Hindawi Publishing Corporation International Journal of Analysis Volume 2014, Article ID 249513, 10 pages http://dx.doi.org/10.1155/2014/249513
Research Article On a Pointwise Convergence of Quasi-Periodic-Rational Trigonometric Interpolation Arnak Poghosyan and Lusine Poghosyan Institute of Mathematics of National Academy of Sciences of Armenia, Bagramian Avenue 24/5, 0019 Yerevan, Armenia Correspondence should be addressed to Arnak Poghosyan;
[email protected] Received 26 November 2013; Accepted 7 March 2014; Published 3 April 2014 Academic Editor: Nicolas Crouseilles Copyright Β© 2014 A. Poghosyan and L. Poghosyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a procedure for convergence acceleration of the quasi-periodic trigonometric interpolation by application of rational corrections which leads to quasi-periodic-rational trigonometric interpolation. Rational corrections contain unknown parameters whose determination is important for realization of interpolation. We investigate the pointwise convergence of the resultant interpolation for special choice of the unknown parameters and derive the exact constants of the main terms of asymptotic errors.
1. Introduction
Let
The quasi-periodic (QP) interpolation πΌπ,π (π, π₯), π β₯ 0 (π is integer) and π₯ β [β1, 1], interpolates function π on equidistant grid π π₯π = , π
|π| β€ π,
π = 0, . . . , π,
β
|π| β€ π
(1)
and is exact for a quasi-periodic function ππππππ₯ ,
π΄ π π (π) = π(π) (1) β (β1)π+π π(π) (β1) ,
2π π= 2π + π + 1
(2)
with period 2/π which tends to 2 as π β β. The idea of the QP interpolation is introduced in [1, 2] where it is investigated based on the results of numerical experiments. Explicit representation of the interpolation is derived in [3β5]. There, the convergence of the interpolation is considered in the framework of the πΏ 2 (β1, 1)-norm and at the endpoints π₯ = Β±1 in terms of the limit function. Pointwise convergence in the interval (β1, 1) is explored in [6]. The main results there, which we need for further comparison, are the following theorems.
(β1)π(π+1) . π+1 π=ββ (2π + π₯)
Ξ¦π,π (ππππ₯ ) = π(ππ/2)(πβ1)π₯ β
(3)
We denote by π
π,π (π, π₯) the error of the QP interpolation as follows:
π
π,π (π, π₯) = π (π₯) β πΌπ,π (π, π₯) .
(4)
Theorem 1 (see [6]). Let π(π+2π) β π΄πΆ[β1, 1] for some π β₯ 1, π β₯ 0, and
π(π) (β1) = π(π) (1) = 0,
π = 0, . . . , π β 1.
(5)
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Then, the following estimate holds for |π₯| < 1 as π β β:
where π
π
π,π (π, π₯) = πΆπ,π (π)
Μ β βπ πΜ πΉπ,π = ππ,π π,β β+π,π ,
π
(β1) ππ+π+1
β=1
Μ = ππ,π
[π/2]
πβπ Γ [sin (π (π + 1) ππ₯) β ( ) π π=0
Γ
(11)
π
β1 2πππ(π β1)/(2π+π+1) ππ,β = π2ππ(β+πβπ)(π+π)/(2π+π+1) βVβ,π π .
π
(β1) β sin (ππππ₯) 22π+1 cos2π+2 (ππ₯/2)
π =1
β1 Here, Vβ,π
are the elements of the inverse of the Vandermonde matrix (Vπ ,β ) as
[π/2]β1
πβπβ2 Γ β ( ) π π=0
Vπ ,β = πΌβπ β1 ,
π
(β1) ] + π (πβπβπβ1 ) , 22π+3 cos2π+4 (ππ₯/2) (6)
Γ
π 1 π β π ( ) πβ2ππππ/(2π+π+1) , 2π + π + 1 π=βπ π
where
πΌβ = π2ππ(β+π)/(2π+π+1) ,
(12)
and have the following explicit form [8]: β1 =β Vβ,π
1
πΌβπ βπ π=1, π =ΜΈ β
π β1
π
βπ½ πΌ , (πΌβ β πΌπ ) π=0 π β
β, π = 1, . . . , π, (13)
where π½π are the coefficients of the following polynomial: π
πΆπ,π (π) = β
π΄ ππ (π) (π + 1)πβπ
πβπ+1 ππβ1 ππβπ+1 π=0 2
Theorem 2 (see [6]). Let π and
(π+1)
(π β π)!
Ξ¦(π) π,π
(β1) .
(7)
β π΄πΆ[β1, 1] for some π β₯ 0
π(π) (β1) = π(π) (1) = 0,
π = 0, . . . , π β 1.
(8)
Then, the following estimate holds for |π₯| < 1 as π β β: π
π,0 (π, π₯) = π΄ 0π (π) [π/2]
Γ β π=0
(9)
In the current paper, we consider convergence acceleration of the QP interpolation by rational corrections in terms of πππππ₯ which leads to quasi-periodic-rational (QPR) interpolation. We investigate the pointwise convergence of the QPR interpolation in the interval (β1, 1) and derive the exact constants of the main terms of asymptotic errors. Comparison with Theorems 1 and 2 shows the accelerated convergence for smooth functions. Some results of this research are reported also in [7]. More specifically, the QP interpolation can be realized by the following formula: π
π=βπ
π=1
π=0
get
(14)
Taking into account that πΌπ β πΌπ = π(1/π), from (13), we β1 Vβ,π = π (ππβ1 ) ,
π σ³¨β β,
(15)
ππ,β = π (ππβ1 ) ,
π σ³¨β β.
