APPROXIMATION OF DIFFERENTIABLE PERIODIC FUNCTIONS BY THEIR BIHARMONIC POISSON INTEGRALS

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APPROXIMATION OF DIFFERENTIABLE PERIODIC FUNCTIONS BY THEIR BIHARMONIC POISSON INTEGRALS

K. M. Zhyhallo and Yu. I. Kharkevych UDC 517.5

We determine the exact values and asymptotic decompositions of upper bounds of approximations by biharmonic Poisson integrals on classes of periodic differentiable functions.

Let f be a 2π-periodic function summable on [–π, π]. The function

P_ρ(f, x)= 1 1

2 1

2 1 ²

π 1 ρ ρ

π π

f t x k

kt dt

k

( + ) +  + − kcos

 

 





( )



=

∞

−

∫ ∑

^{, 0}≤ ρ < 1, –π ≤ x < π,

is called the biharmonic Poisson integral of the function f.

Let W^r, r ∈N, be the set of 2π-periodic functions having absolutely continuous derivatives up to the (r – 1)th order inclusive and such that esssup ( )

x R

fr x

∈

| |

≤ 1, and let W^r be the class of functions conjugate to functions from the class W^r, i.e.,

W^r = f f x f x t t

dt f W^r

: ( ) := − ( + ) , ∈









−

∫

1

2π _π 2

π

ctg . (1)

Denote

(

, P_ρ

)

C = sup ( ) ( , )

f f x P f x C

∈

||

−

||

ρ , 0 ≤ ρ < 1, (2)

where

||

^f

||

C = max ( )

t R f t

∈

| |

^.

If a function g(ρ) = g(; ρ) such that

(

, P_ρ

)

C = g(ρ) + o

(

^g(ρ)

)

as ρ → 1–

has been found in explicit form, then, following Stepanets [1, pp.67–68], we say that the Kolmogorov– Nikol’skii problem is solved for the given class and approximating aggregate P_ρ.

A formal series g_n

n

( )ρ

=

∑

∞ 0

is called (see, e.g., [2, p.21]) an asymptotic decomposition of a function f(ρ) as ρ → 1– if, for all n, we have

|

^gn+1(ρ)

|

^{= o}(

|

^gn(ρ)

| )

Volyn University, Luts’k. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 54, No. 9, pp. 1213–1219, September, 2002.

Original article submitted February 26, 2002.

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and, for any natural N,

f(ρ) = g_n o g_N

n N

( )ρ +

(

( )ρ

)

∑

= 0

, ρ → 1–.

We briefly write this fact as follows:

f(ρ) 艑 g_n

n

( )ρ

=

∑

∞₀ ^.

Quantities (2) and their analogs with the harmonic Poisson integral

A_ρ(f, x) = 1 1

2 ¹

π ρ

π π

f t x ^k kt dt

( + )

{

+ k^∞⁼ cos

}

−

∫ ∑

instead of P_ρ(f, x)

were studied by Timan [3], Sz.-Nagy [4], Kaniev [5], Shtark [6], Zhyhallo and Kharkevych [7–9], Falaleev [10], etc.

The aim of the present work is to determine the exact value of quantity (2) for = W^r, r∈N

\

{1}, and find its asymptotic decomposition for = W^r and = W^r, r ∈N

\

{1}.

Theorem 1. For any l ∈N and 0 < ρ < 1, the following equalities are true:

W ^l P

C 2, _ρ

( )

⁼ ₂ ¹₁ ² ¹ ^{2 1}¹

1 ( )! ₍ ₎ ln

i K _{l i} ⁱ

i l

− ^{− +} ⁻

∑

= _ρ^– ₂¹ ² ² ¹ 1

1

( )!

˜ ₍ ₎ln i K _{l i} ⁱ

i l

= −

∑

− _ρ

+ 1 2

1

2 1

1 1

2

2 1

2

2 1 1

1

2 1

2 0

− 1

− −







− − 

=

−

= − −

∑ ∑

−

ρ

ρ ρ

( )!

˜ ln

( )! ln

( ) ( )

i K

l i i i

l

l i i i

l

– ε_ρ^2l + 1 2

2 2 1

−ρ ε_ρ^l− , (3)

ε_ρ^r =

2 1 1

1 1

1 1 1

1 1

1

π_ρ …2 +

− …

∫

_t _t_r ^t_t ^dt ^dt

t t

r

r K ln ;

W ^l P

C 2+1

(

^, ^ρ

)

⁼ ₂ ¹₁ ² ² ^{2 1}¹

1 ( )! ₍ ₎ ln

i K _{l i} ⁱ

i l

− ^{− +}

−

∑

= _ρ^–

∑

_i₌^l₁ _{( )!}₂¹_i ^K^˜²⁽^{l i}^{− +}⁾ ¹^ln²ⁱ _ρ¹

+ 1 2

1

2 1

1 1

2

2 1

2 1 2 1

1

2 2

0

− 1

− −







− + − 

= −

=

∑ ∑

−

ρ

ρ ρ

( )!

