2r2 t + r22 )λ+1 r2 For −1 ≤ t < 34 , I(t) is bounded uniformly. For 34 ≤ t ≤ 1, we divide I(t) into two parts, (1 − r2 )2 ≥ 1 − t and (1 − r2 )2 ≤ 1 − t, and estimate each part respectively. Since 1 − 2r2 t + r22 = (1 − r2 )2 + 2r2 (1 − t), we have the following inequalities (see [14]): (1 − 2r2 t + r22 )λ+1 ≥ (1 − r2 )2λ+2 , (1 − 2r2 t + r22 )λ+1 ≥ (1 − t)λ+1 , r ≥ 1/2.

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It follows from 1 − r22λ = (1 − r2 )(1 + r2 + r22 + · · · + r22λ−1 ) ≤ 2λ(1 − r2 ) that   Z 1−(1−t) 12 Z r1 1  1 − r22 r12λ − r22λ  |I(t)| = + dr2 2 1 λ+1 1 2λ (1 − 2r2 t + r2 ) r2 1−(1−t) 2 2 Z 1−(1−t) 21 Z 1 ≤4 (1 − r2 )−2λ dr2 + 4(1 − t)−λ−1 (1 − r2 )2 dr2 1 1 2 1−(1−t) ½ 2 C2 + C3 |ln(1 − t)| , λ = 21 , ≤ 1 C2 (λ) + C3 (λ)(1 − t)−λ+ 2 , λ > 12 . Thus as θ → 0+, ½ |k1 (r1 , cos θ)| =

λ = 12 , O(ln(sin θ2 )), θ −2λ+1 O((sin 2 ) ), λ > 12 .

From the above estimations we can see that k1 (r1 , cos θ) is uniformly bounded for r1 . Analogously, by (25), (26) and (28), we have Z |k2 (r2 , cos θ)| ≤ C4 (λ) + This implies that for 1 ≤ q < holds kk1 (1, cos(·))kqq,λ

r2 1 2

r22λ − r12λ |k1 (r1 , cos θ)|dr1 < ∞. r12λ+1

2λ+1 2λ−1 ,

and λ > 1/2, or 1 ≤ q < ∞ and λ = 1/2, there Z Ωn−2 π = |k1 (1, cos θ)|q sin2λ θdθ < ∞. Ωn−1 0

Using (25), (26) and (28), by induction we can prove |kd (1, cos θ)| < ∞. Then by 2λ+1 (27), we can know that for 1 ≤ q < 2λ−1 , and λ > 1/2, or 1 ≤ q < ∞ and λ = 1/2, there holds Z Ωn−2 π q kkd (1, cos(·))kq,λ = |kd (1, cos θ)|q sin2λ θdθ < ∞. Ωn−1 0

4. Main results and their proofs In this section, we first give an estimation of the rate of approximation by the Jackson-type operator (Qdk,s f )(µ). Then, we prove the saturation theorem of the Jackson-type operator on Lp (Sn−1 ). Theorem 1. Let d, s, k be positive integers, 2s > 2d + n − 1, and f ∈ Lp (Sn−1 ), 1 ≤ p ≤ ∞. For the Jackson-type operator (Qdk,s f )(µ) defined above, we have µ ¶ 1 kQdk,s f − f kp ≤ C1 ω 2d f, k p where the constant C1 only depends on d and s.

(29)

Approximation by Jackson-type operator on the sphere

343

Proof. By Definition 1 we have Z (Qdk,s f )(µ) − f (µ) =

π

(−1)d−1 (Sθ − I)d f (µ)Dk,s (θ) sinn−2 θdθ.

0

Therefore Z kQdk,s f − f kp ≤

π

k(Sθ − I)d f kp Dk,s (θ) sinn−2 θdθ

Z0 π

ω 2d (f, θ)p Dk,s (θ) sinn−2 θdθ ¶ µ Z0 π 1 Dk,s (θ) sinn−2 θdθ (kθ + 1)2d ω 2d f, ≤ k p 0 µ ¶ X 2d µ ¶ 1 2d 2d ≤ω f, Cj,s k p j=0 j ¶ µ 1 2d . = C1 ω f, k p ≤

The last inequality follows from Lemma 1, since 2s > 2d + n − 1 > j + n − 1, j = 0, 1, 2, . . . , 2d. This completes the proof of Theorem 1. Theorem 2. Let d be a positive integer, 2s > 2d+n−1. Then (Qdk,s f )(µ) is saturated on Lp (Sn−1 ) with order k −2d and the collection of constants is their invariant class. Proof. For j = 0, 1, 2, . . . , we first prove that lim

k→∞

d 1 − ξk,s (j) d (1) 1 − ξk,s

µ =

j(j + 2λ) 2λ + 1

¶d .

