624.04
Ɇ
.
Ʌ
.
Ɇɨɡɝɚɥɟɜɚ
,
ɉ
.
Ⱥ
.
Ⱥɤɢɦɨɜ
ɎȽȻɈɍ
ȼɉɈ
«
ɆȽɋɍ
»
ȼɕɑɂɋɅȿɇɂȿ
ɋȼȿɊɌɄɂ
ɎɍɇɄɐɂɃ
ȼ
ȻȺɁɂɋȿ
ɏȺȺɊȺ
ɋɜɹɡɚɧɧɵɟɫɛɚɡɢɫɨɦɏɚɚɪɚɜɵɱɢɫɥɟɧɢɹ, ɧɟɨɛɯɨɞɢɦɵɟɞɥɹɪɟɲɟɧɢɹɩɪɚɤɬɢɱɟɫɤɢɯɡɚ -ɞɚɱɫɬɪɨɢɬɟɥɶɧɨɣɦɟɯɚɧɢɤɢɢɦɚɬɟɦɚɬɢɱɟɫɤɨɣɮɢɡɢɤɢ, ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹɩɪɨɫɬɵɦɢɢɷɤɨɧɨ -ɦɢɱɧɵɦɢɚɥɝɨɪɢɬɦɚɦɢ. Ɉɱɟɜɢɞɧɨ, ɱɬɨɩɪɨɛɥɟɦɚɜɵɱɢɫɥɟɧɢɹɫɜɟɪɬɤɢɮɭɧɤɰɢɣ, ɪɚɡɥɨɠɟɧɧɵɯ ɩɨɛɚɡɢɫɭɏɚɚɪɚ, ɜɨɡɧɢɤɚɸɳɚɹɩɪɢɪɚɫɫɦɨɬɪɟɧɢɢɛɨɥɶɲɨɝɨɤɪɭɝɚɩɪɚɤɬɢɱɟɫɤɢɯɩɪɢɥɨɠɟɧɢɣ, ɫɜɨɞɢɬɫɹɤɨɩɪɟɞɟɥɟɧɢɸɫɜɟɪɬɨɤɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯɛɚɡɢɫɧɵɯɮɭɧɤɰɢɣ. ɂɦɟɧɧɨɷɬɨɬɜɨɩɪɨɫ ɢɹɜɥɹɟɬɫɹɩɪɟɞɦɟɬɨɦ ɪɚɫɫɦɨɬɪɟɧɢɹɜ ɪɚɦɤɚɯɧɚɫɬɨɹɳɟɣɫɬɚɬɶɢ. Ⱦɚɧɵɩɨɧɹɬɢɹɨɫɜɟɪɬɤɟ ɮɭɧɤɰɢɣ, ɛɚɡɢɫɟɏɚɚɪɚ, ɩɪɢɜɨɞɹɬɫɹɧɟɤɨɬɨɪɵɟ ɩɪɟɞɫɬɚɜɥɟɧɢɹɮɭɧɤɰɢɣɏɚɚɪɚ, ɩɪɢɜɟɞɟɧɵ ɮɨɪɦɭɥɵɫɜɟɪɬɤɢɮɭɧɤɰɢɣɏɚɚɪɚ.
Ʉɥɸɱɟɜɵɟɫɥɨɜɚ: ɜɟɣɜɥɟɬ-ɚɧɚɥɢɡ, ɛɚɡɢɫɏɚɚɪɚ, ɫɜɟɪɬɤɚɮɭɧɤɰɢɣ, ɡɚɞɚɱɢɫɬɪɨɢɬɟɥɶɧɨɣ
ɦɟɯɚɧɢɤɢ, ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɵɣɦɟɬɨɞɝɪɚɧɢɱɧɵɯɷɥɟɦɟɧɬɨɜ.
-
,
,
[1—3].
,
,
(
)
.
,
.
-
,
-,
[4, 5].
,
,
--
.
1.
ɉɨɧɹɬɢɟ
ɨ
ɫɜɟɪɬɤɟ
ɮɭɧɤɰɢɣ
.
f
(x
)
g
(x
)
«
»
( )
( )
(
) ( )
( ) (
)
.
f x
g x
f x
g
d
f
g x
d
(1.1)
-
( )
x
,
( )
x
f x
( )
f x
( ),
x
∈
(
−∞
,
∞
)
. (1.2)
-
(x x0)0 0
(
x
x
)
f x
( )
f x
(
x
),
x
0∈
(
−∞
,
∞
)
,
x
∈
(
−∞
,
∞
)
. (1.3)
,
(
x
a
)
(
x b
)
(
x
a b
),
(1.4)
a
b
—
.
(1.1)
(
f
g
) ( )
x
f x
( )
g x
( )
f x
( )
g x
( ).
(1.5)
2.
ɉɨɧɹɬɢɟ
ɨ
ɛɚɡɢɫɟ
ɏɚɚɪɚ
.
,
-,
,
1, 0
1/2;
( )
1, 1/2
1;
0,
0
1.
x
x
x
x
x
(2.1)
1
)
(
))
(
),
(
(
1 02
=
=
Ψ
=
Ψ
Ψ
∫
∫
+∞ ∞ −
dx
dx
x
x
x
.
,
(x
)
,
1
1
1, 2
2
2
;
1
1
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2
2
2
; ,
;
2
2
2
0,
2
2
2
,
j j j
j j j j j
k j j j
j j j
k
x
k
x
x
k
k
x
k
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x
k
x
k
(2.2)
Z
—
.
