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Ф1ЗИКО-МАТЕМАТИЧН1 НАУКИ
UDC 519.21
DOI: 10.15587/2313-8416.2016.84180
INVESTIGATION OF THE INTERPOLATION REPRESENTATION OF RANDOM PROCESSES WITH NON-EQUIDISTANCE INTERPOLATION KNOTS
© G. Verovkina
The article deals with some interpolation representations of random processes with non-equidistance interpolation knots. Research is based on observations of the process and its derivatives of the first, second and third orders at some types of knots and observations of the process and its derivatives of the first and second orders at another types of knots
Keywords: random, process, interpolation, representations, series, knot, non-equidistance, separability, convergence, probability
У cmammi розглядаються питання ттерполяцшних представлень випадкових процеав за mpienoeidda-леними вузлами ттерполяци. До^дження базуеться на значеннях процесу та його noxidHux першого, другого та третього порядку у вузлах iнтерполяцii одного типу, а також значеннях процесу та його по-хiдних першого та другого порядку у вузлах iнтерполяцii другого типу
Ключовi слова: випадковий, процес, ттерполящя, представлення, серiя, вузол, нерiвновiддалений, сепа-рабельтсть, збiжнiсть, ймовiрнiсть
1. Introduction
Interpolation representations of a class of random process with non-equidistance interpolation knots are investigated. The necessary results from the theory of entire functions of complex variable are formulated. The function bounded on any bounded region of the complex plane is considered. The estimation 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, it is obtained that the interpolation row converges to the random process uniformly over in any bounded area of parameter changing. The convergence with probability 1 of the corresponding interpolation series to a random process in any bounded domain of parameter changes is proved.
2. Literature review
The one of the fundamental results in the Theory of Information Transmission is a theorem of the function expression with a bounded spectre of values in the periodic sequence of initial moments. The significance of that fact was first introduced by Kotel'nikov;V.;A. in [1]. Further, Shannon;C. was investigated these problems in [2, 3]. The Theorem of Kotel'nikov-Shannon is generally well-known [4]. Later, the work, which summarized the Theorem of Kotel'nikov-Shannon, is appeared [5]. In the present time, the investigations related to the construction of interpolation polynomials are attracting significant interest. Many problems concerning the construction
of a spline approximation as well as a representation of a motion in 3D-modelling with help of interpolation and approximation [6] and the modern theory of signal transmission [7] are based on the Kotel'nikov-Shannon theorem. The problems of constructing interpolation polynomials with non-equidistance interpolation knots are interested. The present work is concerned on the problems stated above.
3. Aim and research problems
The aim of research - to construct the interpolation representations of stochastic processes along non-equidistant interpolation knots. Two types of interpolation knots are considered as the basis. For the first type, the interpolation formula includes the value of the process and its derivatives of the first, second and third orders. For a second type of knots, the interpolation formula includes the value of the process and its derivatives of first and second orders.
Research problems: proof of the interpolation formula and convergence of the corresponding series with probability 1.
4. Assumptions and methods of research
Let's consider the interpolation representation of random processes [8, 9] on non-equidistance interpolation knots of the type
7n
Ko = n—> a
7n n
tnl = n--1—, n e Z
а а
based on observations of the process and its derivatives of the first, second and third orders at knots tn0, n e Z and observations of the process and its derivatives of the first and second orders at knots tnl, n e Z.
Let's 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
5. Results of the research
Consider a random £(t), t e R with M£(t) = 0 and covariance function, which representation is
B(t, s) ={ f(t,l)f(s, n)F(dl, dn), (3)
where A is a set of parameters, F(A, A) is a positive definite additive complex function on Ax A such that
J \F(dl, d^)\ <+œ.
(4)
Г
да
f ( z) = S
n=-a f '(tno )
f (tno)
а (z - tno)4 w n
. 1 f "(tno )
а
( Z - tn0 У
f ' '(tno )
3 sin3
а
iZ - tn0 )2
2 sin3
f I tno +:
( Z - tno )
sin
f I tno +1
а^3 Г n
а IZ - tno -n
n
7 V 7 1
<--
n
Z - tno
sin
n
sin
The function f (t, l) relating to t is an entire function of exponential type with indicator <?(!) such that
sup
sup |f (t,l)\ = Сj. <+да,
sup a(X) = a < .
