ISSN 1063-4541, Vestnik St. Petersburg University. Mathematics, 2008, Vol. 41, No. 2, pp. 102–112. © Allerton Press, Inc., 2008. Original Russian Text © N.Yu. Dodonov, V.V. Zhuk, 2008, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2008, No. 2, pp. 23–33.
TO THE 100th ANNIVERSARY OF BIRTHDAY OF SOLOMON GRIGOR’EVICH MIKHLIN
Directional Approximation of Functions of Two Variables in the Spaces +p(R2) and +p( R 2+ ) N. Yu. Dodonov and V. V. Zhuk Received November 11, 2007
Abstract—Let r ∈ N, α, t ∈ R, x ∈ R2, f : R2
C, and denote
r
r
∆ t, α ( f , x ) =
∑ ( –1 )
r–k
k
C r f ( x 1 + kt cos α, x 2 + kt sin α ).
k=0
In this paper, we investigate the relation between the behavior of the quantity
∫ ∆t, α ( f , · )Ψn ( t ) dt r
, p, G
E
2
as n ∞ (here, E ⊂ R, G ∈ {R2, R + }, and ψn ∈ L1(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: r
ω r, α ( f , h ) p, G = sup ∆ t, α ( f ) 0≤t≤h
p, G .
Here is one of the results obtained. Theorem 1. Let E and A be intervals in R+ such that A ⊂ E, f ∈ Lp(G), α ∈ [0, 2π] when G = R2 and π 2 α ∈ 0, --- when G = R + . Denote ∆n, k = 2 N, we have ∆m, r > 0, ∆m, r + 1 < ∞, and
∫A t
k
ψ n (t)dt. If there exists an r ∈ N such that, for any m ∈
∆ n, r + 1 lim --------------- = 0, n → ∞ ∆ n, r
–1
lim ∆ n, r
n→∞
∫
ψ n = 0,
E\ A
then the relations
lim ∆ n, r ∫ ∆ t, α ( f , · )ψ n dt –1
n→∞
r
E
≤ K, p, G
–r
sup t ω r, α ( f , t ) p, G ≤ K
t ∈ ( 0, ∞ )
are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and 102
DIRECTIONAL APPROXIMATION OF FUNCTIONS OF TWO VARIABLES
103
nt 2 ⎛ sin -----⎞ 2 2 1 σ n, α ( f , x ) = ------ ∫ ∆ t, α ( f , x ) ⎜ -------------⎟ dt. ⎜ t ⎟ πn R+ ⎝ ⎠ πn Then the relations lim -------- σ n, α ( f ) n → ∞ ln n
–1
≤ K and
p, G
sup t ω 1, α ( f, t)p, G ≤ K are equivalent.
t ∈ ( 0, ∞ )
DOI: 10.3103/S1063454108020040
In what follows, , , +, and denote the sets of complex, real, nonnegative real, and positive integer numbers, respectively. Given a set A we denote Am = A × A × … × A (m times). For E ⊂ and 1 ≤ p < ∞, we write p(E) for the set of all measurable functions f: E n
f
⎛ p⎞ = ⎜ f ⎟ ⎝E ⎠
1/ p
∫
p, E
< ∞.