˜ ln

( )! ln

( ) ( )

i K

l i i

i l

l i i

i l

+ δ_ρ²^l⁺¹ – 1 2

2 2

− ρ δ_ρ^l, (4)

δ_ρ^r =

4 1

1 1 1 1

1 1

π_ρ

∫ ∫

…

∫

2 t t t dt …dt

t r t

r

r K arctan ,

where K_n and ˜K_n are the Akhiezer–Krein–Favard constants:

K_n = 4 1

2 1

1 0 1

π

( )

− +

+

= +

∑

∞ ^{m n}ⁿ

m m , n = 0, 1, 2,…, ˜K_n = 4 1

2 1 ¹

π 0

( )

−

+ ⁺

=

∑

∞ ^mnⁿ

m m , n ∈N.

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In the proof of this theorem, we follow the scheme of the proof of Theorems 1 and 2 in [9].

Taking into account the integral representation (1) and the fact that

P_ρ

(

f^,ϕ

)

^{= P f}ρ( , ) = –ϕ 1 1

2 1 ²

π 1 ρ ρ

π π

f t x k ^k ktdt

k

( + )  + ( – ) sin

 

= 

∞

−

∫ ∑

for f ∈W^r, r ∈N

\

{1}, and integrating r times by parts, we get

W P

(

^r, _ρ

)

_C⁼¹_π ^ρ

π π

sup ^{( )}( ) ( , )

f W r

r r

f t V t dt

∈ −

∫

^, ⁽⁵⁾

where

V_r( , )ρ = t 1 1 2 1

1 2

2

1

− + −

 

  + +

 



( )

=

∑

∞

k

k kt r

k

r k

ρ ρ cos ( )π

, 0 ≤ ρ < 1.

If we prove that, for l ∈N, the equalities

sign V₂_l( , )ρ = ±t signsint (6)

and

sign V₂_l ₁ t V₂_l ₁

+ − +  2

 

 

 

( , )ρ ρ π = ±,  signcost (7)

are true, then relation (5) will yield

W P

(

^r, _ρ

)

_C⁼⁴ ¹ ¹ ¹

2 1

2 2 1

0 1

π

ρ ρ ( )

( )

− − + + −

 

+ 

+

= +

∑

∞ ^{k r}

k

k r

k

k . (8)

To prove (6) and (7), we first show that

V₂( , )ρ = t c_k kt

k

sin >

=

∑

∞ ⁰ 1

, c_k :=

1 1

2 1 ²

2

− + −

 

( )



k k

ρ ρk

for t ∈(0, π) and 0 ≤ ρ < 1. For this purpose, we analyze the sequence of the coefficients ck of the decomposition of this kernel. Since

∆c_k = c_k–c_k₊₁ = 1

k2 – 1 1 ²

(

k+

)

–ρ^k k² + ρ^k

k

+

(

+

)

1

12 –1

2 1

− −

+



 



ρ ρ^k ρ^k+

k k =: ξk(ρ), ξk(0) > 0, ξk(1) = 0, and

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ξ′(ρ) =

(

−

)

+ − + −

 

− 

1 1

2

2 1

ρ ρ^k ρ ρ

k k < 0, 0 ≤ ρ < 1, k = 1, 2,…, we get ∆c_k > 0 for any k ∈N and 0 ≤ ρ < 1. Taking into account that

∆²c_k = c_k – 2c_k₊₁ + c_k₊₂

= 1

k2 + 1 2 ²

(

k+

)

– 2 1²

(

k+

)

– ρ^k k² – ρ^k

k

+

(

+

)

2

2 2 + 2 1

1 2

ρ^k k

+

(

+

)

– 1

2 2

2 1

2 2 1

− +

+ −

+









+ +

ρ ρ^k ρ^k ρ^k

k k k =: ηk(ρ) and, for any k ∈N,

ηk(0) > 0, ηk(1) = 0,

η ρ′k( ) =

(

−

)

−

+ − +

+ − −



 

− 

1 2

1 1

2

2 1 2 2

ρ ρk ρ ρ ρ

k k k

( )

< 0, 0 ≤ ρ < 1,

we conclude that ∆²c_k > 0 for all k ∈N and 0 ≤ ρ < 1. Note that the coefficients ck of the kernel V t₂( )= V₂( , )ρ are positive and tend to zero twice monotonically t

(

∆c_k > 0,∆²c_k> 0

)

. Furthermore, it is easy to see that they satisfy the condition c

k

k k=

∑

∞¹ ^<∞. Therefore, according to [11, pp.297–298], we get V₂( , )ρ > 0t for t ∈(0, π) and 0 ≤ ρ < 1.

