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Vestnik KRAUNC. Fiz.-Mat. nauki. 2022. vol. 38. no. 1. P. 131-146. ISSN 2079-6641
MSC 65D30, 65D32 Research Article
Construction of optimal interpolation formula exact for trigonometric functions by Sobolev's method
Kh.M. Shadimetov1'2, A.K. Boltaev2'3, R.I. Parovik3,4
1 Tashkent State Transport University, 1 Odilxojaev str., Tashkent 100167, Uzbekistan
2 V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b University str., Tashkent, 100174, Uzbekistan
3 National University of Uzbekistan named after Mirzo Ulugbek, 4 University str., Tashkent, 100174, Uzbekistan
4 Vitus Bering Kamchatka State University, 683032, Petropavlovsk-Kamchatskiy, Pogranichnaya str., 4, Russia
E-mail: aziz_boltayev@mail.ru, kholmatshadimetov@mail.ru
The paper is devoted to derivation of the optimal interpolation formula in W22'0)(0,1) Hilbert space by Sobolev's method. Here the interpolation formula consists of a linear combination of the given values of a function cp from the space w22'0)(0, 1). The difference between functions and the interpolation formula is considered as a linear functional called the error functional. The error of the interpolation formula is estimated by the norm of the error functional. We obtain the optimal interpolation formula by minimizing the norm of the error functional by coefficients Cp(z) of the interpolation formula. The obtained optimal interpolation formula is exact for trigonometric functions sinx and cosx. At the end of the paper we give some numerical results which confirm the numerical convergence of the optimal interpolation formula.
Key words: extremal function, error functional, Hilbert space, optimal interpolation formula, optimal coefficients, Sobolev's method.
d DOI: 10.26117/2079-6641-2022-38-1-131-146
Original article submitted: 16.02.2022 Revision submitted: 26.03.2022
For citation. Shadimetov Kh.M., Boltaev A.K., Parovik R.I. Construction of optimal interpolation formula exact for trigonometric functions by Sobolev's method. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 38: 1,131-146. d DOI: 10.26117/2079-6641-2022-38-1-131-146
The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)
© Shadimetov Kh. M., Boltaev A.K., Parovik R.I., 2022
Funding. The work was done without financial support
Introduction and statement of the Problem
It is well-known that most problems of science and technology are studied according to mathematical models. Such models are often expressed as differential, integro-differential, integral or functional equations. Solutions of such equations are sought in Hilbert spaces and they are mainly solved approximately using various type of approximation and interpolation methods of computational mathematics. In order to obtain effective approximate methods it is needed to study structures of the considered spaces. The internal properties of a space can be studied, for instance, by reproducing kernel functions in the theory of reproducing kernels in Hilbert spaces [4, 5] and by extremal functions in the theory of optimal formulas in Hilbert spaces [18, 22]. Presently, the reproducing kernel Hilbert space methods are widely used in numerical analysis, in particular, in numerical solution of fractional differential and integro-differential equations, see, for instance, [1, 2, 3] and references therein. The reproducing kernel Hilbert spaces are studied in probability and statistics [5]. In the recent work [14], authors studied the reproducing kernel functions in the standard Sobolev space W| and their application to tractability of integration. Using the extremal functions in various Hilbert spaces the optimal integration formulas were studied in the works [10, 13, 15, 17, 20, 21] and the interpolation formulas and splines were obtained in [7, 9, 12, 16, 19].
In the present work we study the problem of construction of an optimal interpolation formula based on variational method.
Assume, we have the table of the values (p(xp), p = 0,1,...,N of a function pp at the points xp e [0,1]. It is required to approximate the function pp by another more simple function Pp, i.e.
N
<p(x) = P<p(x)=£ Cp(x) ■ cp(xp) (1)
P=0
which satisfies the following interpolation conditions
(p(xp) = Pp(xp), p = 0,1,...,N.