with the norm
Notation C(E) is used for the space of uniformly continuous bounded functions f: E f
∞, E
such that
= sup f ( x ) . E
We assume that ∞(E) = C(E). If x ∈ , then x1, …, xn are the coordinates of x; i.e., x = (x1, …, xn). n
Let r ∈ , α, t ∈ , x ∈ , and f: 2
. Then
2
r
r ∆ t, α (
f , x) =
∑ ( –1 )
r–k
C r f ( x 1 + kt cos α, x 2 + kt sin α ), k
k=0 r
r δ t, α (
f , x) =
∑ ( –1 ) C k
k r
k=0
r r f ⎛ x 1 + ⎛ --- – k⎞ t cos α, x 2 + ⎛ --- – k⎞ t sin α⎞ . ⎝ ⎝2 ⎠ ⎝2 ⎠ ⎠
In this paper, we investigate the relation between the behavior of the quantities
∫∆
r t, α (
f , · )ψ n ( t ) dt
, p, E
F
∫δ
r t, α (
f , · )ψ n ( t ) dt p, E
F
as n ∞ (here, F ⊂ , E is either or + , and ψn is a positive kernel) and structural properties of function f characterized by its “directional” moduli of continuity. The paper consists of two sections. The first section contains notation and auxiliary results; in Section 2, we establish general facts and apply them to particular methods of approximation. 2
2
PRELIMINARIES 1. Let us introduce some notation. Given 1 ≤ p ≤ ∞, f ∈ p(E), (where E ∈ { , + }), h > 0, r – 1 ∈ , and x ∈ E, we denote 2
2
hh
1 S h, 1 ( f , x ) = -----2 f ( x 1 + u, x 2 + v ) du dv , h 00
∫∫
S h, r ( f ) = S h, 1 ( S h, r – 1 ( f ) ).
Function Sh, r( f ) is called the Steklov function of order r with step h for a function f. VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
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For 1 ≤ p ≤ ∞, f ∈ p(E), where E ∈ { , + }, h ≥ 0, r ∈ , and α ∈ , we denote 2
2
ω r, α ( f , h ) p, E = sup ∆ t, α ( f ) r
p, E .
0≤t≤h
The quantity ωr, α(f, h)p, E is call the modulus of continuity of order r with step h along the direction (cosα, sinα) of function f in the space p(E). (r)
By C p (E) we denote the set of functions f ∈ C(E) whose partial derivatives of orders up to r inclusive belong to C(E) ∩ p(E). 2. Let us remind some known results. Lemma A. Let 1 ≤ p ≤ ∞, E ∈ { , + }, f ∈ p(E), h > 0, r ∈ , and x ∈ E. Then 2
2
(1) the following relation holds: rh rh
S h, r ( f , x ) =
∫ ∫ f (x
1
+ u, x 2 + v )ϕ h, r ( u )ϕ h, r ( v ) du dv ,
(1)
0 0
where rh
ϕ h, r ∈ C ( [ 0, rh ] ),
ϕ h, r ( t ) ≥ 0 at t ∈ [ 0, rh ],
∫ϕ
h, r
= 1;
0
(2) the following equality is satisfied lim f – S h, r
p, E
h → 0+
= 0;
(2)
(3) the following inclusion is valid: (r + 1)
S h, r + 2 ( f ) ∈ C p
( E ).
(3)
Lemma B. (generalized Minkowskii inequality for integrals, see, e.g., [1, pp. 23, 24]). Let 1 ≤ p < ∞, m be the space n of points x, y be the space m of points y, and ϕ(x, y) be a measurable function n x
defined on x × y . Then n
m
∫ ϕ ( ·, y ) d y m
y
≤ p,
n x
∫
ϕ ( ·, y )
n
p, x
dy.
(4)
m
y
3. Now let us establish several auxiliary facts. Lemma 1. Let 1 ≤ p < ∞, α ∈ , h > 0, x ∈ 2, E and A be intervals in , A ⊂ E, ψn: E 1(E), and f ∈ p(2). We denote U n, α ( f , x ) =
∫∆
r t, α (
f , x )ψ n ( t ) dt,
∆ n, k =
E
∫ t ψ ( t ) dt , k
n
δ n, k =
A
∫t
k
+, ψn ∈
ψ n ( t ) dt.