It is obvious that V₂_l( , )ρ 0 = V₂_l( , )ρ π = 0 for l ∈N. Hence, assuming that V₂_l( , )ρ = 0 for one moret point t₀∈(0, π) and using the Rolle theorem 2l – 2 times, we establish that, for the function V₂( , )ρ , theret exists t_2l_–₂∈(0, π) such that V₂

(

ρ,t₂_l₋₂

)

= 0. However, as shown above, this is impossible. Thus, equality (6) is true. Further, if we assume that V₂_l₊₁

(

ρ,t₀

)

–V₂_l ₁

+  2

 

ρ π,  = 0, t₀∈(0, π), t₀≠ π

2, then, according to the Rolle theorem, there exists t₁∈(0, π) such that ′V₂_l₊₁

(

ρ,t₁

)

= 0, whence ′V₂_l

(

ρ,t₁

)

= 0. But this is impossible by virtue of (6). Equality (7) is proved.

Finally, using relation (8), we get

W P

(

^r, _ρ

)

_C⁼⁴ ¹ ¹₂ ₁² ¹¹

π 0( ) ρ

( )

− −

+

= +

∑

∞ ^kr ^k^r

k k – 2 1

1 1

2 1

2

0

(

−

)

−

= +

∑

∞

ρ

π ( )

( )

kr

k k r + 2 1

1 1

2 1

2 2 1

0

(

−

)

− −

+

=

∑

∞

ρ

π ( ) ρ

( )

kr k

k k r. (9)

Using relation (9) and taking into account that the functions

ϕn(ρ) = 4 1

2 1

2 1 1

π 0 −ρ

+

= +

∑

∞ ^kⁿ

k ( k ) ,

ψn(ρ) = 4

1 1

2 1

2 1 1

π 0( ) ρ

( )

− −

+

= +

∑

∞ ^k ^kⁿ

k k

satisfy the equalities (see [3, p.20])

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ϕn(ρ) = ( )

! ( ) ln

− ⁻

=

−

∑

₁¹ ¹^k ¹ − ⁰ ¹ k

n

n k k

k ϕ

ρ + (−1)ⁿ⁻¹ε_ρⁿ,

ψn(ρ) = ( )

! ( ) ln

− ⁻

=

−

∑

₁¹ ¹^k ¹ − ⁰ ¹ k

n

n k k

k ψ

ρ + (−1)ⁿ⁻¹δ_ρⁿ, we obtain the statement of Theorem 1.

Remark 1. Note that, by virtue of (8), we have

W P

(

^r, ₀

)

_C^{= ˜}^K^r, r = 2, 3,… . Remark 2. Taking into account that

δ_ρ^r = O

(

(1 – ρ)^r

)

^,ε_ρ^r = O (1 ) ln^r 1

− 1

−



 

ρ 

ρ (10)

as ρ → 1– (see [3, p.18]), by virtue of Theorem 1 we get

W P

(

², _ρ

)

_C^{= O (}^_ ¹⁻^ρ^{) ln}² ₁₋¹_ρ^_ ⁽¹¹⁾

and

W P

(

^r, _ρ

)

_C^{= K}^_ ^r⁻¹⁺^K^˜^r₂⁻²^_⁽¹⁻^{ρ + O (}⁾² ^_ ¹⁻^ρ^{) ln}³ ₁₋¹_ρ^_, r = 3, 4,…, (12)

as ρ → 1–. Comparing (11) and (12) with the estimates

W A

(

^r, _ρ

)

_C^{= K}^r^–¹⁽^{1 –}^ρ^{) + O (}^_¹⁻^ρ^{) ln}² ₁₋¹_ρ^_ ⁽¹³⁾

obtained in [9] (here, A_ρ is the harmonic Poisson integral), we conclude that, in the case where r ∈N

\

{1} and ρ → 1–, the right-hand sides of (11) and (12) are smaller by an order of magnitude than the right-hand side of (13).