(16)
In this section, we consider convergence acceleration of the QP interpolation by rational trigonometric corrections which leads to the QPR interpolation. π Consider a vector π = {ππ }π=1 . By Ξππ (π, ππ ), we denote generalized finite differences defined by the following recurrent relations: Ξ0π (π, ππ ) = ππ ,
+ π (πβπβ1 ) .
πΌπ,π (π, π₯) = β πΉπ,π ππππππ₯ ,
π
2. Quasi-Periodic-Rational Interpolation
(β1)π sin (πππ₯) 2π+1 ππ+1 cos (ππ₯/2)
β (β1)π 22π (β1)π β (π β 2π)!π2π+1 π =ββ (2π + 1)2π+1
π
β (π₯ β πΌπ ) = β π½π π₯π .
π β π, π β₯ 0,
(10)
πβ1 Ξππ (π, ππ ) = Ξπβ1 π (π, ππ ) + ππ Ξ πβ1 (π, ππ ) ,
π β₯ 1,
(17)
for some sequence ππ . When π β‘ 1, we put Ξππ (π, ππ ) = Ξππ (ππ ) .
(18)
It is easy to verify that Ξ1π (π, ππ ) = ππ + π1 ππβ1 , Ξ2π (π, ππ ) = Ξ1π (π, ππ ) + π2 Ξ1πβ1 (π, ππ ) = ππ + (π1 + π2 ) ππβ1 + π1 π2 ππβ2 , Ξ3π (π, ππ ) = ππ + (π1 + π2 + π3 ) ππβ1 + (π1 π2 + π1 π3 + π2 π3 ) ππβ2 + π1 π2 π3 ππβ3 . (19)
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In general, we can prove by the mathematical induction that
where
π
Ξππ (π, ππ ) = βπΎπ (π) ππβπ ,
πΌ
π π(π) (β1) π₯ ( + 1) , π! π π=0
(20)
πleft (π₯) = β
π =0
where πΎπ (π) are the coefficients of the following polynomial: π
π
π =0
π =1
βπΎπ (π) π₯π = β (1 + ππ π₯) .
πΌ
π π(π) (1) π₯ πright (π₯) = β ( β 1) . π! π π=0
(21) π {π π }|π|=1 ,
+
According to definition of πβ , we can write
Consider the following vectors: π = π = π π {π π }π=1 , and πβ = {π βπ }π=1 . By πΏππ (π, ππ ), we denote modified finite differences defined by the following recurrent relations: πΏπ0 (π, ππ ) = ππ , πΏππ
(π, ππ ) =
πΏππβ1
(π, ππ ) +
πβ1 π βπ πΏπβ1
β
πβ (π₯) = β ππβ πππππ₯ , π=ββ
β
π (π₯) = β ππβ ππππππ₯ ,
π β₯ 1, (22)
π=ββ
(23)
π
π
(24)
π =0
where
π
π
π
π=1
π=0
π
π
π=1
π=0
|π|>π
The following transformation is easy to verify (see details in [9] for similar transformation): π
π,π (π, π₯) = π β1
β (1 + π π π₯) = β πΎπ (π+ ) π₯π ,
πΉβπβ1,π πβπππππ₯ β πΉπ,π ππππ(π+1)π₯ (1 + π β1 πππππ₯ ) (1 + π 1 πβππππ₯ )
+ π1
(25)
β (1 + π βπ π₯) = β πΎπ (πβ ) π₯π . +
It is easy to verify that
πΉπ+1,π ππππππ₯ β πΉβπ,π πβπππ(π+1)π₯ (1 + π β1 πππππ₯ ) (1 + π 1 πβππππ₯ ) 1
(1 + π β1
πππππ₯ ) (1
+ π 1 πβππππ₯ )
(34)
π
πΏππ (ππ ) = Ξ2π π+π (ππ ) ,
(26)
Γ β πΏπ1 (π, ππβ β πΉπ,π ) ππππππ₯
(27)
+
π=βπ
π
π Ξππ (ππ ) = β ( ) ππβπ , π π =0 2π (ππ ) = β ( ) ππ+πβπ . π π =0
πleft (π₯) , π₯ β [β1, βπ) , { { { π₯ π₯ β [βπ, π] , π (π₯) = {π ( ) , { { π {πright (π₯) , π₯ β (π, 1] ,
1 (1 + π β1 πππππ₯ ) (1 + π 1 πβππππ₯ )
Γ β πΏπ1 (π, ππβ ) ππππππ₯ .
2π
(28)
We assume that π β πΆπΌ [β1, 1] for some πΌ β₯ 1 and we denote β
(32)
π=βπ
πΏππ (π, ππ ) = β πΎπ (π+ ) βπΎπ (πβ ) ππ+πβπ ,
πΏππ
π₯ β [β1, 1] .