(30)

In fact, for any 0 < δ < π, there exists β such that 2s > β + n − 1 > 2d + n − 1, then it follows from (11) that Z

Z

π

Dk,s (θ) sin δ

2λ

π

θ ( )β Dk,s (θ) sin2λ θdθ δ δ Z π ≤ δ −β θβ Dk,s (θ) sin2λ θdθ

θdθ ≤

0

≤ Cβ,s (δk)−β . Since β > 2d, then Z

π δ

Dk,s (θ) sin2λ θdθ = o(k −2d ),

k → ∞.

(31)

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F. Cao and X. Guo

And using (11) again, we have Z d 1 − ξk,s (1) =

µ ¶d Gλ1 (cos θ) Dk,s (θ) 1 − sin2λ θdθ λ (1) G 0 1 Z π 2d θ d = 2 Dk,s (θ) sin sin2λ θdθ 2 Z π0 ≤ Dk,s (θ)θ2d sin2λ θdθ π

0

≤ Ck −2d . So we obtain d 1 − ξk,s (1) = O(k −2d ).

Let

à Tjλ (cos θ)

= 1−

1−

Gλj (cos θ)

(32) !d

Gλj (1)

.

We deduce from (8) that lim

θ→ 0

1 − Tjλ (cos θ) 1 − T1λ (cos θ)

µ =

j(j + 2λ) 2λ + 1

¶d .

Hence for any ε > 0, there exists δ > 0 such that for 0 < θ < δ ¯ ¯ ¶d µ ¯ ¯ j(j + 2λ) ¯ ¯ λ λ (1 − T1 (cos θ))¯ ≤ ε (1 − T1λ (cos θ)). ¯(1 − Tj (cos θ)) − ¯ ¯ 2λ + 1 Then ¯ ¯ µ ¶d ¯ ¯ j(j + 2λ) ¯ ¯ d d (1 − ξk,s (1))¯ ¯1 − ξk,s (j) − ¯ ¯ 2λ + 1 ¯  à ! d ¯Z π ¶d µ µ ¶d λ ¯ Gλj (cos θ) (cos θ) G j(j + 2λ)  1− 1 λ − = ¯¯ Dk,s (θ)  1 − 2λ + 1 Gλj (1) G1 (1) ¯ 0 ¯ ¯ ¯ × sin2λ θdθ¯ ¯ µ ¶d Z δ Gλ (cos(θ)) ≤ Dk,s (θ) ε 1 − 1 λ sin2λ θdθ G (1) 0 1 à µ ¶d ! Z π j(j + 2λ) sin2λ θdθ + Dk,s (θ) 1 + 2λ + 1 δ ≤ ε · O(k −2d ) + o(k −2d ) = o(k −2d ),

k → ∞.

Therefore (30) holds and is not equal to zero. According to Lemma 3, we finally finish the proof of Theorem 2.

Approximation by Jackson-type operator on the sphere

345

From Lemma 5, it is easy to see that the saturation class of (Qdk,s f )(µ) is H(X, −(j(j + 2λ))d ). Next we give an equivalent theorem about the saturation class. Theorem 3. Let f ∈ X. The following statements are equivalent. (i) f ∈ H(X, −(j(j + 2λ))d ). (ii) For f ∈ Lp (Sn−1 ) and g ∈ Lp (Sn−1 ) (1 < p < ∞) f (x) = Y0 (f ; x) −

1

Z

Ωn−1

Sn−1

kd (1, (x · y))g(y)dy

(33)

holds almost everywhere, and for f ∈ C(Sn−1 ) and g ∈ L∞ (Sn−1 ), (33) follows too. For f ∈ L1 (Sn−1 ) and the measure µ ∈ M (Sn−1 ) f (x) = Y0 (f ; x) −

1 Ωn−1

Z Sn−1

kd (1, (x · y))dµ(y)

(34)

holds almost everywhere, where kd (1, t)(−1 ≤ t ≤ 1) is given in Section 3. Proof. We first prove that (i) implies (ii). From (26), we obtain the Gegenbauer coefficient of kd (1, t): j+λ 1 kˆd (1, j) = , λ (j(j + 2λ))d

j = 1, 2 . . . ,

kˆd (1, 0) = 0.

If f ∈ H(X, −(j(j + 2λ))d ), and X = Lp (Sn−1 ), 1 < p < ∞ or X = C(Sn−1 ), then for any j = 1, 2 . . . , we have Yj (f ; x) = −

1 λ ˆ Yj (g; x) = − kd (1, j)Yj (g; x). (j(j + 2λ))d j+λ

If f ∈ H(X, −(j(j + 2λ))d ) and X = L1 (Sn−1 ), then we also have for j = 1, 2 . . . Yj (f ; x) = −

λ ˆ kd (1, j)Yj (dµ; x). j+λ

Therefore, by (10), Lemma 4 and the uniqueness theorem of Laplace series expansion, there hold (33) and (34). Using (9), (10) and Lemma 4, we can deduce (i) from (ii). This completes the proof.

Acknowledgement The author would like to thank the referees for their helpful suggestions.

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