A
If there exists an r ∈ such that, for any m ∈ , we have ∆m, r > 0, δm, r + 1 < ∞, and δ n, r + 1 lim ------------- = 0, n → ∞ ∆ n, r
∫ψ
lim ∆ n, r –1
n→∞
n
= 0,
E\ A
then the inequality lim ∆ n, r U n, α ( f ) –1
n→∞
p,
2
≤K
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DIRECTIONAL APPROXIMATION OF FUNCTIONS OF TWO VARIABLES
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implies the relation r
j r– j j ∂ S h, r + 2 ( f ) - cos α sin α C r --------------------------j r– j ∂x 1 ∂x 2 j=0 r
∑
Proof. Instead of ·
p,
≤ K. p,
2
and Sh, r + 2 we shall write ||·|| and S, respectively. If g:
2
, then
r
r ∆ t ( g,
∑ ( –1 )
x) =
r–k
C r g ( x + kt ) k
k=0
is its finite difference of order r. We set γ = r + 2. First of all, we note that, by Hölder’s inequality, relation δn, r + 1 < ∞ implies that the quantities ∆n, r are finite. It is clear that U n, α ( S ( f ), x ) =
∫∆
r t, α ( S (
f ), x )ψ n ( t ) dt =
∫ S(∆
E
r t, α (
f ), x )ψ n ( t ) dt.
E
With due account for (1), this gives us γh γh
⎛ ⎞ r U n, α ( S ( f ), x ) = ⎜ ∆ t, α ( f , x 1 + u 1, x 2 + u 2 )ϕ h, γ ( u 1 )ϕ h, γ ( u 2 ) du 1 du 2⎟ ψ n ( t ) dt. ⎝ ⎠ E 0 0
∫ ∫∫
(5)
Let us demonstrate that function l ( x, u, t ) = ∆ t, α ( f , x + u )ϕ h, γ ( u 1 )ϕ h, γ ( u 2 )ψ n ( t ) r
is summable with respect to variables u and t on the set [0, γh]2 × E = F for almost all x ∈ . Indeed, applying Lemma B, we obtain 2
∫ l ( ·, u, t ) ≤ ∫ l ( ·, u, t ) ≤ ∫ ∆ F
F
≤2 f r
r t, α (
f , · + u ) ϕ h, γ ( u 1 )ϕ h, γ ( u 2 )ψ n ( t ) du 1 du 2 dt
F
∫ϕ
h, γ ( u 1 )ϕ h, γ ( u 2 )ψ n ( t ) du 1 du 2 dt
r
=2 f
∫ψ
n
< ∞.
E
F
Thus, we established the inequality
∫ l ( ·, u, t )
< ∞,
E
which implies that
∫
l (x, u, t)dudt is finite for almost all x ∈ . Taking this fact into account and applying 2
E
Fubini’s theorem about iterated integrals, in view of (5), for almost all x ∈ , we have 2
U n, α ( S ( f ), x ) = S ( U n, α ( f ), x ). Using this equality and applying Lemma B, we obtain U n, α ( S ( f ) ) = S ( U n, α ( f ) ) ≤ U n, α ( f ) .
(6)
Let us set g x ( t ) = S ( f , x 1 + t cos α, x 2 + t sin α ). VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
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Taking into account inclusion (3), one can easily verify that ∆ t, α ( S ( f ), x ) = ∆ t ( g x, 0 ) r
t
=
t
∫ ∫ …
0
r
t
(r) g x ( y1
+ … + y r ) dy 1 … dy r =
(r) r g x ( 0 )t
0
t
⎛ + … ⎜ ⎝ 0 0
y1 + … + yr
∫ ∫
∫
(r + 1)
gx
0
⎞ ( z ) dz⎟ dy 1 … dy r . ⎠
Applying Lemma B once again, we obtain t
t
⎛ … ⎜ ⎝ 0 0
y1 + … + yr
∫ ∫
∫
(r + 1)
g·
0
⎞ r+1 (r + 1) ( z ) dz⎟ dy 1 … dy r ≤ r g · (0) t . ⎠
From what was said above, it follows that
∫∆
t, α ( S (
(r)
(r + 1)
f ), · )ψ n ( t ) dt ≥ g · ( 0 ) ∆ n, r – r g ·
( 0 ) δ n, r + 1 .
A
Now, combining this relation with inequality (6), we obtain
∫∆
r t, α ( S (
∫∆
f ), · )ψ n ( t ) dt ≤ U n, α ( S ( f ) ) +
A
r t, α ( S (
f ), · )ψ n dt
E\ A
∫ψ.
