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ALGORITHMS FOR APPROXIMATING A FUNCTION BASED ON INACCURATE OBSERVATIONS
G.Sh. Tsitsiashvili1, M.A. Osipova1,2 •
*Insitute for Applied Mathematics, Far Eastern Branch of Russian Academy Scinces, Russia
2Far Eastern Federal University, Russia guram@iam.dvo.ru, mao1975@list.ru
Abstract
This paper is devoted to the approximation of a function by a trigonometric polynomial based on its inaccurate values at selected points. Two methods of observation are considered. The first method is to make observations at points evenly distributed on the segment where the function is specified. The second method is to take observations at the points of division into a finite number of equal parts of the neighbourhoods of the selected points. Upper estimates of the standard deviation of the function from trigonometric polynomials are constructed and the rate of their convergence is estimated. Differences were found in the computational complexity of these approximations and in the number of observations of the function values at the selected points. Thus, the problem of approximating a function from inaccurate observations of their values at selected points is a multi-criteria one and its solution depends on the choice of observation points.
Keywords: trigonometric functions, inaccurate observations, error in function estimation, experimental plan
1. Introduction
This paper is devoted to the approximation of the periodic function from inaccurate observations. To solve this problem, Chebyshev, Hermite, Jacobi, Laguerre polynomials, trigonometric polynomials are used for the exact values of the function at the selected points (see, for example, [1], [2]). However, the task becomes significantly more complicated if it is necessary to evaluate the function based on inaccurate observations at selected points. In this case, there are many different solutions that need to be compared by various indicators (solution error, computational complexity, number of observation points). In this paper, two solutions to this problem are proposed and compared.
Trigonometric polynomials were used to approximate the function from inaccurate observations. The first method of approximation consists in observing the function at points evenly distributed over the segment of its assignment. In the second method, observations are considered at the points of division into a finite number of equal parts of the neighbourhoods of the selected points. In both cases, upper estimates of the standard deviation of the approximation of the function from its exact value are constructed. Despite the proximity of the upper estimates obtained, differences in the computational complexity of these approximations are found in the number of observations of the function values at the selected points.
The proposed algorithm for estimating the value of a function by a trigonometric polynomial using inaccurate deterministic or stochastic observations, unlike classical algorithms, allows us to estimate the rate of convergence of the estimates obtained to the estimated parameters.
ALGORITHMS FOR APPROXIMATING A FUNCTION BASED RT&A No 3 (79)
ON INACCURATE OBSERVATIONS Volume 19, September 2024
And considering a small interval of time observation makes it possible to build an experiment planning procedure. The authors previously used this idea to solve the problem of estimating the parameters of a number of ordinary differential equations and their systems, partial differential equations [3], [4].
The paper considers a function f (x) that is continuously differentiable on the segment [0,2n]. This function decomposes into a Fourier series in the space L2 [0,2n]. Denote
a n
fn (x) = y + Y («k cos(kx) + bk sin(kx)), (1)
k=1
1 r2n 1 r2n
ak = — f(x)cos(kx)dx, bk = — f(x)sin(kx)dx, k = 0,1,...,n.
n u 0 n •) 0
It is known (see, for example, [5]) that under given conditions for the function f (x) there exists a number C such that |ak | < C/k, |bk | < C/k, k = 1,..., and, therefore,
^ p2n
n E («2 + &2)=/ (f (x) - fn(x))2dx = D(n) = O(n-1). (2)
k=n+1 1/0
In this paper, we will consider two different estimates of the function f (x) based on inaccurate observations. The first estimate of fn (x) of the function f (x) is constructed as follows. Let tp fc, tp k, £p fc, £p fc, P = 1,..., k = 0,..., independent random variables. Moreover, the random
variables tpfc, tpfc have a uniform distribution on the segment [0,2n], and random variables £pk, £pk characterizing measurement errors have zero mean and variance a2. Then we define random variables
2 m—1 2 m—1
«k = m Y (f (tp,k) + £P,k) cos(ktp,k), = - Y (f (tp,k) + £P,k) sin(ktp,k) (3)
m p=0 m p=0
and we will make the first assessment
fn (x) = ^ + E («k cos(kx) + ' sin(kx)). (4)
2
k=1
The second estimate /(x) of the function f (x) is based on inaccurate observations in the following way. Let xp = 2np/m, p = 0,..., m — 1,and random variables £p,j, p = 0,..., m — 1, j = — (2N + 1),... ,2N + 1, characterizing measurement errors are independent, and have zero mean and variance a2.Let's assume random measurements of quantities f (xp) equal
1 N
f(xp) = ^TT Y (f(xp + jh) + £p,j), h = N—<*.