АеЛ
(5)
(6)
7
f ' I tno
а^3 Г n
а VZ - tno -n
2 sin4
. 4 а, \ . 3а Г n\
x sin - (z - tno )sin -1z - tno--y.
\Rn ( z)| < LG( z)Cf where L is a constant, Cf = sup I f (t)|, G(z) =
а 1
The following theorem holds true.
Theorem 1. Let Ç(t) is a separable random process that satisfies conditions (3)-(6). Then for any a, a>a with probability 1, the following representation holds true
Г
& ) = S
Гo )
Г o )
а
7, (t - tno)4 2sin37 ГаJ (t - tno)3
Г (tno)
r(tn o )
Sin
(1)
In
where tn0 = n—, n e Z, provided that the interpola-
a
tion series (1) converges uniformly in any 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
n Г
7 V Л) 1
Г tno Vn0
+-I
а/
3
11 - ' tno -
(t - tno )2
n
2 si 7 VyJ (t - tno )
sin
n
Г tno
sin
n
7 V 7
t - tno -
Г ' I tno +
n
а
sin
n
t - tno -
2 sin'
n
(2)
. 4 а , \ . 3 а Г n\
x sin - (t - tn o ) sin - It - tn o -~j .
(7)
Proof: according to the theorem about spectral representation of random processes [9], we will write the process £(t) as follows:
. 4 ^ . I w /I
sin — z x sin —(z--)
7 7 а
is a function bounded on any bounded region of the complex plane.
(8)
m = j f (t,X) z ( dx),
A
where Z(dX) is a random measure on A, such that
MZ (A ) • Z (A ) = F (A, A )• For any natural n consider a
ЛхЛ
1
+
x
x
4
а
+
n=-OT
1
1
+
x
x
+
x
7
3
а
n
+
x
x
x
process 4 (t), which we will define as a partial sum with a number n of row (7).
E4(t) -4 (t )|2
(
4 (t )=E
4 0 )
4 о)
7, (t -1, о )4 2Sln3T ^ (t - tk о )3
i 4(tk о)
x--1--—^——
1
4"(h о)
1
s'"' f ф' (t - tk о )2 1 ф4 (t -1„) srn> 1
1
4tk о +-
41 hо
a)3 ( n
f (t - tk о -f
. 4 n
SinT (yl 1t-tkо -
n 4n
n 1 2 sin —
• 4 a , . 3 a f n|
x sin -[t -tk0)sin -^ - tk0.
Using the representation (8) and the statement of the lemma, we will write 4(t) as follows:
(
n
4 (t) = EJ
f (tk о)
f '(tk о)
k=-n л
f "(tk о)
(^ 4(t - tk о)4 2s'n' f О 4(t - tk о) . 1 f '"(tk о) 1
3 sin3
n
f (t - tkо)2 2s'n' f 1f
n
n
7 (7
(t - tkо) sin
n
f (tk о +- )
a
(t - k о-f)2 sin4 a
n
f (tk о +-)
f '" I tkо +
n
a
f] V - tk о--) sin4
7 J a
n
t tk о
2 sin'
n
• 4 a, . ' a ( я! r-,, ^
x sin — (t - ti0) x sin — I t - ti0--I x Z(aa).
7 7 ( aJ
Then, based on the representation (1), (8), (9) and the estimation (2), we obtain
MI s(t)) - 4 (t) r < ^n2 (t) J |F (dx,, d<u)\ =
(9)
= l2g 2 rt;c2
a - a J n
-1 J \F (dk, du)|. (10)
From the inequality (10) and considering the condition (4), we obtain the following: an interpolation row (7) converges to 4(t) in the mean square.
Considering the separability of the process ¿;(t) and the convergence of a row
obtain that the interpolation row (7) converges to the random 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 random process £(t) in any bounded domain of changes of parameter t.
Consider the interpolation representation of random processes [8] on non-equidistance interpolation knots of the type
7n
a
7n n
a a
7n 2n
tn2 = n--1--, n e Z
a a
based on observations of the process and its derivatives of the first and second orders at knots t„o, Ki, n e Z and observations of the process at knots tn2, n e Z .
Let's formulate the necessary results from the theory of entire functions of complex variable.
Lemma. Let f(z) is an entire bounded on the real axis function of exponential type with indicator c.