π
π,π (π, π₯) = β (ππβ β πΉπ,π ) ππππππ₯ + β ππβ ππππππ₯ . (33)
Similar to (20), we can show that
π=0
(31)
Therefore,
for some sequence ππ . When π β‘ 1, we put πΏππ (π, ππ ) = πΏππ (ππ ) .
π₯ β (β1, 1) .
Hence,
(π, ππ )
πβ1 (π, ππ ) + π βπ πΏππβ1 (π, ππ )) , + π π (πΏπ+1
(30)
|π|>π
Reiteration of it up to π times leads to the following expansion of the error: π
π,π (π, π₯)
(29)
π
= β π βπ π=1
πβ1 πβ1 πΏβπβ1 (π, πΉπ,π ) πβπππππ₯ β πΏπ (π, πΉπ,π ) ππππ(π+1)π₯
βππ =1 (1 + π βπ πππππ₯ ) (1 + π π πβππππ₯ )
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International Journal of Analysis π
+ βππ
πβ1 πβ1 πΏπ+1 (π, πΉπ,π ) ππππππ₯ β πΏβπ (π, πΉπ,π ) πβπππ(π+1)π₯
βππ =1
π=1
+
(1 + π βπ
πππππ₯ ) (1
+ ππ
πβππππ₯ )
Let π π be chosen as in (38) and let πΎπ (π) be the coefficients of the following polynomial:
1
π
3. Convergence Analysis
βπ =1 (1 + π βπ πππππ₯ ) (1 + π π πβππππ₯ )
Γ β πΏππ (π, ππβ β πΉπ,π ) ππππππ₯ |π|β€π
+ Γ
βπ =1 (1 + π βπ πππππ₯ ) (1 + π π πβππππ₯ ) β πΏππ |π|>π
where the first two terms can be assumed as corrections of the error. This observation leads to the following QPR interpolation:
π=0
(39)
π1 π ) π = Ξ1π (ππ ) β 1 ππβ1 . (40) π πβ1 π
For π = 2, we have Ξ2π (π+ , ππ ) = Ξ1π (π+ , ππ ) + (1 β = Ξ1π (ππ ) β
π2 ) Ξ1πβ1 (π+ , ππ ) π
π1 π π πβ1
π2 π ) (Ξ1πβ1 (ππ ) β 1 ππβ2 ) π π π + π2 1 ππ = Ξ2π (ππ ) β 1 Ξ πβ1 (ππ ) + 1 22 Ξ0πβ2 (ππ ) . π π (41)
π
πΌπ,π (π, π₯)
+ (1 β
= πΌπ,π (π, π₯) π
+ β π βπ
πβ1 πβ1 πΏβπβ1 (π, πΉπ,π ) πβπππππ₯ β πΏπ (π, πΉπ,π ) ππππ(π+1)π₯
βππ =1 (1 + π βπ πππππ₯ ) (1 + π π πβππππ₯ )
π=1
+ βππ
π =1
Ξ1π (π+ , ππ ) = ππ + (1 β
(π, ππβ ) ππππππ₯ , (35)
π
π
where π = {π1 , . . . , ππ }. Let us modify (20) in view of (38). For π = 1, we write
1
π
π
β (1 + ππ π₯) = β πΎπ (π) π₯π ,
πβ1 πΏπ+1
πππππ₯
(π, πΉπ,π ) π
π=1
βππ =1
(1 + π βπ
β
πβ1 πΏβπ
βπππ(π+1)π₯
(π, πΉπ,π ) π
πππππ₯ ) (1
+ π π πβππππ₯ )
(36)
In general, we can prove by the mathematical induction the following expansion [10]: π
π
π =0
π =0
Ξππ (π+ , ππ ) = βπΎπ (π+ ) ππβπ = β(β1)π
with the error π
π
π,π (π, π₯) =
Now, let us modify (24) in view of (38). According to (20) and (42), we get (note that π+ = πβ )
1
π
βπ =1 (1 + π βπ πππππ₯ ) (1 + π π πβππππ₯ )
+
π
1
(37)
Γ β πΏππ (π, ππβ ) ππππππ₯ .
π =0
π
(43)
Similar to (42), we can show that
The QPR interpolation is undefined until parameters π π are unknown. Hence, determination of these parameters is a crucial problem for realization of the QPR interpolation. First, we assume that π = 1, . . . , π,
π=0
πΎ (π) πβπ = β(β1)π π π β πΎπ (π+ ) Ξ π+πβπ (ππ ) . π π =0 π=0
|π|>π
π βπ
π
π
βπ =1 (1 + π βπ πππππ₯ ) (1 + π π πβππππ₯ )
π = ππ = 1 β π , π
π
πΏππ (π, ππ ) = β πΎπ (π+ ) βπΎπ (πβ ) ππ+πβπ
Γ β πΏππ (π, ππβ β πΉπ,π ) ππππππ₯ |π|β€π
πΎπ (π) πβπ Ξ (π ) . ππ πβπ π (42)
(38)
where ππ are some new parameters independent of π. In the next section, we investigate convergence of the QPR interpolation independent of the choice of parameters ππ . Then, we discuss some choices of these parameters. We also consider an approach connected with the idea of the FourierPade interpolation which leads to quasi-periodic FourierPade interpolation.
