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7PHYSICS AND MA THEMA TICS
THE INTERPOLATION REPRESENTATIONS OF STOCHASTIC PROCESSES WITH NON-EQUIDISTANCE INTERPOLATION KNOTS IN GROUPS WITH TWO POINTS
Ganna Verovkina
Ukraine, Kyiv, Taras Shevchenko National University of Kyiv, Department of Mathematical Physics, Associate Prof.
Abstract. Paper deals with some interpolation representations of stochastic processes with non-equidistance interpolation knots in groups with two points. Research is based on observations of the process and its derivatives of the first and second orders. The function bounded on any bounded region of the complex plane is considered. The estimate of the residual of the interpolation series is obtained. The interpolation formula that uses the value of the process and its derivatives at the knots of interpolation is proved. Considering the separability of the process and the convergence of a row that the interpolation row converges to the stochastic process uniformly over in any bounded area of changing of parameter is obtained. The main purpose of our work is the obtained convergence with probability 1 of the corresponding interpolation series to a stochastic process in any bounded domain of changes of parameter. Obtained results may be applied in the modern theory of information transmission.
Keywords: interpolation, representation, stochastic, process, series, non-equidistance, knot.
Introduction. The one of the fundamental results in the Theory of Information Transmission is a theorem of expression of the function with a bounded specter of values in the periodic sequence of initial moments. The significance of that fact was first introduced in [1]. Further these questions were studied in [2, 3]. The Theorem of Kotel’nikov-Shannon is generally well-known [4]. In the present time, the investigations related to the construction of interpolation polynoms are attracting significant interest. Many of the questions concerning the construction of a spline approximation as well as a representation of a motion in 3D-modelling with help of interpolation and approximation [5] are investigate. Many questions in modern physics [6] as well as physics of materials [7] and the modern theory of signal transmission [8] are based on the Kotel’nikov-Shannon theorem. The questions of interest are constructing interpolation polynoms with non-equidistance interpolation knots. The present work is concerned on the questions stated above.
Research. Consider the interpolation representation of stochastic processes [9] on nonequidistance interpolation knots of the type
7 n
tn0 = n—,
a
7n n
Kx = n— + -, a a
n E Z
based on observations of the process and its derivatives of the first, second and third orders at knots tn0, n є Z and observations of the process and its derivatives of the first and second orders at knots tnl, n є Z.
Let us formulate the necessary results from the theory of entire functions of complex variable.
Lemma. Let f (z) be an entire bounded on the real axis function of exponential type with indicator a.
Then for any a, a > a, the representation holds true
(
f (z) =Z
f o„) і
f '(t,o)
-x-L- + ГЫ X_L
_ . 3 /iz
(z - t„o)4 2s>ny
n
-\ (z -^)3 sin У ly| (z - О
_t 'i2 2sin3
f " (t no)
-Z. , П X
f (t nO + _)
a
■x-
+ -
a і (z -tno) sin
n
7 ,fj (z - tno -П sin4
n
a
f '(tnO +-) ____________________a
3
■ x ■
+
7 17 ) (z - tno-П) sin4
\
f "(tnO + )
+ -
a
■ x--------
a V n '•> • 4n
a] (z - tno--) 2sin 7
a 7
• 4 a , \*3 a , —X
x sln -(z - tno)sln -(z - tn 0-),
7 7 a
(1)
4
n=-w
7
v
1
1
4
1
where tn0 = n—, n є Z , provided that the interpolation series (1) converges uniformly in any a
bounded region of the complex plane.
Proving Lemma, as in [10-12] we obtain estimate of the residual of the interpolation series (1), which has the following form
a 1
\Rn (z)| < LG(z)Cf-
a -a n
(2)
where L is a constant,
G (z)
Cf = sup f (t )|,
t^R
. 3 a . 3 a . n. a . 2n.
sin — z x sin —(z-----) x sin — (z----)
7 7 a 7 a
is a function bounded on any bounded region of the complex plane.
Results of the Research. Consider a stochastic (j(t), t є R with M£(t) = 0 and covariance function which the representation is
B(t, s) = [ f (t,A)f(s,p)F(dA,dp) (3)
AxA
where Л is a set of parameters, F(.,.) is a positive definite additive complex function on AxA such that
[ |F(dA, dp)\ <+да (4)
AxA
The function f (t, A) with respect to t is an entire function of exponential type with indicator a(A) such that
sup sup \f (t, A)I = Cf <+да (5)
A є Л -да < t < +да>
sup а(Л) = a < (6)
ІєЛ
The following theorem holds true.
