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

DISCOVERY IN THE APPROXIMATION THEORY

AND NUMERICAL ANALYSIS.

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# 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

ADDRESS: Sergey Babenko

Stasova 145-b, 36

350058 Krasnodar

RUSSIA

PHONE: (8612) 33 17 21