π
π
π=0
π=0
β πΎπ (π+ ) ππ+π = β (β1)π
πΎπ (π) πβπ Ξ (π ) . ππ π+π π
(44)
Then, from (43), we have π
πΏππ (π, ππ ) = β(β1)π π =0
π
πΎπ (π) πΎ (π) πβπ β (β1)π π π Ξπβπ π+π (Ξ πβπ (ππ )) . ππ π=0 π (45)
This leads to the following needed expansion: π
πΏππ (π, ππ ) = β(β1)π π =0
πβπ
πβπ
π
πΎπ (π) πΎ (π) β (β1)π π π Ξ2πβπβπ π+πβπ (ππ ) (46) π π π=0 π
2πβπβπ
as Ξ π+π (Ξ πβπ (ππ )) = Ξ π+πβπ (ππ ).
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Then,
and, consequently,
πΏππ€ (πΏππ (π, ππ )) π
= β(β1)π π =0
π
πΎπ (π) πΎ (π) 2πβπβπ β (β1)π π π Ξ2π€ π+π€ (Ξ π+πβπ (ππ )) . ππ π=0 π (47)
πΏππ€ (πΏππ (π, ππβ )) =
π =0
πΎπ (π) πΎ (π) β (β1)π π π Ξ2π€+2πβπβπ π+πβπ +π€ (ππ ) . ππ π=0 π
Γ πΏππ€ (πΏππ (π, π΅π (π))) + π (πβπβ2πβπβ1 ) . (55) Taking into account (49) and estimating Ξππ (π΅π (π)) π(πβπβ1βπ ) (see details in [9]), we get
(49)
1 1 ππ = β« π (π₯) πβππππ₯ ππ₯. 2 β1
πΏππ€ (πΏππ (π, (β1)π ππππ½π/(2π+π+1) ))
(50)
=(
π
(51)
π=0
(β1)π(π+1) . 2πβπβπ +π+1 π=ββ (2π Β± 1)
π
Lemma 3. Let π(π+2π+π) β π΄πΆ[β1, 1] for some π β₯ 0, π, π β₯ 1, and (β1) = π
(1) = 0,
π = 0, . . . , π β 1.
(52)
Let parameters π π be chosen as in (38). Then,
|π| > π, π σ³¨β β. (53) Proof. We have (see details in [5]) π+2π+π
β π=π
= β(β1)π π =0
π
πΎπ (π) πΎ (π) β (β1)π π π π π π=0 π
(59)
π πππ½π/(2π+π+1) Γ Ξ2π€+2πβπβπ ). π+πβπ +π€ ((β1) π
In view of (27), we write Ξππ ((β1)π ππππ½π/(2π+π+1) )
πΏππ€ (πΏππ (π, ππβ )) = π (πβ2π€β1 πβ2πβπ ) + π (πβπβ2πβπβ1 ) ,
ππβ =
π
ππ½ 2π 2 π+π ) βπΎπ (π) β (β1)π πΎπ (π) ( ) . (58) 2 πππ½ π =0 π=0
πΏππ€ (πΏππ (π, (β1)π ππππ½π/(2π+π+1) ))
First, we prove some lemmas.
π
βπ (π½, π) = (
Proof. From (49), we have
Γ β
(π)
(57)
where
π
β
(π)
ππ½ 2π€ (β1)π ππππ½π/(2π+π+1) βπ (π½, π) ) 2 π2π€+2π
+ π (πβ2π€β2πβ1 ) ,
π =ΜΈ 0, π΅0 (π) = 0,
Β± ππ,π,π (π) = β(β1)π πΎπ (π) β πΎπ (π) (2π β π β π + π)!
(56)
which completes the proof.
Let (β1)π+1 , π΅π (π) = 2(πππ)π+1
=
Lemma 4. Let π π be chosen as in (38), and let π½ β R be a constant. Then, the following estimate holds for π β Z as π β β:
We will frequently use the latest formula. We denote by ππ the πth Fourier coefficient of π as
π =0
(π β π)!
πΏππ€ (πΏππ (π, π΅π (π))) = π (πβ2π€βπβ1 πβ2π )
π
π
π΄ ππ (π) (π + 1)πβπ (2π + π + 1)π
π=0
πΏππ€ (πΏππ (π, ππ )) = β(β1)π
1 2π ππ
π=π π
(48)
we find that
π
β
Γβ
Taking into account that 2πβπβπ 2π€+2πβπβπ Ξ2π€ π+π€ (Ξ π+πβπ (ππ )) = Ξ π+πβπ +π€ (ππ ) ,
π+2π+π
π
π = (β1) β ( ) (β1)π ππππ½(πβπ)/(2π+π+1) π π
π=0
π
π = (β1)π ππππ½π/(2π+π+1) β ( ) (β1)π π
(60)
π=0
π π΄ ππ (π) (π + 1)πβπ (2π + π + 1)π 1 π΅π (π) β 2π ππ π=0 (π β π)!