Theorem. Let %(t) be a separable stochastic process that satisfies conditions (3)-(6). Then for any a, a>a with probability 1 the following representation holds true
(
m =1
£(Ц y 1
f'(t,o)
4"(tn0)
a
(t - t„o)4 2sin3 - fa
7 I 7
(t - t„o)3
• 3
sin
-
7 I 7
a
(t -t,f 2sin
-
Г'(tn0)
t(t„o + -) a
14 I (t - tn0 )
sin3
-
a
7 I 7
-
—
(t - tn0------)
■x-----
2 sin4
+-
Г' (tno +—) a
+
a
a ' - — sin4
n | (t - tn0-------)
7 ) a
Г''(tno + -)
+ -
a
■x-
-
aI (t - tno --) 2sin47
a 7
. 4 . 4 . 3 a , - x
x sin -(t - tn0)sin ~(t - tn0-----------) •
a
7
a
(7)
Proof: according to the theorem about spectral representation of stochastic processes [9], we will write the process <T(t) as follows:
1
1
x
4
n=-W
7
К
1
1
X
4
3
1
)
r(t) = f f (t,A) Z(dA) (8)
Л
where Z(dA) is a stochastic measure on Л such that MZ(A) • Z(A) = F(A, A). For any natural n consider a process Г (t), which we will define as a partial sum with a number n of row (7).
f
r (t)=£
^(t* o)
1
- x -
0] (t-tko)4 2sin3- at (t-1*o)
r'(t* o)_x + __гъ^х _j_
•з 7Г f /-y 7 _ . з -
1 a ' (t -1 )2 2 sin3 —
(t t*o) 7
3 sin , _ *o) 7 I 7
r% o)
r(t* o +-)
a
-
a
-
■x-----
2 sin4
+-
r'(t* o +-) a
+
(t t*o) sin 7 \ q ] (t t*o ^) 7
a ' - ■ - sin4
(t t*o )
a
-,
r (t* o + - )
+
a
1
a] (t-1*o--) 2sin4-
5 ) a 7
a
a
- 4
x sin — (t - ti0)sin —(t - ti0------).
a
Using the representation (8) and the statement of the Lemma 1, we will write Г(t) as follows:
1
1
1
x
x
4
3
3
x
(
t. о) = Z j
k=-n л
f (tk o)
(t - tko)4 2sln3 f 1^1 (t - tko)3
f '(tk o) .. 1 I f "(tk o) x_
7
ж
sln7 |yI (t - tko)
'i2 2sln3
f (tk o)
1
ґ' / Ж \
f (o + )
a
+ -
(t -tko) sin 7 [ у I (t -tko---)
+-
f '(tk o +-) a
+
a \ ,. . f 2 sin4
ж
a
a \ '■ ж sin4
(t tk o )
a
Ж,
f "(tk o + )
a
1
a \ (t -1 -f) 2sin4 (t lk o ) 7
a 7
• 4 a , . .3 a , жч _. 7 x
x sin —(t - tkQ) x sin —(t - tk0-) x Z(da).
7 7 a
Then, based on the representation (1), (8), (9) and the estimation (2) we obtain
MI m - у (t)\2 < R2 (t) j \F (dX„ dp)\= L2G2 (t)Cf {-О-1 j- xj \F (dX, dp)\
(9)
(10)
1
1
X
V.
1
1
X
X
X
4
3
3
x
J
ЛхЛ
ЛхЛ
From the inequality (10) and considering the condition (4) we obtain the following: an interpolation row (7) converges to £(t) in the mean square.
Considering the separability of the process £(t) and the convergence of a row
да
Z l£(') -$n (t)2
п=-да
obtain that the interpolation row (7) converges to the stochastic process £(t) almost surely uniformly over t in any bounded area of changing of t.
We obtain that the interpolation series (7) converges with probability 1 to a stochastic process 4(t) in any bounded domain of changes of parameter t.
Conclusions. The research focuses on the modern questions in the theory of stochastic processes. The results in this work are principally new and they are related to the interpolation representations of stochastic processes with non-equidistance interpolation knots. The type of representation knots group was constructed. For this type of knots, the interpolation formula includes the value of the process and its derivatives of the first and second orders. The interpolation formula that uses the value of the process and its derivatives at the knots of interpolation was constructed. The convergence of the interpolation series to the considered stochastic process with probability 1 has been proved. The work is a continuation and supplement of previously considered problems [10, 11, 12].
Obtained results can be applied in the construction of spline-approximation and in the modern theory of information transmission. The further research on this problem is planned in order to obtain new schemes of interpolation representations of stochastic processes with non-equidistance interpolation knots.
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