Here Cp(x) and xp (e [0,1]) are called coefficients and nodes of the interpolation formula (1), respectively. By W(2,0)(0,1) we denote the set of all functions pp defined on [0,1] which posses an absolutely continuous first derivative on [0,1] and whose second derivative is in L2(0,1). The set w(2,0)(0,1) under the pseudo -inner product
(P,4) =
(p"(x) + p(x))«(x) + ^(x)) dx.
is a Hilbert space if we identify functions that differ by (x) and (x). Here we consider the norm
1
1/2
I^W(2'0)(0,1)
(p "(x) + (p(x))2dx
For the fixed z e [0,1] the error
(t,p) = p(z)- Pp(z) (2)
of the interpolation formula (1) is a linear functional. Here, in fixed z e [0,1],
N
t(x,z) = 6(x - z)-£ Cp (z)6(x - xp),
p=0
and is the error functional of the interpolation formula (1) belongs to the space W22,0)*(0,1). Here w22,0)*(0, 1) is the conjugate space to the space w22,0)(0,1), 6 is Dirac's delta-function.
By the Cauchy-Schwarz inequality the absolute value of the error (2) is estimated as follows
|(t,P)! < |MIW(2,0) ■ P||W(2,0)*,
where
||«|W(2,0)* = SUP j-r-.
W p, ||p|=0 ||p|w22,0) Therefore, in order to estimate the error of the interpolation formula (1) on functions of the
space W22,0) it is required to find the norm of the error functional t in the conjugate
(2 0)*
space W2 ' . That is we get the following problem.
Problem 1. Find the norm of the error functional t of the interpolation formula (1) in the space w22,0)*.
It is clear that the norm of the error functional t depends on the coefficients Cp(z) and the nodes xp. The problem of minimization of the quantity ||t|| by coefficients Cp(z) is the linear problem and by nodes xp is, in general, nonlinear and complicated problem. We consider the problem of minimization of the quantity ||t|| by coefficients Cp(z) when nodes xp are fixed.
The coefficients Cp(z) (if there exist) satisfying the equality
(20U = inf p||w(2,0)* (3)
W(2,0)* Cß (zj W2 , )
are called the optimal coefficients and corresponding interpolation formula Pp(z) =
N (2,0)
Cp(z)p(xp) is called the optimal interpolation formula in the space W( , There-
p=0
fore, for construction of the interpolation formula we should solve the next problem.
Problem 2. Find the coefficients Cp(z) which satisfy equality (3) when the nodes xp are fixed.
The rest of the paper is organized as follows. In Section 2, using the extremal function, the norm of the error functional is found. Existence and uniqueness of the optimal interpolation formula of the form (1) is discussed in Section 3. In Section 4 some preliminaries are given. Section 5 is devoted to calculation of coefficients of the optimal interpolation formula (1). Finally, in Section 6 some numerical results which confirm the theoretical results of the paper are presented.
The extremal function and the norm of the error functional I
Here we find explicit form of the norm of the error functional For finding the explicit form of the norm of the error functional I in the space w22'0) we use its extremal function which was introduced by Sobolev [18, 19]. The function —« from W22,0) space is called the extremal function for the error functional I if the following equality is fulfilled
= P||W(2,0)*- |WHW(2,0)
Further, for convinience we denote p||w(2,0)* by ||«||. According to the Riesz theorem
'w2
any linear continuous functional « in a Hilbert space is represented in the form of a inner product. So, in our case, for any function p from w22'0) space we have
(£,p) = <—«,p). (4)
Here —« is the function from w22'0) is defined uniquely by functional I and is the extremal function.
It is easy to se satisfies the following equalities
It is easy to see from (4) that the error functional defined on the space w22'0),
(£,sin(x)) = 0, (5)
(£,cos(x))= 0. (6)
The equalities (5) and (6) mean that our interpolation formula is exact for the functions (x) and (x).
The equation (4) was solved in [6] and for the extremal function —« was obtained the following expression
—«(x) = («* G2)(x) + di sin(x) + d2cos(x), (7)
where
x
G2(x) = 4 [sin(x)-xcos(x)], (8)
It should be noted that formula (7) was also found in the work [11] and used in the construction of the optimal quadrature formulas and interpolation splines minimizing the semi-norm.