β
Γβ
π‘
(πππ½) (β1)π‘ ππ‘
π‘=0 π‘!(2π
+ π + 1)π‘
(54)
π‘
(β1)π‘ (πππ½) π‘ ππ,π‘ , π‘=0 π‘!(2π + π + 1) β
+ π (πβπβ2πβπβ1 ) ,
= (β1)π ππππ½π/(2π+π+1) β
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where
Then, in view of (54), we get π
π ππ,π‘ = β ( ) (β1)π π π‘ . π π =0
(61)
β β ππ+π(2π+π+1)
π =ΜΈ 0
π+2π+π
=
Taking into account that ππ,π = 0, 0 β€ π < π and ππ,π = (β1)π π! (see [10], Lemma 2.1), we get
β π=π π
Γβ
Ξππ ((β1)π ππππ½π/(2π+π+1) )
1 2π ππ
π΄ ππ (π) (π + 1)πβπ (2π + π + 1)π (π β π)!
π=0
π
(β1)π (πππ½) πππ½π/(2π+π+1) = π + π (πβπβ1 ) . 2π ππ
Γ β π΅π+π(2π+π+1) (π) + π (πβπβ2πβπβ1 ) ,
(62)
π =ΜΈ 0
π
β
β=1
π=ββ
β β ππ,β β ππ+β+π(2π+π+1)
Then, π πππ½π/(2π+π+1) ) Ξ2π€+2πβπβπ π+πβπ +π€ ((β1) π
= (β1)π+π+π +π€ (
=
πππ½ 2π€+2πβπβπ πππ½(π+πβπ +π€)/(2π+π+1) π ) 2π
(β1)π+1 2π + π + 1 π+2π+π
Γ β π=π
+ π (πβ2π€β2π+π+π β1 )
Γ πβππ(πβ1)π/(2π+π+1) β Ξ¦π,π (π2ππ(π+β)/(2π+π+1) )
πππ½ 2π€+2πβπβπ ) 2π
β=1
π
β1 2π€ππ(π β1)/(2π+π+1) Γ βVβ,π π + π (πβπβ2πβ2 ) ,
+ π (πβ2π€β2π+π+π β1 )
π =1
(63)
where we used estimate (16). According to the Taylor expansion
which completes the proof together with (59). Lemma 5. Let π(π+2π+π) β π΄πΆ[β1, 1] for some π β₯ 0, π, π β₯ 1, and π
(β1) = π
(π)
(1) = 0,
π = 0, . . . , π β 1.
Ξ¦π,π (π2π€π(π+β)/(2π+π+1) ) 2π+π
= β
(64)
Let parameters π π be chosen as in (38). Then, the following estimate holds as π β β: πΏππ€ (πΏππ (π, πΉπ,π β ππβ )) = π (πβπβ2π€β2πβ1 ) + π (πβπβ2πβ2 ) , β π β€ π β€ π + π. (65) Proof. First, we estimate πΉπ,π βππβ . We have (see details in [5], at the beginning of the proof of Lemma 5)
π‘=0
π‘ 1 (π‘) Ξ¦π,π (β1) (π2ππ(π+β)/(2π+π+1) + 1) π‘!
β
π
β
π=ββ
β=1
π=ββ
π
β
β=1
π=ββ
β β β β ππ,β β ππ+β+π(2π+π+1) . πΉπ,π β ππβ = β ππ+π(2π+π+1)
(66)
(68)
+ π (πβ2πβπβ1 ) and relations π
π
π
β1 2π€ππ(π β1)/(2π+π+1) π β (π2ππ(π+β)/(2π+π+1) + 1) βVβ,π π =1
β=1
2πππ/(2π+π+1)
= (π
π
+ 1) ,
π = 0, . . . , π β 1,
we derive πΉπ,π β ππβ =
β β β β ππ,β β ππ+β+π(2π+π+1) , πΉπ,π = β ππ+π(2π+π+1)
π =ΜΈ 0
πβπ
π
π΄ ππ (π) (π + 1) 1 β π π π=0 2πβπ (ππ)π+1 (π β π)! π
= (β1)π+π+π +π€ ππππ½π/(2π+π+1) (
(π)
(67)
π+2π+π
β π=π π
Γβ
1 2π ππ
π΄ ππ (π) (π + 1)πβπ (2π + π + 1)π
π=0
(π β π)!
Γ β π΅π+π(2π+π+1) (π) π =ΜΈ 0
(69)
International Journal of Analysis +
7
(β1)π πβππ(πβ1)π/(2π+π+1) 2π + π + 1
Proof. We proceed as in the proof of Lemma 5 and derive
π π΄ ππ (π) (π + 1)πβπ 1 Γ β β π πβπ (ππ)π+1 (π β π)! π=π π π=0 2 π+2π+π
πβ1 Ξ¦(π‘) π,π
Γ [β
(β1)
π‘!
π‘=0 π+2π
+ β
(π2πππ/(2π+π+1) + 1)
π=ββ
π+2π+π
2ππ(π+β)/(2π+π+1)
+
π‘
+ 1)
π
πβ1 Ξ¦(π‘) π,π
Γ [β
π
π+2π
+ β π
π
β1 Γ βVβ,π π =1
(72)
π
Γ πΏΒ±π (π, (β1)π ππππ(2π β1βπ)/(2π+π+1) ) ]
Let parameters π π be chosen as in (38). Then, (β1)π+1 2π+1 ππ+2π+1
+ π (πβπβ2πβ2 ) . (76)
π΄ ππ (π) (π + 1)πβπ 2π (π β π)!(ππ)π+1
π=0
Γ [ (β1)π
Β± ππ,π,π (π)
π!