2N + 1 j=—N
Let's construct a second estimate of the function f( x)
a n
fn (x) = «2 + Y («k cos(kx) + bk sin(kx)). (5)
2 k=1
2 m— 1 2 m— 1
«k = mm Y fp cos(kxp), bk = - Y fp sin(kxp), k = 0,1,..., n.
m p=0 m p=0
Our task is to build upper bounds
r 2n ^ 2n
Mj (f (x) — fn(x))2dx, Mj (f (x) — fn (x))2dx
and compare them.
n
2. The error of the first estimate
Obviously, the inequality is fair
r 2n _ i- 2n ^
Mj (f (x) - fn (x))2dx = j M[( f (x) - fn (x)) + (fn (x) - fn (x))]2 dx <
2
f 2n !■ 2n ^
I (f (x) - fn(x))2dx + Mj (fn(x) - fn(x))2dx
In turn,
r 2n ^
M J0 (fn(x) - fn(x))2dx = nM
ao - ao
+ E (ak - h)2 + E (bk - h)2 k=1 k=1
Then it is not difficult to get from the formulas (4)
Mho = ao ^ Mfaa°^)2 = m (a2 + t f2(x)dx - ¿2) ,
2 i m \ m jo
4 ( 1 f2n \
Mak = ak ^ M(ak - ak )2 < — ¡a2 + — f2(x)dx - «2 , k = 1,2,...,
v 2 n j o j
- - 4 ( 1 f 2n \
Mbk = bk ^ M(bk -bk )2 < m fa2 + — J f2(x)dx - bfj , k = 1,2,....
It follows from this and from the formula (2) that
r2n
M Jo (f (x) - fn(x))2dx < 2
^Sn + l) (> + 2L f2n f2(x)dx m V 2n Jo
and, therefore, due to the formulas (2), (6) we have
,-2n
o
¡■2K ^
M J (f (x) - fn(x))2dx = O(n-1) + O(nm-1).
In particular, for n = [m1/2] we have (here [a] is the integer part of the real number a)
i-2n
M J (f (x) - fn (x))2dx = O(m-1/2).
(6)
(7)
(8)
2
n
n
3. The error of the second estimate
Let Afp = f(xp) - f (xp), first evaluate M(Afp)2. From the continuous differentiability of the function f (x) by [o, 2n] we have
max( sup If (x)|, sup |f'(x)|) = C < <x>.
o<x<2n o<x<2n
Then
M(Afp )2 = M
1
N
2N + 1
E (f (xp + jh) + £p,j) - f (xp)
j=-N
M
1
N
2N + 1
E (f (xp + jh) + £p,j - f (xp))
j=-N
M
1 N 1 N
E (f (xp + jh) - f (xp)) + 2N-+T E £p,j
j=-N V + j=-N
2N +1
1
N
<
2N + 1 j=-N 1N
1 E C|j|h
E (f(xp + jh) - f (xp))
+
a
2
2N +1
<
2N +1
j=-N
Therefore, the relation is fulfilled
+
a
2
2N +1
O(h2 N2 ) + O( N-1 ).
sup M(Afp )2 = O(h2 N2 ) + O( N-1 ).
0<p<m-1
In particular, for h = N 3/2 we get
sup M(Afp )2 = O(N-1 ).
0<p<m-1
(9)
Using this definition of the estimate fp, we construct an estimate of the error of the function fn (x). Denote
fn (x) = «20 + E « cos(kx) + bfc smCfcx)^
(10)
k=1
m—1
m—1
= m Efpcos(kxp), bfc = m Efpsin(kxp), k = 0,-,...,«.
p=0
m
p=0
Consider
r 2n ^ „ /• 2n _
M/0 (f (x) - (x))2dx = /q M[(f (x) - fn(x)) + (fn(x) - ffc (x)) + (ffc (x) - fn(x))]2dx <
< 3
/•2n c 2n c 2n ^
/ (f (x) - fn (x))2 dxW (.fn (x) - f (x))2dxW M(ffc (x) - fn (x))2dx 0 0 0
f2n,
(11)
Let's focus first on the assessment f^"(f (x) — fn(x))2dx.
f2n 2 f2n «0 — a* n n
(fn (x) — fn* (x))2 dx = dx —--0 + Y («k — ) cos(kx) + Y (bk — b*) sin(kx) 0 0 2 k=1 k=1
n
Y
(«0 - «fc)2 + 2 E («k - «fc )2 + (bk - bfc)2 k=1
We have
(1 r2n 2 m-1 \ / 1 m-1 rxp+1 2 m-1 \ '
n i f (x)dx - m Ef (xp >) = (1 El; f(x)dx - m E / (xp )j
/ 1 m-1 /• xp+1 \2 /1 m-1 /■ x
H 1E /p (f (x) - f (xp))dx < ^E
p=^xp / \n p=^xp
2 m-1 rxp+1 2nC , \2 16n2C2 -dx <-~—.