* is the operation of convolution which for the functions f and g is defined as follows
(f * g)(x) =
f(x - y)g(y)dy =
f(y)g(x - y)dy. (9)
Now we obtain the norm of the error functional Since the space w22'0) is the Hilbert space, then by the Riesz theorem about general form of linear continuous functional, we have
(«,—«) = ||i||-|M = ||«|2. (10)
Hence, using (7) and (8), taking into account (9) and (10), we get
= (W =
-oo
X
N
5(x-z)-^Cp(z)6(x-xp) I =
p=0
( N \
= x I G2(x-z)-^Cp(z)G2(x-xp) I dx.
V p=0 )
Hence, keeping in mind that G2(x), defined by (8), is the even function, we have
N N N
II«!2 = XX Cp(z)CT(z)G2(xp - xT)- 2 X Cp(z)G2(z - xp). (11)
p=0y=0 p=0
Thus, Problem 1 is solved.
Further, we solve Problem 2.
Existence and uniqueness of the optimal interpolation formula
Assume that the nodes xp of the interpolation formula (1) are fixed. The error functional (2) satisfies the conditions (5) and (6). The norm of the error functional « is a multivariable function with respect to the coefficients Cp(z) (p = 0,N). For finding the point of the conditional minimum of the expression (11) under the conditions (5) and (6) we apply the Lagrange method. Consider the function
¥(C0(z),Ci(z),...,CN(z),di,d2) = II«!2 - 2 (di(«,sin (x))+ d2(«,co s(x))).
Equating to 0 the partial derivatives of the function ¥ by Cp(z) (p = 0,N), di and d2, we get the following system of N + 3 linear equations of N + 3 unknowns
N
^CT(z)G2(xp - xT ) + di sin(xp) + d^œ s(xp) = G2(z - xp), p = 0,N, (12)
Y=0
N
(z^n(xT) =sin(z), (13)
Y=0
N
(z^s(xT) ^os(z), (14)
Y=0
where G2(x) is defined by equality (8).
The system (12)-(14) has a unique solution and this solution gives the minimum to ||«||2 under the conditions (13) and (14).
2
The uniqueness of the solution of the system (12)-(14) is proved as the uniqueness of the solution of the system (24)-(26) of the work [22].
Therefore, in fixed values of the nodes xp the square of the norm of the error functional being quadratic function of the coefficients Cp(z), has a unique minimum in some concrete value Cp(z) = Cp(z).
Remark 1. It should be noted that by integrating both sides of the system (12)-(14) by z from 0 to 1 we get the system (29)-(30), when w = 1 of the work [11]. This means that by integrating the optimal interpolation formula (1) in the space w22'0) we get the optimal quadrature formula of the form (1) in the same space (see [11]).
Remark 2. It is clear from the system (12)-(14) that for the optimal coefficients the following are true
Cp(hY) = | J, ^ y = 0,1,...,N, p = 0,1,...,N.
Below for convenience the optimal coefficients Cp(z) we remain as Cp(z).
Preliminaries
Below, mainly the concept of discrete argument functions is used. The theory of discrete argument functions is given in [18, 22]. For completeness we give definitions about functions of discrete argument.
Assume that the nodes xp are equally spaced, i.e., xp = hp, h = N, N = 1,2,.... Definition 1. The function p(hp) is a function of discrete argument, if it is given on some set of integer values of p.
Definition 2. The inner product of two discrete argument functions p(hp) and — (hp) is given by
oo
[p(hp),-(hp)] = ^ p(hp) ■ —(hp),
p=-oo
if the series on the right hand side converges absolutely.
Definition 3. The convolution of two functions p(hp) and —(hp) is the inner product
oo
p(hp) *-(hp) = [p(hY),-(hp -hY)]= ^ p(hY) ■ -(hp -hY).
