# DISCOVERY IN THE APPROXIMATION THEORY AND NUMERICAL ANALYSIS

[ Follow Ups ] [ Post Followup ] [ Netlib Discussion Forum ] [ FAQ ]

Posted by Babenko Sergey Nikolaevich on September 01, 1998 at 06:58:03:

DISCOVERY IN THE APPROXIMATION THEORY
AND NUMERICAL ANALYSIS.
========================================

# NEW TYPE OF CONVERGENCE IN FUNCTIONS SPACES-
CONVERGENCE OF APPROXIMATIONS OF MULTI-VARIABLE
FUNCTIONS WHEN VARIABLE NUMBER INCREASE AND
APPROXIMATIONAL POLINOMIAL POWER DOES NOT
CHENGES.

# COEFFICIENTS NUMBER OF APPROXIMATIONAL MULTI-
VARIABLE POLINOMIAL DEPENDENS ON POWER OF
APPROXIMATIONAL POLINOMIAL, BUT DOES NOT
DEPENDES ON VARIABLES
NUMBER.

New method was created for approximation task
solving. This method is based on metrical analysis in
semi-ordered spaces. Metrical polinomials were used
for approximation task solving instead of algebraic
polinomials.

MAIN RESULTS.

1. It was numerical verified metrical polinomials
convergence to the original multi-variable function
when variables number is increasing and metrical
polinomials degree does not change.
It is new comvergence type for approximations of
multi-variable functions.
For different functions convergence speeds are
different because their propertis are different.

2. Coefficients number of approximation metrical
multi-variable polinimial does not dependes on the
variables number.
Metrical polinomials formulas are analogue to one-
variable algebric polinomial one. It makes it easy to
use them in theoretical analysis.

3. Metrical approximation formulas require much less
actions for them calculations than multi-variable
algebraic polynomial one.
It allows use not very powerful computers for
numerical building approximations containing hundreds
and thousands variables.

4. Metrical polinomials formulas require much less
datum for them calculations than multi-variable
algebraic polynomial one.
For example, it needs not less 2^1024 for for
constracting regular net for building interpolating
approximation by multi-variable algebraic polinomial
contains 1024 variables. For building metrical
interpolating approximation contans 1024
variables it was used not more 9 points.
It needs not less 4097 points for building
uniform approximation by multi-variable algebraic
polinomial contains 4096 variables. For
building metrical approximation contains 4096
variables it was it used not more 12 points.
It gives chance for building models for very
complex, multi-dimensional technical, social,
biological systems when it is difficult or impossibe
to obtain much datum.

5. Interpolating metrical approximations avoid
regular net constracting for bilding approximations.
It is sufficiently that interpolating approximation
points was ordered with given order relation. It
makes more easy experemental datum use for
interpolation approximations constracting.

6. Metrical approximations avoid variables mutual
influence problem because all variables are using
together. It corresponds to the reality because in
the real systems all variables ussialy change
together.

The main results were published in All-RUSSIAN
institute of scientific and technical information.

The bases of metrical approximations was given in
the papers:

1. Babenko S.N. On metrical interpolating of
operators in the semi-ordered spaces.(rus.) 1988.
7 p. 15.03.89, N 1698-B89.
2. Babenko S.N. On uniform metrical approximation
of operators in the semi-ordered spaces.(rus.) 1997.
6 p. 18.08.97, N 2701-B97.

Numerical researches of metrical approximations
were given in the papers:

3. Babenko S.N. On metrical interpolating of muti-
variale functions.(rus.) 1998. 14 p. 06.05.98,
N 1403-B98.
4. Babenko S.N. On uniform metrical approximation of
multi-variable functions.(rus.) 1998. 16 p.
06.05.98, N 1402-B98.

Complete description of this method was published in
the paper:

5. Babenko S.N. Metrical approximation of
operators.(rus.) 1998 141 p. 06.07.98 N 2108-B98

METRICAL APPROXIMATION OF OPERATORS.
-----------------------------------

CONTENTS.
1. INTRODUCTION ................................. 4
1.1 Difficalts of the econometrical modeling ..... 4
1.2 The task of constracting of analitical form of econometrical dependence and approximation theory. 7
1.2.1 Common analitical form of approximation dependence ...................................... 7
1.2.2 The analitical form of interpolating approximation ..................................... 11
1.2.3 The analitical form of uniform
approximation ..................................... 16
1.2.4 The task of constracting analitical form of approximation dependence .......................... 19

2. METRICAL ANALYSIS IN THE SEMI-ORDERED SPACES.. 20
2.1 Metrical divided difference of operator....... 20
2.2 Metrical derivation of operator............... 27
2.3 Metrical integral of operator................. 46
2.4 Relationship between metrical divided difference, metrical derivation and metrical integral
of operator ....................................... 55

3. INTERPOLATIONS METRICAL APPROXIMATIONS OF OPERATORS IN THE SEMI-ORDERED SPACES .............. 61
3.1 Metrical interpolating of operators............ 61
3.2 Convergence of interpolating metrical approximations of operators ....................... 76

4. UNIFORM METRICAL APPROXIMATIONS IN THE SEMI-ORDERED SPACES. ................................... 86
4.1 Basis metrical functionals .................... 86
4.2 Linear dependence of the basis metrical
functional ........................................ 90
4.3 The best approximation operators metrical polinomials .......................................100
4.4 Convergence of uniform metrical approximations of operators .........................................108

5. NUMERICAL RESEARCHS OF METRICAL APPROXIMATIONS OF MULTI-VARIABLE FUNCTION ...........................128
5.1 The task of numerical researches...............128
5.2 Interpolating metrical approximation of multi-variable function. ................................130
5.3 Uniform metrical approximation of multi-variable function. .........................................132
5.4 Convergence of metrical approximations of multi-variable functions when variables number
is increase .......................................135

6. CONCLUSIONS ....................................137
7.REFERENCES......................................139

EMAIL: nemol@kubstu.ru

Stasova 145-b, 36
350058 Krasnodar
RUSSIA
PHONE: (8612) 33 17 21