p=0-> xp y p=0-/xp m y
It is not difficult to get when xp < x < xp+1, k = 0,1,..., n - 1,
|f (x) cos(kx) - f (xp) cos(kxp)| < |f (x)| ■ | cos(kx) - cos(kxp)| +
2nC(k +1)
+ | cos(kxp)|- |f(x) - f (xp)| <
m
2
2
2
2
n
It follows that
m—1
lak — aH
1 t-2n 2 m—1
- / f (x) cos(kx)dx — - E /(Xp)cos)
n j 0 — ^_n
1 1 rXp+1
- E (/(x) cos(kx) — f (xp) Cos(kxp))
n „—n j xp
p=0
' 4nC(k +1)
n p=o-j xp
dx
m
Then
n ^^16C2n2(n + 1)(n + 2)(2n + 3) _ 3 _2
E(ak — 4)2 < — ^ -y '" ' = O(n ),
k=o 6m
By analogy, we have
^n 1*^2^16C2n2(n + 1)(n + 1)(2n + 3Ï. 3 _2,
E(bk — bk) <---6-2---- = 0(n3m 2).
k=0 6m
r 2n
Jo (fn(x) — fn (x))2dx = O(n3m—2). (12)
Therefore,
r2n
fn (^ -2^ r 2n _ 2
Let's now move on to the evaluation of M J (f* (x) - f (x))2dx, by calculating first
r2n ((a* — a^)2 n n \
M (f*(x) - fn(x))2dx = nM i ( o 4 o) + E M(a* - ak)2+ E M(b* - bk)2 . o 4 k=1 k=1
And then let's use the equalities
n n 2 m - 1 2
A = E M(a* - ak )2 = E m[— E Afp cos(kxp ) =
k=1 k=1 m p=o
4 n / m-1 m-1 \
= m¡2 E M E Afp cos(kxp) E Afq cos(kxq^ =
m k=1 p=o q=o
4 m- 1 m- 1 n
= EE MAfp Afq E cos(kxp) cos(kxq).
m p=o q=o k=1
Therefore, we have
(13)
4n m—1 m—1 2n m—1 m—1
IA < mn EE iMf f I < m2 EE M({Mp )2 + (f )2) < 4n sup M(A/p )2.
m p=0 q=0 m p=0 q=0 0<p<m—1
From these relations and the formula (9) we obtain |AI = O(nN—1 ). In turn,
/ \ 2 4 /m—1
M(a*0 — a0)2 = mM ( E A/p) = O(N—1 ).
Then
m2 p=0
E M(a*k— ak)2 = O(nN—1 ). (14)
k=0
Similarly, it is not difficult to obtain equality
E M(bkk — bk)2 = O(nN—1 ) (15)
k=0
Combining the formulas (13) - (15), we get
r 2n
My (fn* (x) — fn (x))2dx = O(nN—1). (16)
Finally from the formulas (2), (11), (12), (16) we come to the ratio 1
Mj0 (f (x) — fn (x))2 dx = O(n—1) + O(n3m—2) + O(nN—1). (17)
In particular, for n = m1/2 and N = m we get
f 2n
M J (f (x) — fn (x))2dx = O(m—1/2). 4. Conslusion
From formulas (8), (17) it follows that the estimation error fn (x), as well as the estimation error fn(x) are equal to O(m—1/2). In turn, the number of observations in the first case is equal to O(nm) = O(m3/2), and in the second case is equal to O(Nm) = O(m2). However, it should be noted that the formula for calculating the Fourier coefficients (5) can be made more economical using the fast Fourier transform method (see, for example, [6]).
This work was supported by the Ministry of Science and Higher Education of Russian Federation (Agreement No 075-02-2024-1440).
References
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[2] Suetin P. K. Classical orthogonal polynomials, Nauka, Mockow, 1979.
[3] Tsitsiashvili, G. Sh., Osipova, M. A., Kharchenko, Yu. N. (2022). Estimating the Coefficients of a System of Ordinary Differential Equations Based on Inaccurate Observations. Mathematics, 10(3), 502.
[4] Tsitsiashvili, G. Sh., Osipova, M. A., Gudimenko, A. I.(2023)and Statistical Aspects of Estimating Small Oscillations Parameters in a Conservative Mechanical System Using Inaccurate Observations. Mathematics, 11(12), 2643.
[5] Ivanov G.E. Lectures on mathematical analysis: in 2 parts: textbook, MIPT, Moscow, 2011.
[6] Nussbaumer G. Fast Fourier transform and convolution calculation algorithms, Radio and Communications, Moscow, 1985.