Y=-oo
Furthermore, in our computations we need the discrete analogue D2(hp) of the
d4 d2
differential operator d^r + 2-^1^x2 +1 which satisfies the equality
D2(hp) * G2(hp)= 5d(hp), (15)
where G2(hp) is the discrete argument function corresponding to the function G2(x) defined by (8), 6d(hp) is the discrete delta-function, i.e., 6d(hp) = 0 for p = 0 and 6d(0) = 1.
In [8] the discrete analogue D2(hp) was constructed, when w = 1 and the following was proved.
d4 d2
Theorem 1 The discrete anal°gue of the differential operator -xt + 2|? +1 satisfying the equation (15) has the form
D2(hß)= pi
A■ A|ß|—1, |ß| > 2, I + A, |ß| = I,
P2 + A, ß = 0.
(16)
where
2
Pi =
sinh—h-cosh'
p2 =
2hcos2h—sin 2h sinh—h-cosh '
A=
4h2sin4(h) ■ A2
(A2 — 1)(sin h—hcos h)2'
A=
2h—sin(2h) — 2sin(h) ■ yh2 — sin2(h) 2(hco s(h) — sin(h))
|A| < 1,h is small parameter.
Several properties of the discrete argument function D2(hp) were proved in [8]. Here we give the following.
d4
Theorem 2. The discrete analogue D2(hp) of the differential operator + 2-d22 +1 satisfies the following equalities
1) D2(hß) *sin(hß) = 0,
2) D2(hß) *cos(hß) = 0,
3) D2(hß) * (hß)cos(hß) = 0,
4) D2(hß) * (hß) sin(hß) = 0.
Optimal coefficients of the optimal interpolation formula (1)
In this section we give solution of Problem 2 and we find explicit formulas for optimal coefficients Cß(z), ß = 0,1,...,N of the optimal interpolation formula (1) using
-4 -2
the discrete analogue D2(hß) of the differential operator + 2—^ +1 • The following hold
Theorem 3. Coefficients of the optimal interpolation formula (1) with equal spaced nodes in the space W(2,0) have the following form
Q(z) = pi
P2G2(z) + G2(z—h) + G2(z + h) — (a—+ cos(z)+4zsin(z)') • sin(h) +
+A
N
X AYG2(z — hy) + M1 + AnN1
Y=0
(17)
Cß(z) = pi
+A
G2(z — h(ß — 1))+ P2G2(z — hß) + G2(z — h(ß +1)) + N
X A|ß—yiG2(Z—hy)+ AßM1 + AN—ßN1
Y=0
, ß = 1,2,..,N — 1,
Cn(Z)= pi
r> i 1 a I r> i 1 i a i 4at^ras(z)-Mn(z) -ha,
P2G2(z-1) + G2(z-1 - h) +--1—4—ï--sin(h)+
+G2(z -1 + h) + A
N
^AN-YG2(z - hy)+ AnM1 + N1
Y=0
1
2
dk = ö(ak + a+), k = 1,2,
where
M1 =
A
A2 - 2Aco s(h) +1
1
- 4 (cos(z)+ zsin (z))sin(h)
G2(z + h) + hcos(z + h) - AG2(z) - a- sin(h)-
(19)
(20)
4
XAY ■ (hy) -cos(z + hy), (21)
Y=1
N1 =
A
4 cos 1
A2 - 2Aco s(h) +1
+ hcos(z-1 - h) - AG2(z -1 )
4a+-cos(z)-zsin^ -,
1 -sin(h) + G2(z -1 - h)+
4
X AY ■ (hy) -cos(z-1 - hY), (22)
Y=1
and pi,p2,A are defined by Theorem 1, a- and a+ are defined from (32)-(33). To prove this theorem, it is necessary to perform the following calculations. Suppose that C((z) = 0 when (3 < 0 and (3 > N. Thus we have the following problem. Problem 3. Find the discrete functions C((z), (3 = 0,1,...,N, di and d2 which satisfy the system (12)-(14).
Further we investigate Problem 3 which is equivalent to Problem 2. Instead of C((z) we introduce the following functions
V2(hß) = G2(hß) * Cß(z),
u2(hß) = v2 (hß) + d^n(hß)+ d2cos(hß).