In view of relation (10) and the estimate ln1

ρ ⬃ (1 – ρ) as ρ → 1–, equalities (3) for r = 2l and (4) for r = 2l + 1 enable one to specify only the first r – 1 terms of asymptotics with the corresponding constants (Kolmogorov–Nikol’skii constants). The theorems presented below give asymptotic decompositions of quantities (2) for = W^r and = W^r that enable one to calculate the Kolmogorov–Nikol’skii constants corresponding to asymptotic terms of arbitrarily high order of smallness. In the proof of these theorems, we use the following lemmas from [8] (for technical reasons independent of the authors, several misprints were made in their formulation in [8]):

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Lemma 1. For the functions

ϕn(ρ) = … … +

− …

∫

₁ ¹ ¹1

1 1 1 0

0

1 2

t t

t

t dt dt

n n

t t_n

ln

ρ

, n ∈N,

the following asymptotic decomposition is true:

ϕn(ρ) 艑 α ρ

ρ β ρ

k

n k

k

n k

k

(1 ) ln 1 ( )

1 1

1

− − + −









=

∑

∞ ^,

where

αk

n = ( )

!

−1^k

n k

k a , (14)

βk

n = ( )

! ( ) ln

− +  +

 

 +







− 

=

−

∑

1

∑

0 2 1

1 1

k 1

n i i

k n k

i k

i n

k n

k a a

i S

ϕ , (15)

S_kⁿ = 0

2

1 1 1

, ;

, ,

k n

aⁱ A a k n

k

i n i

k n k i i

k n

i n k

≤

+ >





 ⁻

− −

=

−

= +

∑ ∑

a_i^j = 0

1 1 1

1

2 1

1 1 1

1

1 1

1

1 1

1

1 1

, ;

( ) ( )!, ;

( ), ;

( ) ( ), ,

i j

j i

a a j i j n

a a j n i j

i n a a j i n n i j

j

i j

>

− − =

− − ≤ ≤

− − + = ≤

− − − − − + − + < ≤







−− −







A_kⁿ = n n n k k

( − … − +1) ( 1)

, ϕn(0) = π π 2 2

K n

n

,

˜ ,

is odd, is even.





 Lemma 2. For the functions

ψn(ρ) =

∫ ∫

…

∫

^{arctan t}t …t dt …dt

n n

t t_n

1

1 1

0 0

1 2

ρ

, n ∈N,

the following asymptotic decomposition is true:

ψn(ρ) = γkn ρ k k

(1 )

1

= −

∑

∞ ^,

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where, for k ∈N,

γk

n = ( )

! ( )

−  +





− 

∑

=

1 0

1 k

n i i

k k n i

n

k ψ b σ , (16)

ψn(0) = π π 4 4

K n

n

,

˜ ,

is odd, is even,





 σk

n = 0

12

, ,

k n

bⁱ k n

k i n i n

k

≤

>





 −

= +

∑

b_i^j = 0

1 1 1

1

2 1

2 1 2 2 1

1

1 1

1

1 1

1

1 1

, ,

( ) ( )!, ,

( ), ,

( ) ( ), .

i j

j i

b b j i j n

b b j n i j

i n b b j i n n i j

j

i j

>

− − =

− − ≤ ≤

− − + = ≤

− − − − − + + < ≤







−− −











Theorem 2. The following asymptotic decompositions are true:

W P

(

², _ρ

)

_C^艑¹_π⁽¹⁻^ρ^{) ln}² ₁₋¹_ρ^{+ K}^_ ¹⁺^ln_π² ⁻₂¹_π^_⁽¹⁻^ρ⁾²

+ 2 1

2 1 1

1

2 1 1

2 1

π 3 α α α ρ

k k k ρ

k k

+ −

 

 −

−



 ⁻ ⁻

=

∑

∞ ⁽ ^{) ln} ⁺^_^β^k² ⁺^β¹^k⁻¹⁻¹₂^β¹^k⁻²^_⁽¹⁻^ρ⁾^k

}

^,

W P

(

^r, _ρ

)

_C^{艑 K}^_ ^r⁻¹⁺¹₂ ^K^˜^r⁻²^_{ −}⁽¹ ^{ρ +}⁾² ² ¹¹ ¹₂ ¹² ¹ ₁¹

π 3 α α α ρ

k ρ

r k r

k

r k

k

+ −

 

 −

−



 ⁻

− −−

=

∑

∞ ⁽ ^{) ln}

+ βk β β ρ

r k r

k

r k

+ −

 

 −

}

−−

−− 1

1

2

1 1

2 (1 ) , r = 2l + 2, l ∈N,

W P

(

^r, _ρ

)

_C^{艑 K}^_ ^r⁻¹⁺¹₂^K^˜^r⁻²^_{ −}⁽¹ ^{ρ +}⁾² ⁴ ¹¹ ¹₂ ¹² ¹

π 3 γkr γ γ ρ

k r

k

r k

k

+ −

 

 −

−−

=

∑

∞ ⁽ ) , r = 2l + 1, l ∈N, where the coefficients αk

r, βk

r, and γk

r are calculated according to formulas (14), (15), and (16), respec- tively.