≤ U n, α ( f ) + 2 S ( f ) r
n
E\ A
Hence, (r)
g·
⎞ –1 ⎛ r (r + 1) ≤ ∆ n, r ⎜ ∆ t, α ( S ( f ), · )ψ n ( t ) dt + r g · ( 0 ) δ n, r + 1⎟ ⎝ A ⎠
∫
r –1 ⎛ ≤ ∆ n, r ⎜ U n, α ( f ) + 2 S ( f ) ⎝
∫
(r + 1)
ψ n + r g·
E\ A
⎞ ( 0 ) δ n, r + 1⎟ . ⎠
Thus, (r)
g · ( 0 ) ≤ lim ∆ n, r U n, α ( f ) + 2 S ( f ) lim ∆ n, r r
–1
n→∞
n→∞
∫ψ
(r + 1)
n
+ r g·
E\ A
=
–1 lim ∆ n, r n→∞
δ n, r + 1 ( 0 ) lim ------------n → ∞ ∆ n, r
U n, α ( f ) ≤ K.
Now it remains to take into account that r
(r) g· ( 0 )
=
j r– j j ∂ S h, γ ( f ) - cos α sin α . C r --------------------j r– j ∂x 1 ∂x 2 j=0
∑
r
Remark 1. Lemma 1 remains valid for p = ∞. The proof of Remark 1 is similar to that of Lemma 1 (and is a little simpler). Remark 2. In Lemma 1 and in Remark 1, the difference ∆r t, α ( f ) can be replaced with δ t, α ( f ). The proof remains unchanged. r
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DIRECTIONAL APPROXIMATION OF FUNCTIONS OF TWO VARIABLES
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Lemma 2. Let 1 ≤ p ≤ ∞, α ∈ [ 0; π/2 ] , h > 0, x ∈ + , E and A be intervals in +, A ⊂ E, ψn: E 2
+, ψn ∈ 1(E), and f ∈ p( + ). We denote 2
U n, α ( f , x ) =
∫∆
r t, α (
f , x )ψ n ( t ) dt,
∫ t ψ ( t ) dt.
∆ n, k =
k
n
E
E
If there exists an r ∈ such that, for any m ∈ , we have ∆m, r > 0, ∆m, r + 1 < ∞, and ∆ n, r + 1 lim -------------- = 0, n → ∞ ∆ n, r
∫ψ
lim ∆ n, r –1
n→∞
n
= 0,
E\ A
then the inequality lim ∆ n, r U n, α ( f ) –1
n→∞
2
p, +
≤K
implies the relation r
j r– j j ∂ S h, r + 2 ( f ) - cos α sin α C r --------------------------j r– j ∂x 1 ∂x 2 j=0 r
∑
≤ K. 2
p, +
In other words, Lemma 1 and Remark 1 remain valid if, in their statement, is replaced with + and it is required that α ∈ [ 0; π/2 ] . The proof of Lemma 2 is similar to that of Lemma 1. Lemma 3. Let 1 ≤ p ≤ ∞, r ∈ , E ∈ { , + }; α ∈ [0, 2π] if E = and α ∈ [ 0; π/2 ] if E = + ; f ∈ p(E), and 2
2
2
2
r
r j r– j j ∂ S h, r + 2 ( f ) - cos α sin α lim C r --------------------------j r – j h → 0+ ∂x 1 ∂x 2 p, E j=0
∑
≤ K.
Then sup t ω r, α ( f , t ) p, E ≤ K. –r
0
set
Proof. Instead of ||·||p, E, Sh, r + 2, and ωr, α( f, h)p, E, we shall write ||·||, Sh, and ωr( f, h), respectively. Let us g x, h ( f , u ) = S h ( f , x 1 + u cos α, x 2 + u sin α ). For u ≥ 0, we have u r ∆ u, α ( S h (
f ), x ) =
r ∆ u ( g x, h (
f ), 0 ) =
u
∫ …∫ g 0
(r) x, h ( y 1
+ … + y r ) dy 1 … dy r .