(23)
Now we should express the coefficients Cp(z) by the function u(hp). For this, using (15) and (16) equalities for optimal coefficients, we have
Cß(z) = D2(hß) *u2(hß).
(24)
Thus if we find the function u2(hß) then the optimal coefficients Cß(z) will be found from equality (24).
In order to calculate the convolution (24) it is required to find the representation of the function U2(hß) at all integer values of ß. From equality (23) we get that U2(hß) = G2(z — hß) when hß e [0,1]. Now we find the representation of the function U2(hß) when ß < 0 and ß > N.
Since Cß(z) = 0 when hß e [0,1] then
Cß(z) = D2(hß) *U2(hß)= 0, hß / [0,1].
1
1
Now we calculate the convolution V2(hp) = G2(hp) * Cp(z) when hp ^ [0,1]. Suppose p < 0 and p > N then taking into account equalities (8), (13) and (14), we
have
a—sin(hß) + a—cos(hß) + ^ ■caste—hß), ß < 0,
U2(hß) = 4 G2(z—hß),
0 < ß < N,
(25)
k a+sin(hß) + a+cos(hß) — hß ■coste—hß), ß>N,
where a- , a+, a— and a+ are unknowns. From (25) when p = 0 and p = N we get
a2 = G2(z),
+
1
a2 =-T
2 cos1
G2(z — 1) +
cos(z — 1) + .
— a+ sin 1
4
(26) (27)
Thus, putting (26) and (27) to (25) we have the following explicit form of the function
U2(hp):
' a— sin(hß) + G2(z)cos(hß) + hßcos(z—hß), ß < 0,
U2(hß) = <
G2 ( z - hß ) ,
+ sin(h.ß—1) . cos(hß)
a
cos 1
+
cosl
[ — hfœs(z—hß),
G2(z — 1) +
cos(z—1 )
4
0 < ß < N,
ß >N.
(28)
In the last expression of the function u2(Hß) we have only two unknowns a- and a+. Hence using (16) and (28) we get the following problem. Problem 4. Find the solution of the equation
D2(hß) *u2(hß) = 0, hß / [0,1],
having the form (28). Here a- and a+ are unknowns.
Unknowns a- and a+ we find from the equation
D2(hß) *u2(hß) = 0,
(29)
when p = — 1 and p = N +1. From the last equation
From the system (28) in the case p = — 1 and after some simplifications we have the following system of equations
M11a1 -cos1 — M12a+ = —T1 -cos1
(30)
where
M11 = sin(2h) + P2 sin(h) + A(A2_A£Lhh)+1), M12 = ÄA^I^rh^, T1 ^os(z)+4z"n(z) ■ (M11 + Mf) — S11 — S12 — S13,
S11 = P2G2(z + h) + G2(z+2h) + A(G2(zA+(^rirr+h+—AG2(z)) — Ah £ ayycos(z + hY),
_ AAN + ^n(h)
A(A2—2Aco s(h)+1)
Y=1
N
S12 = G2(z) + AY_ AYG2(z—hY),
Y=0
oo
c ,N+1 G2(z -1 - h) + hcos(o -1 - h) + AG2(z -1) , , , A
S'3 = A +--A2-2Aeo-h) + l--4 LA +TYCOS<z- 1 - hY).
Y=1
Now, from (28) in the case p = N +1 doing some calculations we get the next system of equations
M12a- - cos 1 - M11a+ = -T2 -co s1 (31)
where
T2 = cos^sm^ . (M12 + Mf) - S21 - S22 - S23,
521 = AN+1 - G2(0+A2+-12Ca0cos0+)+-AG2(0) - 4 £ AN+Y^s(o + hY),
Y=1
N
522 = G2(o- 1) + A £ AN-YG2(o- hY),
523 = P2G2 (0 -1 - h) + G2(o -1 - 2h) + A(G2 (0-1-hA+^ 0^^+AG2(0-1))-
00
- 4Ah £ AYYCO s(o - 1 - hY).