The validity of Theorem 2 follows from equality (9), Lemmas 1 and 2, and the relations

… …

+

− …

∫

¹₀ ₀²_t₁ ¹_t_n ¹₁ ^t_t¹₁^dt¹ ^dtⁿ

t_n t

ln

ρ

= 2 1

2 1

2 1 0 1

− +

+

= +

∑

∞ ^ρ ^kⁿ

k ( k ) (17)

and

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…

∫

… …

∫

^{arctan t}t t dt dt

n n

t t_n

1

1 1

0 0

1 2

ρ

= ( )

( )

− −

+

= +

∑

∞₀ ¹^k ₂¹ ²₁^kⁿ¹¹

k k

ρ . (18)

Theorem 3. The following asymptotic decompositions are true:

(

^W^r^{, P}ρ

)

C 艑 ˜K_r₋ + K_r₋ ( )

 

 −

1 2

2 1 ρ + 2 1

2 1 1

1 1

1

2 1

π 3 α α α ρ

k ρ

r k r

k

r k

k

+ −

 

 −

−



 ⁻

− −−

=

∑

∞ ⁽ ^{) ln}

+ βk β β ρ

r k r

k

r k

+ −

 

 −

}

−−

−− 1

1

2

1 1

2 (1 ) , r = 2l + 1, l ∈N, (19)

(

^W^r^{, P}_ρ

)

C 艑 ˜K_r₋ + K_r₋ ( )

 

 −

1 2

2 1 ρ + 4 1

2 1

1 1

2 1

π 3 γk γ γ ρ

r k r

k

r k

k

+ −

 

 −

−−

=

∑

∞ ⁽ ) , r = 2l, l ∈N, (20)

where the coefficients αkr, βkr, and γkr are calculated according to formulas (14), (15), and (16), respec- tively.

Proof. As in the proof of Theorem 1, it is easy to show that

(

^W^r^{, P}ρ

)

C = 1

π ρ

π π

sup ^{( )}( ) ( , )

f W

r

r r

f t V t dt

∈ −

∫

^,

where

V_r(ρ, t) = 1 1 2 1

2

1

− + −

 

  +

 



( )

=

∑

∞

k

k kt r

k

k r

ρ ρ cos π ,

and, furthermore,

signV_r(ρ, t) = ±signsint for r = 2l + 1,

sign V _r( , )ρ t −  V_r ρ π,

 

2  = ±signcost for r = 2l. Therefore, for r ≥ 2, we get

(

^W^r^{, P}ρ

)

C = 4 1

1 1 2 1

2 1

1

2 2 1

1

π 0

ρ ρ

( )

− − + + −

 

+ 

+

= +

∑

∞ ^{k r}

k

r k

k

= 4

1 1

2 1

1 2 1

1

π 0( ) ρ

( )

− −

+

+ +

= +

∑

∞ ^{k r} ^k^r

k k – 2 1

1 1

2 1

0

(

−

)

−

+

=

∑

∞

ρ

π ( )

( )

( ) k r

r

k k

+ 2 1

1 1

2 1

2 1 2 1

0

(

−

)

− −

+

+ +

=

∑

∞

ρ

π ( ) ρ

( )

k r k

k k r. (21)

(9)

Taking into account relations (17) and (18) and Lemmas 1 and 2, we obtain the asymptotic decompositions (19) and (20) from (21). Theorem 3 is proved.

Note that asymptotic decompositions of the upper bounds of approximations on the class of differentiable functions W¹ by their harmonic Poisson integrals were obtained in [6], and on the classes W^r, r ∈N

\

{1}, and W^r, r ∈N, in [8]. An asymptotic decomposition of

(

^W¹^{, P}_ρ

)

C was obtained in [7].

REFERENCES

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Dokl. Akad. Nauk SSSR, 74, 17–20 (1950).

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Akad. Nauk Ukr., No. 5, 18–23 (2002).

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