0
Applying Lemma B, we obtain that u r ∆ u, α ( S h (
f )) =
∫ …∫ g 0
u
u (r) ·, h ( y 1
+ … + y r ) dy 1 … dy r
0
u
∫ ∫
(r)
(r)
≤ … g ·, h ( y 1 + … + y r ) dy 1 … dy r ≤ u g ·, h ( 0 ) . 0
r
0
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Thus, (r)
ω r ( S h ( f ), t ) ≤ t g ·, h ( 0 ) . r
(7)
Now, by virtue of relation (2), we have ω r ( f , t ) – ω r ( S h ( f ), t ) ≤ ω r ( f – S h ( f ), t ) ≤ 2 f – S h ( f ) r
h → 0+
0.
(8)
Combining (7) and (8), we arrive at the relations (r)
ω r ( f , t ) = lim ω r ( S h ( f ), t ) ≤ t lim g ·, h ( 0 ) . r
h → 0+
h → 0+
It remains to take into account that r
(r) g ·, h ( 0 )
r j r– j j ∂ S h, r + 2 ( f ) - cos α sin α C r --------------------------j r– j ∂x 1 ∂x 2 j=0
∑
=
.
2. MAIN RESULTS 1. Let us present our main results. Theorem 1. Let 1 ≤ p ≤ ∞, G ∈ { , + }; E and A be intervals in +, A ⊂ E, ψn: E 2
2
x ∈ G, f ∈ p(G), a ∈ [0, 2π] if G = and α ∈ [ 0; π/2 ] if G = 2
U n, α ( f , x ) =
∫∆
r t, α (
f , x )ψ n ( t ) dt,
2 +.
∆ n, k =
E
+, ψn ∈ 1(E),
We denote
∫t
k
ψ n ( t ) dt.
A
If there exists an r ∈ such that, for any m ∈ , we have ∆m, r > 0, ∆m, r + 1 < ∞, and ∆ n, r + 1 lim -------------- = 0, ∆ n, r
∫ψ
lim ∆ n, r –1
n→∞
n→∞
= 0,
n
(9)
E\ A
then the relations lim ∆ n, r U n, α ( f )
≤K
(10)
sup t ω r, α ( f , t ) p, G ≤ K
(11)
–1
n→∞
p, G
–r
t ∈ ( 0, ∞ )
are equivalent; i.e., the validity of any of these relations implies the validity of the other one. Proof. Suppose that inequality (11) holds. Then U n, α ( f )
p, G
≤
∫∆
r t, α (
f)
p, G ψ n ( t ) dt
E
∫
∫
E
A
≤ ω r, α ( f , t ) p, G ψ n ( t ) dt ≤ K t ψ n ( t ) dt + 2 f r
r
p, G
∫ ψ ( t ) dt. n
E\ A
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DIRECTIONAL APPROXIMATION OF FUNCTIONS OF TWO VARIABLES
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With due account for relations (9), this gives us lim ∆ n, r U n, α ( f )
≤K+2 f
–1
r
p, G
n→∞
p, G
∫ψ
lim ∆ n, r –1
n→∞
n
= K.
E\ A
On the other hand, combining Lemmas 1–3 and Remark 1, we readily obtain that inequality (10) implies inequality (11). Remark 3. If, in the statement of Theorem 1, we have G = , then the difference ∆ t, α ( f, x) can be r
2
replaced with the quantity δ t, α ( f, x). r
Remark 4. If, in the statement of Theorem 1, we additionally require that G = and r/2 ∈ , then the requirement that E and A are intervals in + can be replaced with the requirement that E and A are intervals in . 2
Remark 5. Remark 4 remains true if the difference ∆ t, α ( f, x) is replaced with the quantity δ t, α ( f, x). r
r
2. Let us formulate one-dimensional analogs of propositions presented in Section 1. Suppose that G ∈ {, +}. For a function f of one variable, we denote r
∑ ( –1 )
∆t ( f , x ) = r
r–k
C r f ( x + kt ), k
k=0 r
δt ( f , x ) = r
∑ ( –1 ) C k
k r
k=0
r f ⎛ x + ⎛ --- – k⎞ t⎞ , ⎝ ⎝2 ⎠ ⎠
ω r ( f , h ) p, G = sup ∆ t ( f ) r
0≤t≤h
p, G .