Y=1
1 linlrnATITTIO rt ^ and U-|
Thus for the unknowns a- and a+ we solving the system (30)-(31) of equations and
get
M12T2 - M11T1
+ M11T2 - M12T1
a+= M2.-M2. œs1- (33)
11 '12
Then taking into account (20), using (26) and (27) we have
dk = 2(ak+ a+), k = 1,2.
Proof. Now we find the coefficients Cp(z), P = 0,1,...,N . From (23), taking into account (28), we have
Cp(z) = Z. °2(hp + Hy) [G2(z-Hy)- a-sin(HY)-^HYl(cos(z) + zsin(z))] +,
Y=1 L J
OO r + . , , -|
+ H D2(H(N + Y k p)) G2(z k 1 k hy) + ¿+rSm(HY)-sYHcos(z) + zsin(z)) +
Y=1 J
N
+ D2(Hp - HY)G2(Z- HY),
Y=0
where P = 0,1,...,N.
From here, using (16) and taking into account (21) and (22), after some calculations we arrive at expressions of the coefficients Cp(z), P = 0,1,...,N which are given in the assertion of the theorem. Theorem 3 is proved. □
Numerical results
In this section we give some numerical results.
First, when N = 5 using Theorem 3, we get the graphs of the coefficients of the optimal interpolation formulas
5
p(z) = Pp(z) = Y_ Cp(z)p(hp), z e [0,1].
p=0
They are presented in Fig 1. These graphical results confirm Remark 2 for the case N = 5 , i.e. for the optimal coefficients the following hold
Cp(hY)= 6|3Y, P,Y = 0,1,...,5,
where 6pY is the Kronecker symbol.
Now, in numerical examples, we interpolate the functions
p1 (z) = z2, p2(z) = ez and p3(z) = sin(z)
by optimal interpolation formulas of the form (1) in the cases N = 5, 10, using Theorem 3. For the functions pt, i = 1,2,3 the graphs of absolute errors |pt(z) — Ppi(z)|, i = 1,2,3, are given in Fig 2, Fig 3, Fig 4. In these Figures one can see that by increasing value of N absolute errors between optimal interpolation formulas and given functions are decreasing.
C_0(z)
C_l(z)
C_2(z)
8 1
C_3(z)
C_4(z)
C_5(z)
Fig. 1. Graphs of coefficients of the optimal interpolation formulas (1) in the case N = 5
The Figure 4 shows exactness our optimal interpolation formula for the function ( x) .
Fig. 2. Graphs of absolute errors for N = 5 and N = 10: |z2 - Pzi (z)|
0.0025 0.0020 0.0015 0.0010 0.0005 0
0.2 0.4 0.6 0.8 1
z
0.2 0.4 0.6 0.8 1
z
Fig. 3. Graphs of absolute errors for N = 5 and N = 10: |exp(z)- PeXp(z)(z)
-45
1.4 X 10
-45
1. X 10
-46
6. X 10
-44,
1.5 x 10
-44
1. x 10
-45
5. x 10
0
-45
-5. x 10
Fig. 4. Graphs of absolute errors for N = 5 and N = 10: |sin(z) — PSin(z)(z)| Conclusion
The paper has been devoted to construction of the optimal interpolation formula exact for the trigonometric functions sin(x) and cos(x). For construction of the optimal interpolation formula we used Sobolev method wich is based on the discrete ana-
-4 -2
logue D2(Hp) of the differential operator + 2-X2 +1. Applying the discrete analogue D2(Hp) we have obtained the explicit expressions for optimal coefficients (see Theorem 3) which are very useful in applications. At the end, numerical results that show the reliability of optimal interpolation formula we constructed.
Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.
Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.
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Shadimetov Kholmatvay MakhkambaevichA - D. Sci. (Phys. & Math.), Professor, Chief of the Department of Computer Science and computer graphics, Tashkent State Transport University, ©ORCID 0000-0002-4183-6184.
Boltaev Aziz KuziyevichA - Ph. D. (Phys. & Math.), senior staff scientist at the laboratory of Computational Mathematics, V. I. Ro-manovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, ORCID 0000-0002-8329-4440.