Then for any a, a> c, the representation holds true
(
f (*) = E
f (tn о)
f' (tn о)
(* - tnо)3
(* -tnо)2
fCtnü^i a) % 2 a (* - tnо)
. 3 n . 2 2n sin —sin —
f I tn о +-
f I tnо +-
a)3 ( n
a) (*- tnо-n
z - tn о --
a)3 ( n
a (* - tn0 -n
sin
n
f I tn0 +
+-
2n
a
a
7
x sin3
2n
a
- . 3n . 3 2n
2 sin —sin —
• 3 a i ч . 3 a ( n)
in у(*- к о)x sin у ( *- tn о —J
2n
a
n=-w
7
a
+
n=-w
1
+
x
3
+
x
a
+
3
2
a
n
1
+
+
x
7
ЛжЛ
2
1
x
x
ЛжЛ
where
Г
tno = n
7n
n e Z,
г z) =S
provided that the interpolation series (11) converges uniformly in any bounded region of the complex plane.
Proving Lemma, we obtain estimation of the residual of the interpolation series (11), which has the following form
R (z )| <
< LG(z)Cf —— -1, (12)
a-a n
^(tno )
Г (tno)
Г'(tno )
(t - tno )3
(t - tno )2
(t - tno )
. 3 n . 2 2n sin —sin — 7 7
fl'no+J
Г I tno +
n
а
Î-^no +J
а\3 Г n
а Vt - tno -n
t - 'п o -
а
t - tno -
а
Г tno +
2n
а
-+-7~
.4 n а Г 2n
sin — —It -1 n--
7 7 l no а
1
- . 3n . 3 2n 2 sin —sin — 77
• 3 a/ \ • 3 а Г n\
x sin - (t - tno )x sin - I' - tno -~j
where L is a constant,
Cf = sup| f (t),
t eR
G( z) =
. 3 а . 3 а n .а, 2nч
sin — z x sin —( z--) x sin—( z--)
7 7 а 7 а
is a function bounded on any bounded region of the complex plane.
Consider a random £(t), t e R with M£(t) = 0 and co-variance function, which representation is
B(t, s) = j f (t,l)f (s, M)F (dl, dn), (13)
where A is a set of parameters, F (.,.) is a positive definite additive complex function on AxA such that
J |F(dl, d^)| <+œ.
(14)
The function f (t, l) relating to t is an entire function of exponential type with indicator a(l) such that
sup sup \f(t, i)\ = с < +œ,
sup a(l) = a <+да.
leA
(15)
(16)
The following theorem holds true.
Theorem 2. Let Ç(i) is a separable random process that satisfies conditions (13)—(16). Then for any a, a>a with probability 1 the following representation holds true
/
а
x siny I t - tno -
2n а
(17)
We obtain that the interpolation series (17) converges with probability 1 to a random process £(t) in any bounded domain of changes of parameter t.
6. Conclusions
The work is devoted to investigation of interpolation representations of a class of random processes. The main results are theorems on the convergence of interpolating series to a random process with probability 1. The following results were obtained:
1. We constructed two types of representation knots' groups.
2. For the first type of knots, the interpolation formula includes the value of the process and its derivatives of the first, second and third orders. For the second type of knots, the interpolation formula includes the value of the process and its derivatives of first and second orders. The interpolation formula that uses the value of the process and its derivatives at the knots of interpolation is constructed.
3. We have proved the convergence of the interpolation series to the considered stochastic process with probability 1.
References
1. Kotel'nikov, V. A. On the transmission capacity of 'ether' and wire in electric communications [Text] / V. A. Kotel'nikov // Uspekhi Fizicheskih Nauk. - 2006. - Vol. 176, Issue 7. - P. 762. doi: 10.3367/ufnr.0176.200607h.0762
2. Shennon, C. E. Communication in the presence of noise [Text] / C. E. Shennon // Proceedings of the IRE. -1949. - Vol. 37, Issue 1. - P. 10-21. doi: 10.1109/jrproc. 1949.232969
3. Shennon, C. E. Mathematical Theory of Communication [Text] / C. E. Shennon // Bell System Technical Journal. -1948. - Vol. 27. - P. 379-423.
4. Hirurhin, Y. I. Methods of the theory of entire functions in radiophysics, radio and optics theory [Text] / Y. I. Hirurhin, V. P. Yakovlev. - Moscow, 1962. - 220 p.