πβ1 Ξ¦
β β
π‘=0
(π‘) π,π
This completes the proof in view of Lemma 4 and the following estimate: (β1)
π‘!
β
(73)
π€ πΏΒ±π (πΏππ (π, β π΅π+π(2π+π+1) (π))) π=ββ
π‘
π‘ Γ β ( ) πΒ±πππ(2πβπ+1)/(2π+π+1) π π=0
=
+ π (πβπβ2πβ2 ) , π
πΏΒ±(π+1) (π, πΉπ,π ) = βπΏΒ±π (π, πΉπ,π ) + π (πβπβ2πβ2 ) , π
π
(β1)π+π+π€+1 Β± ππ,π,π+2π€ (π) 2(πππ)π+1 π2π€+2π π!
(77)
+ π (πβ2π€β2πβπβ2 ) .
Γ βπ (2π β π + 1, π) ]
π
π‘
β=1
Lemma 6. Let π(π+2π+π) β π΄πΆ[β1, 1] for some π β₯ 0, π, π β₯ 1, and
Γβ
π‘!
Γ β (π2ππ(π+β)/(2π+π+1) + 1)
+1 = π(1/π).
π = 0, . . . , π β 1.
Ξ¦(π‘) π,π (β1)
π‘=π
(71) We used also the fact that π
π‘
π‘ β( ) π π=0
Γ πΏΒ±π (π, (β1)π ππππ(2πβπ+1)/(2π+π+1) )
π =ΜΈ 0
2ππ(π+β)/(2π+π+1)
(π β π)!
(β1)
π‘!
π‘=0
πΏππ€ (πΏππ (π, β π΅π+π(2π+π+1) (π))) = π (πβ2π€β2πβπβ1 ) .
(1) = 0,
π΄ ππ (π) (π + 1)πβπ
πβπ (ππ)π+1 π=0 2
This completes the proof in view of (49), Lemma 4, and the following estimate [9]:
π
1 1 β 2π + π + 1 π=π ππ
Γβ
(70)
π
β
π‘!
+ π (πβπβ2πβ2 ) .
πΏΒ±π (π, πΉπ,π ) =
(π β π)!(π + 1)πβπ
π
β1 2π€ππ(π β1)/(2π+π+1) ΓβVβ,π π ] π =1
(β1) = π
π΄ ππ (π) (2π + π + 1)π
Γ πΏΒ±π (π, β π΅π+π(2π+π+1) (π))
π
π
1 2π ππ
π=0
β=1
(π)
π=π π
Γ β (π
(π)
β
Γβ
π‘
Ξ¦(π‘) π,π (β1)
π‘=π π
π+2π+π
π
πΏΒ±π (π, πΉπ,π ) =
(74)
πΏβπ (π, πΉπ,π ) = (β1)π+1 πΏπ (π, πΉπ,π ) + π (πβπβ2πβ2 ) . (75)
The proof of (77), for π = 0, can be found in [9]. General case can be proved similarly and we omit it. Estimates (74) and (75) can be proved similarly. Now, we present the main results of the paper.
8
International Journal of Analysis
Theorem 7. Let π(π+2π+π) β π΄πΆ[β1, 1] for some π β₯ 0, π, π β₯ 1, and π(π) (β1) = π(π) (1) = 0,
π = 0, . . . , π β 1.
(78)
Let parameters π π be chosen as in (38). Then, the following estimate holds for |π₯| < 1: π
π
π,π (π, π₯) =
(β1)π sin (ππ (π + (1/2)) π₯ β (ππ/2)) cos2π+1 (ππ₯/2) (2π)π+2π+1 π
Γβ
π΄ ππ (π) (π + 1)πβπ 2π
π=0
Hence, the last term in the right-hand side of (80) is π(πβπβ2πβ1 ). Then, by Lemma 5, πΏπ1 (πΏππ (π, πΉπ,π β ππβ )) = π (πβπβ2πβ3 ) + π (πβπβ2πβ2 ) , β π β€ π β€ π + π, and the fifth term is also π(πβπβ2πβ1 ). Therefore, π
π
π (π₯) π
π,π (π, π₯) = πβπππππ₯ πΏβπβ1 (π, πΉπ,π )
(π β π)!ππβπ ππ+1
π
β ππππ(π+1)π₯ πΏπ (π, πΉπ,π )
π+ (π) πβ1 Ξ¦(π‘) π,π (β1) π π,π,π Γ ((β1) ββ π! π‘! π‘=0
π
+ ππππππ₯ πΏπ+1 (π, πΉπ,π )
π σ³¨β β. (79)
Proof. We have from (37) by the Abel transformation (see transformation from (33) to (34) with π β‘ 1) π βπππππ₯ πΏβπβ1
π
(π, πΉπ,π ) π (π₯)
π
π,π (π, π₯) = π
π
βπ
(π, πΉπ,π ) π (π₯)
(80)
π
πΏβπ (π, πΉπ,π ) π (π₯)
+
1 π 1 π β πΏ (πΏ (π, ππβ β πΉπ,π )) ππππππ₯ π (π₯) π=βπ π π
+
1 β πΏ1 (πΏπ (π, ππβ )) ππππππ₯ , π (π₯) |π|>π π π
where π
πππ₯ π (π₯) = 4cos ( ) β (1 + π βπ πππππ₯ ) (1 + π π πβππππ₯ ) . 2 π =1 (81) 2
It is easy to verify that lim π (π₯) = 22π+2 cos2π+2 (
πββ
ππ₯ ). 2
(82)
According to Lemma 3, πΏπ1 (πΏππ (π, ππβ )) = π (πβ3 πβ2πβπ ) + π (πβπβ2πβ2 ) , |π| > π,
π
π
π (π₯) π
π,π (π, π₯) = πΏπ (π, πΉπ,π ) Γ ((β1)π (πβπππππ₯ + πβπππ(π+1)π₯ ) (86)
which concludes the proof in view of Lemma 6.