Theorem 1'. Let 1 ≤ p ≤ ∞, G ∈ { , + }, E and A be intervals in +, A ⊂ E, ψn: E x ∈ G, and f ∈ p(G). We denote 2
2
Un( f , x ) =
∫ ∆ ( f , x )ψ ( t ) dt, r t
∆ n, k =
n
E
∫t
k
+, ψn ∈ 1(E),
ψ n ( t ) dt.
A
If there exists an r ∈ such that, for any m ∈ , we have ∆m, r > 0, ∆m, r + 1 < ∞, and Eqs. (9) are satisfied, then the relations lim ∆ n, r U n ( f )
≤ K,
(10')
sup t ω r ( f , t ) p, G ≤ K
(11')
–1
n→∞
p, G
–r
t ∈ ( 0, ∞ )
are equivalent; i.e., the validity of any of them implies the validity of the other one. To prove Theorem 1', it is sufficient to apply Theorem 1 to function γ(x) = f(x1)g(x2), where function g(x2) belongs to the class p(G) and has the property ||g||p, G = 1 (for α = 0). Remark 3'. If, in the statement of Theorem 1', we have G = , then the difference ∆ t ( f, x) can be r
replaced with the quantity δ t ( f, x). r
Remark 4'. If, in the statement of Theorem 1', we additionally require that G = and r/2 ∈ , then the requirement that E and A are intervals in + can be replaced with the requirement that E and A are intervals in . Remark 5'. Remark 4' remains true if the difference ∆ t ( f, x) is replaced with the quantity δ t ( f, x). r
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It should be noted that the results presented above were earlier established for 2π-periodic functions in paper [2]; their multidimensional analogs (of another type) were established in papers [3, 4]. 3. Let us give an example illustrating the application of Theorem 1. Take n, r ∈ and denote nt 2r ( 2r – 1 )!2 2 - ⎜ -------------⎟ , Φ n, r ( t ) = --------------------------------2r – 1 ⎜ t ⎟ πλ 2r n ⎝ ⎠ -⎞ 2r – 1 ⎛ sin ----
where r–1
∑ ( –1 ) C
λ 2r =
l
l 2r ( r
– l)
2r – 1
.
l=0
Function Φn, r is called the generalized Fejér–Jackson–de la Vallée-Poussin kernel. It is clear that nt 2 ⎛ sin -----⎞ 2 2 Φ n, 1 ( t ) = ------ ⎜ -------------⎟ , ⎜ πn t ⎟ ⎝ ⎠
nt 4 ⎛ sin -----⎞ 12 2 Φ n, 2 ( t ) = --------3 ⎜ -------------⎟ . ⎜ πn ⎝ t ⎟⎠
Kernels Φn, 1 and Φn, 2 are classical objects (see, e.g., [5, p. 150]). In the literature, they are referred to as the Fejér–de la Vallée-Poussin kernel and the Jackson–de la Vallée-Poussin kernel, respectively. They are used in the Fejér σn( f ) and Jackson–de la Vallée-Poussin In( f ) approximation methods: σn ( f , x ) =
∫ f ( x + t )Φ
n, 1 ( t ) dt,
∫ f ( x + t )Φ
In( f , x ) =
n, 2 ( t ) dt.
These methods play an important role in approximation theory: Theorem 2. Let 1 ≤ p ≤ ∞, G ∈ { , + }, x ∈ G; α ∈ [0, 2π] if G = and α ∈ [ 0; π/2 ] if G = + ; n, r, m ∈ , f ∈ p(G), and 2
2
2
2
U n, α, m, r ( f , x ) =
∫∆
m t, α (
f , x )Φ n, r ( t ) dt.