5. Gagerman, G. L. Some General Aspects of the Sampling Theorem [Text] / G. L. Gagerman, L. G. Fogel // IEEE Transactions on Information Theory. - 1956. - Vol. 2, Issue 4. - 139-146. doi: 10.1109/tit.1956.1056821
а
n=-rM
1
x
3
3
2
n
x
x
AxA
6. Shikin, E. V. Curves and surfaces on a computer screen. Guide for users of splines [Text] / E. V. Shikin, L. I. Plis. - Moscow: Dialog-MIFI, 1996.
7. Meijering, E. A Chronology of Interpolation From Ancient Astronomy to Modern Signal and Image Processing [Text] / E. Meijering // Proceedings of the IEEE. - 2002. -Vol. 90, Issue 3. - P. 319-342. doi: 10.1109/5.993400
8. Yaglom, A. M. Spectral representations for various classes of random functions. Vol. 1 [Text] / A. M. Yaglom // Proc. 4-th USSR Math. Congr. Izd. Akad. Nauk SSSR. - Leningrad, 1963. - P. 250-273.
9. Yaglom, A. M. Spectral representations for various classes of random functions. Vol. 1 [Text] / A. M. Yaglom // Trudy 4 Vsesouzn. Mat. Congr. - Leningrad: Izd. Akad. Nauk SSSR, 1963. - P. 132-148.
10. Verovkina, G. V. The interpolation representation of some classes of random processes [Text] / G. V. Verovkina // Mechanics and Mathematics. - 2013. - Vol. 2. - P. 9-11.
11. Verovkina, G. V. The interpolation representation of one class of random fields [Text] / G. V. Verovkina, V. N. Nagornyi // Bulletin of Kiev University. Series of physical and mathematical sciences. - 2005. - Vol. 1. - P. 31-34.
12. Verovkina, G. V. The interpolation representation of some classes of random fields [Text] / G. V. Verovkina // XVII International Scientific Conference in Honour of Acad. M. Kravchuk. - 2015. - Vol. III. - P. 14-16.
References
1. Kotel'nikov, V. A. (2006). On the transmission capacity of 'ether' and wire in electric communications. Uspekhi Fizicheskih Nauk, 176 (7), 762. doi: 10.3367/ufnr.0176.200607h.0762
2. Shannon, C. E. (1949). Communication in the Presence of Noise. Proceedings of the IRE, 37 (1), 10-21. doi: 10.1109/jrproc.1949.232969
3. Shennon, C. E. (1948). Mathematical Theory of Communication. Bell System Technical Journal, 27, 379-423.
4. Hirurhin, Y. I., Yakovlev, V. P. (1962). Methods of the theory of entire functions in radiophysics, radio and optics theory. Moscow, 220.
5. Jagerman, D., Fogel, L. (1956). Some general aspects of the sampling theorem. IEEE Transactions on Information Theory, 2 (4), 139-146. doi: 10.1109/tit. 1956.1056821
6. Shikin, E. V., Plis, L. I. (1966). Curves and surfaces on a computer screen. Guide for users of splines. Moscow: Dia-log-MIFI.
7. Meijering, E. (2002). A chronology of interpolation: from ancient astronomy to modern signal and image processing. Proceedings of the IEEE, 90 (3), 319-342. doi: 10.1109/ 5.993400
8. Yaglom, A. M. (1963). Spectral representations for various classes of random functions. Vol. 1. Proc. 4-th USSR Math. Congr. Izd. Akad. Nauk SSSR. Leningrad, 250-273.
9. Yaglom, A. M. (1963). Spectral representations for various classes of random functions. Vol. 1. Trudy 4 Vsesouzn. Mat. Congr. Leningrad: Izd. Akad. Nauk SSSR, 132-148.
10. Verovkina, G. V. (2013). The interpolation representation of some classes of random processes. Mechanics and Mathematics, 2, 9-11.
11. Verovkina, G. V., Nagornyi, V. N. (2005). The interpolation representation of one class of random fields. Bulletin of Kiev University. Series of physical and mathematical sciences, 1, 31-34.
12. Verovkina, G. V. (2015). The interpolation representation of some classes of random fields. XVII International Scientific Conference in Honour of Acad. M. Kravchuk, III, 14-16.
Рекомендовано до публкаци д-р фгз.-мат. наук Моклячук М. П.
Дата надходження рукопису 14.10.2016.
Ganna Verovkina, PhD, Associate Professor, Department of Mathematical Physics, Taras Shevchenko National University of Kyiv, Volodymyrska str., 64/13, Kyiv, Ukraine, 01601 E-mail: ganna.verov@gmail.com