π πππππ₯ πΏπ+1
β πβπππ(π+1)π₯
Taking into account estimates (74) and (75), we get
+ π (πβπβ2πβ1 )
π (π₯)
+π
+ π (πβπβ2πβ1 ) .
β (ππππππ₯ + ππππ(π+1)π₯ ))
π πππ(π+1)π₯ πΏπ (π, πΉπ,π )
(85)
β πβπππ(π+1)π₯ πΏβπ (π, πΉπ,π )
π‘
π‘ Γ β ( ) π2πβπ+1 βπ (2π β π + 1, π)) π π=0 + π (πβπβ2πβ1 ) ,
π σ³¨β β (84)
π σ³¨β β.
(83)
Let us compare the results of Theorems 1 and 7. Theorem 1 investigates the pointwise convergence of the QP interpolation on (β1, 1) and states that for π(π+2π) β π΄πΆ[β1, 1] the convergence rate is π(πβπβπβ1 ) for π β₯ 1. Theorem 7 explores the pointwise convergence of the QPR interpolation and shows that convergence rate is π(πβπβ2πβ1 ) for π(π+2π+π) β π΄πΆ[β1, 1] and π β₯ 1. We see that for π = 2π both theorems are provided with the same rates of convergence by putting the same smoothness requirements on π, although the exact constants of the asymptotic errors are different. Then, we see that for π > π/2 the QPR interpolation has improved accuracy compared to the QP interpolation and improvement is by factor π(π2πβπ ). In this case, Theorem 7 puts additional smoothness requirement on π and comparison is valid if only the interpolated function has enough smoothness (for example, if it is infinitely differentiable). It is worth recalling that parameter π indicates the size of the Vandermonde matrix (12) that must be inverted for realization of the QP and QPR interpolations. It is wellknown that the Vandermonde matrices are ill-conditioned and standard numerical methods fail to accurately compute the entries of the inverses when the sizes of the matrices are big. Hence, from practical point of view, it is more reasonable to take π small (π β€ 6) and additional accuracy obtain by increasing π.
International Journal of Analysis
9 Γ10β25
Γ10β25 8
1.5
6
1
4 0.5
2
β0.6
β0.4
β0.2
0.2
0.4
0.6
β0.6
β0.4
β0.2
(a)
Figure 1: The graphs of the absolute errors (94)) (b) while interpolating (90).
π
(87)
Another choice is based on the asymptotic estimates of Theorems 7 and 8. If it is possible to vanish the following expressions by the choice of parameters ππ : (β1)π
Similarly, the case π = 0 can be analyzed. We present the corresponding theorem without the proof which can be performed as the above one.
π = 0, . . . , π β 1.
sin (πππ₯) (β1)π+π π+2π+1 cos2π+1 (ππ₯/2) (2π) π΄ ππ (π) 2π
π
Γβ π=0
ππ ππ+1
+ π (π
(π β π)!π!
βπβ2πβ1
+ ππ,0,π (π)
π!
πβ1 Ξ¦(π‘) π,π
ββ
π‘=0
(β1)
π‘!
π‘
or + ππ,0,π (π) = 0,
π = 0, . . . , π,
(89)
π
π
π,π (π, π₯) = π (πβπβ2πβ1 ) ,
|π₯| < 1.
(93)
For example, when π = π = 4 and π = 2, we find π1 = β0.3047909 β 0.4539705π,
Comparison with Theorem 2 shows improvement by factor π(π2π ) for any π β₯ 1 if π has enough smoothness.
π2 = 0.9640011 β 0.1481304π, π3 = 6.375385 + 0.000739π,
4. Parameter Determination in Rational Corrections
(94)
π4 = 12.877189 β 0.000022π.
Till now, we did not discuss the problem of parameters ππ determination as Theorems 7 and 8 are valid for all choices. Now, let us consider some choices with the corresponding numerical results. Let 4
(92)
then, Theorems 7 and 8 will be provided with improved estimate
).
π (π₯) = (1 β π₯2 ) sin (π₯ β 1) .