+
Then (1) for m ≤ 2r – 2 and 2r
( 2r – 1 )!2 sin u - du, K ( r, m ) = --------------------------- ------------2r – m πλ 2r u m
∫
+
the relations m
n lim ------------------- U n, α, m, r ( f ) n → ∞ K ( r, m )
p, G
≤ K,
sup t ω m, α ( f , t ) p, G ≤ K –m
t ∈ ( 0, ∞ )
are equivalent; (2) relations K (r) n lim ------------------------ U n, α, 2r – 1, r ( f ) ln n n→∞ 2r – 1
sup t
t ∈ ( 0, ∞ )
– 2r + 1
p, G
≤ K,
ω 2r – 1, α ( f , t ) p, G ≤ K,
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where ( 2r – 1 )!2 K ( r ) = --------------------------2 π λ 2r 2r
π/2
∫ sin
2r
t dt,
0
are equivalent. Proof. One can easily see that ∞
π
∫
∫t Φ
K1(r) Φ n, r ( t ) dt ≤ -------------, 2r – 1 n π
β
n, r ( t ) dt
0
K 1 ( r, β ) ≤ ------------------2r – 1 n
for β > 2r – 1; π
β
2r
( 2r – 1 )!2 sin u β - ------------- du t Φ n, r ( t ) dt ∼ -------------------------β 2r – β πλ n u 2 0
∫
∫
(n
∞)
+
for 0 < β < 2r – 1; and π
∫t
2r – 1
0
π
( 2r – 1 )!2 ln n 2r - sin t dt Φ n, r ( t ) dt ∼ -----------------------------------2 2r – 1 π λ 2r n 0 2r
∫
(n
∞ ).
Taking into account these relations and applying Theorem 1 (with A = [0, π] and E = +) to the operator U n, α ( f , x ) = U n, α, m, r ( f , x ), we obtain the desired result. Corollary. 1. Let 1 ≤ p ≤ ∞, G ∈ { , + }, x ∈ G; α ∈ [0, 2π] when G = and α ∈ [ 0; π/2 ] when 2
2
2
G = + ; f ∈ p(G), and 2
nt 2 ⎛ sin -----⎞ 2 2 1 σ n, α ( f , x ) = ------ ∆ t, α ( f , x ) ⎜ -------------⎟ dt. ⎜ t ⎟ πn ⎝ ⎠ +
∫
Then the relations πn lim -------- σ n, α ( f )
n → ∞ ln n
p, G
≤ K,
sup t ω 1, α ( f , t ) p, G ≤ K –1
t ∈ ( 0, ∞ )
are equivalent. Corollary 2. Suppose that, in the conditions of Corollary 1, A = [0, 2π] for G = and A = [ 0; π/2 ] for 2
G = + . Then the relations 2
πn lim -------- sup σ n, α ( f ) n → ∞ ln n α ∈ A
p, G
≤ K,
sup t sup ω 1, α ( f , t ) p, G ≤ K –1
t ∈ ( 0, ∞ )
α∈A
are equivalent. VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
Vol. 41
No. 2
2008
112
DODONOV, ZHUK
ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project no. 05-01-00742. REFERENCES 1. O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems (Nauka, Moscow, 1996; Wiley, New York, 1979). 2. A. S. Zhuk and V. V. Zhuk, Zap. Nauch. Sem. POMI 337, 134–164 (2006). 3. N. Yu. Dodonov and V. V. Zhuk, Zap. Nauch. Sem. POMI 337, 51–72 (2006). 4. N. Yu. Dodonov and V. V. Zhuk, Problems of Calculus, No. 33, 79–90 (2006). 5. N. I. Akhiezer, Theory of Approximation (Nauka, Moscow, 1965; Frederick Ungar, New York, 1956).
VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
Vol. 41
No. 2
2008