Parovik Roman IvanovichA - D. Sci. (Phys. & Math.), Associate Professor, Professor of the Depart. Math.& Phys., Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatskiy, Russia, ©ORCID 0000-0002-1576-1860.
Вестник КРАУНЦ. Физ.-мат. науки. 2022. Т. 38. №. 1. С. 131-146. ISSN 2079-6641
УДК 519.652 Научная статья
Построение оптимальной интерполяционной формулы методом Соболева точных для тригонометрических функций
Х.М. Шадиметов1'2, А. К. Болтаев2'3, Р. И. Паровик3,4
1 Ташкентский государственный транспортный университет, Одилжожаев ул. 1, Ташкент 100167, Узбекистан
2 Институт математики имени И.В. Романовского, АН Уз, ул. Университетская, 4б, Ташкент, 100174, Узбекистан
3 Национальный университет Узбекистана имени Мирзо Улугбека, ул. Университетская, 4, Ташкент, 100174, Узбекистан
4 Камчатский государственный университете имени Витуса Беринга, 683032, Петропавловск-Камчатский, ул. Пограничная, 4, Россия.
E-mail: aziz_boltayev@mail.ru, kholmatshadimetov@mail.ru
Работа посвящена построению оптимальной интерполяционной формулы методом Соболева в гильбертовом пространстве Wf'0)(0,1). Здесь
N
интерполяционная формула состоит из линейной комбинации Y. Cßф(xß)
ß=0
заданных значений функции ф из пространство Wj2'0)(0,1). Отличие функций от интерполяционной формулы рассматривается как линейный функционал, называемый функционалом погрешности. Погрешность интерполяционной формулы оценивается нормой функционала погрешности. Мы получаем оптимальной интерполяционной формулы путем минимизации нормы функционала погрешности на коэффициенты Cß (z) интерполяционной формулы. Полученная оптимальная интерполяция формула точна для тригонометрических функций sinx и cosx. В конце статьи мы приводим некоторые численные результаты, которые подтверждают наши теоретические результаты.
Ключевые слова: экстремальная функция, функционал погрешности, гильбертово пространство, оптимальная интерполяционная формула, оптимальные коэффициенты, метод Соболева.
d DOI: 10.26117/2079-6641-2022-38-1-131-146
Поступила в редакцию: 16.02.2022 В окончательном варианте: 26.03.2022
Для цитирования. Shadimetov Kh. M., Boltaev A.K., Parovik R.I. Construction of optimal interpolation formula exact for trigonometric functions by Sobolev's method // Вестник КРАУНЦ. Физ.-мат. науки. 2022. Т. 38. № 1. C. 131-146. d DOI: 10.26117/20796641-2022-38-1-131-146
Финансирование. Работа выполнена без финансовой поддержки
Конкурирующие интересы. Конфликтов интересов в отношении авторства и публикации нет.
Авторский вклад и ответсвенность. Все авторы участвовали в написании статьи и полностью несут ответственность за предоставление окончательной версии статьи в печать.
Контент публикуется на условиях лицензии Creative Commons Attribution 4-0 International (https://creativecommons.Org/licenses/by/4.0/deed.ru)
© Шадиметов Х. М., Болтаев А. К., Паровик Р. И., 2022
Шадиметов Холматвай МахкамбиевичА - доктор физико-математических наук, профессор, заведующий кафедрой информатики и компьютерной графики, Ташкентский государственный транспортный университет, Ташкент, Узбекистан, СЖСГО 0000-0002-4183-6184.
Болтаев Азиз Кузиеви^ - кандидат физико-математических наук, старший научный сотрудник лаборатории вычислительной математики, Институт математики имени В.И. Романовского, академии наук Узбекистана, СЖСГО 0000-0002-8329-4440.
Паровик Роман Иванович А - доктор физико-математических наук, доцент, профессор кафедры математики и физики, Камчатский государственный университет имени Витуса Беринга, Петропавловск-Камчатский, Россия, СЖСГО 0000-0002-1576-1860.