(91)
π = 0, . . . , π
(88)
Let parameters π π be chosen as in (38). Then, the following estimate holds for |π₯| < 1 as π β β: π
+ ππ,π,π (π)
π‘ Γ β ( ) π2πβπ+1 βπ (2π β π + 1, π) = 0, π π=0
Theorem 8. Let π(π+2π+1) β π΄πΆ[β1, 1] for some π β₯ 0, π β₯ 1, and
π
π,0 (π, π₯) =
0.6
on interval [β0.7, 0.7] for π = 2 and π = 4 when ππ = π (a) and with optimal ππ (see
π σ³¨β β, π = 0.
π(π) (β1) = π(π) (1) = 0,
0.4
(b) π |π
256,π (π, π₯)|
Note that for π = 1 the second term in the brackets of + (1, π) = 0. Hence, estimate (79) vanishes and also ππ,1,0 π
π,1 (π, π₯) = π (πβ2πβ1 ) ,
0.2
(90)
One choice is ππ = π which shows satisfactory numerical results (see Figure 1).
4 With this choice π
256,2 (π, π₯) = π(πβ13 ) for (90). Figure 1 compares the choices ππ = π (a) and optimal choice (104) (b). As it was expected the precision of interpolation is higher in (b) compared to (a). The third choice is not connected with (38) and allows determining parameters π π immediately along the ideas of the Fourier-Pade interpolation ([11]). This approach is more complex as π π must be recalculated for each function π and for each π and as a consequence leads to nonlinear interpolation but, however, is much more precise when |π₯| < 1.
10
International Journal of Analysis Γ10β35 8
[3] 6
[4]
4 2
[5] 0.7
β0.7
π Figure 2: The graph of |π
256,π (π, π₯)| on interval [β0.7, 0.7] for π
[6]
=2 and π = 4 while interpolating (90) by the quasi-periodic FourierPade interpolation (see (96)). [7]
More specifically, parameters π can be determined from the following system: πΏππ (π, πΉπ,π ) = 0,
|π| = π β π + 1, . . . , π.
(95)
[8]
We will refer to this interpolation as quasi-periodic FourierPade interpolation. For example, when π = 2, π = 4, and π = 256, then, for (90), we get
[9]
π 1 = 0.922 + 0.010π,
π 2 = 0.952 + 0.010π,
π 3 = 0.975 + 0.010π,
π 4 = 0.994 β 0.011π,
π β1 = 0.994 + 0.011π,
π β2 = 0.975 β 0.010π,
π β3 = 0.952 β 0.010π,
π β4 = 0.922 β 0.010π.
[10]
(96)
4 Figure 2 shows the graph of |π
256,2 (π, π₯)| for function (90) with π π from (96). Comparison with Figure 1 shows high precision of the quasi-periodic Fourier-Pade interpolation for |π₯| < 1 compared to other choices of parameters π π . Theoretical analysis of convergence of the quasi-periodic Fourier-Pade interpolation will be carried out elsewhere.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments The authors acknowledge the reviewers and the editors for their valuable suggestions and constructive comments that helped to improve the paper.
References [1] A. Nersessian and N. Hovhannisyan, βQuasiperiodic interpolation,β Reports of the National Academy of Sciences of Armenia, vol. 101, no. 2, pp. 115β121, 2001. [2] A. Nersessian and N. Hovhannisyan, βMinimization of errors of the polynomial-trigonometric interpolation with shifted
[11]
nodes,β 2007, Preprint of the Institute of Mathematics of the National Academy of Sciences of Armenia, http://math.sci .am/upload/preprints/toPrprt.pdf. L. Poghosyan, βOn a convergence of the quasi-periodic interpolation,β in Proceedings of the 2012 International Workshop on Functional Analysis, Timisoara, Romania, October 2012. L. Poghosyan, βOn L2 -convergence of the quasi-periodic interpolation,β Reports of the National Academy of Sciences of Armenia, vol. 113, no. 3, pp. 240β247, 2013. L. Poghosyan and A. Poghosyan, βAsymptotic estimates for the quasi-periodic interpolations,β Armenian Journal of Mathematics, vol. 5, no. 1, pp. 34β57, 2013. L. Poghosyan and A. Poghosyan, βOn a pointwise convergence of the quasi-periodictrigonometric interpolation,β Proceedings of the National Academy of Sciences of ArmeniaβMathematics. In press. L. Poghosyan and A. Poghosyan, βConvergence acceleration of the quasi-periodic interpolation by rational and polynomial corrections (abstract),β in Proceedings of the 2nd International Conference Mathematics in Armenia: Advances and Perspectives, Tsaghkadzor, Armenia, August 2013. I. Gohberg and V. Olshevsky, βThe fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices,β Journal of Complexity, vol. 13, no. 2, pp. 208β234, 1997. A. Poghosyan, βOn a fast convergence of the rationaltrigonometric-polynomial interpolation,β Advances in Numerical Analysis, vol. 2013, Article ID 315748, 13 pages, 2013. A. Nersessian and A. Poghosyan, βOn a rational linear approximation of Fourier series for smooth functions,β Journal of Scientific Computing, vol. 26, no. 1, pp. 111β125, 2006. G. A. Baker and P. Graves-Morris, Pade Approximants, Encyclopedia of Mathematics and Its Applications, vol. 59, Cambridge University Press, Cambridge, UK, 2nd edition, 1966.
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