0
0
( )
x
( );
x
(2.3)
1
0 1
1,
2
;
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(
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1, 2
1
;
;
0,
1
,
k
k
x
k
x
x
k
k
x
k
k
x
k
x
k
(2.4)
1 1 01, 0
2
;
1
1
( )
1, 2
2 ;
;
2
2
2
0,
0
2 .
j
j j j
j
j j
j
x
x
x
x
j
x
x
(2.5)
3.
ɇɟɤɨɬɨɪɵɟ
ɩɪɟɞɫɬɚɜɥɟɧɢɹ
ɮɭɧɤɰɢɣ
ɏɚɚɪɚ
.
,
)
(x
1
1
1
1
( )
sign( )
sign
sign
sign(
1) ,
2
2
2
2
x
x
x
x
x
1,
0;
sign( )
1,
0.
x
x
x
(3.1)
1
1
( )
sign( )
2sign
sign(
1) ,
2
2
x
x
x
x
(3.2)
x
1
/
2
.
(3.1)
1
1
1
( )
sign( ) sign
( )
.
2
2
2
x
x
x
x
x
(3.3)
(3.3)
(1.4).
1, 0
1 / 2;
1
1
( )
sign( ) sign
0,
0
1 / 2,
2
2
x
x
x
x
x
x
(3.4)
1/2]
,
0
[
.
,
(3.3) (3.4)
1
( )
( )
( )
,
2
x
x
x
x
(3.5)
,
,
1
( )
( )
.
2
d
x
x
x
dx
1
1
1, 2
2
2 ;
1
( )
,
;
2
0,
2
2
2 .
2
j j j
j
k j j j j j
k
x
k
x
x
k
j k
x
k
x
k
(3.7)
:
00
( )
x
( );
x
(3.8)
1 0
1
1,
2
;
( )
(
)
;
0,
2
,
k
k
x
k
x
x
k
k
x
k
x
k
(3.9)
1
0 1
1, 0
2
;
1
( )
;
2
0,
0
2
.
2
j j j j jx
x
x
j
x
x
(3.10)
(3.4) (3.7),
1 1
1
( )
sign
sign
.
2
2
2
2
2
j
k j j j
x
x
x
k
k
(3.11)
,
1
( )
( )
( )
2
.
j j j
k
x
kx
x
x
(3.12)
,
1
1
( )
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2
( )
2
1 1
1
1
sign
sign
sign
sign
1
2
2
2
2
2
2
2
2
1 1
1
sign
2sign
sign
1
2
2
2
2
2
2
j j j j j
k k k
j j j j
j
j j j
j
x
x
x
x
x
x
x
x
x
k
k
k
k
x
x
x
k
k
k
.
,
(2.2), (3.1)
1
( )
2
2
j
k j j
x
x
k
1 1
1
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sign
1
.
2
2
2
2
2
2
j j j jx
x
x
k
k
k
,
(3.12)
.
,
(3.11)
1
1 1
( )
sign
( )
2
.
2
2
2
j j
k j j
x
x
k
x
x
(3.13)
,
(3.12)
1
1
1 1
( )
sign
( )
2
( )
2
,
2
2
2
j j j
k j j
x
x
k
x
x
x
x
,
(1.4),
1
1 1
( )
sign
( ) 2
2
(
2 ) ,
2
2
2
j j j
k j j
x
x
k
x
x
x
(3.14)
x
2
j1.
sign
2
2
2
j
j j
x
x
k dx
k
(3.15)
2 1
( )
2
( ) 2
2
(
2 ) .
2
j j j j
k j
d
x
x
k
x
x
x
dx
4.
Ɏɨɪɦɭɥɵ
ɫɜɟɪɬɤɢ
ɮɭɧɤɰɢɣ
ɏɚɚɪɚ
.
( )
( ).
j p
k
x
qx
(3.14),
1
1
1 1
1 1
( )
( )
sign
( ) 2
2
(
2 )
2
2
2
1
1
sign
( ) 2
2
(
2 )
2
2
2
1
1
( ) 2
2
(
2 )
( ) 2
2
4
2
j p j j
k q j j
p p
p p
j j p
j p
x
x
x
k
x
x
x
x
q
x
x
x
x
x
x
x
x
(
2 )
sign
sign
.
2
2
p
j p
x
x
x
k
q
[7, 8]
1
(
)
,
|
|
x
x
Z
/ {0},
Z
,
(4.1)
,
:
1 1
sign
sign
2
sign
2
2
2
2
1
2
sign
2
2
2
2
2
2
2
2
2
|
2
|
2
(
2
2
2
j
j p j p
j j
j p j p p
j p p j p p
j p
x
x
d
x
x
k
q
k
q
dx
x
d
x
x
x
k
q
k
q
dx
x
x
k
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x
k
x
q
)
2 |
2
j|
(
2
p)
2
2
j2
p,
x
k
x
q
x
k
q
. .
sign
sign
2
2
2
.
2
2
j p
j p
x
x
k
q
x
k
q
(4.2)
(4.2)
,
1
1
1
1
( )
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2
(
2 )
2
2
( ) 2
2
(
2 )
2
2
.
j p j j
k q
j p
p p j p
x
x
x
x
x
x
x
x
x
k
q
(4.3)
Ɂɚɦɟɱɚɧɢɹ
.
:
1)
2.3.8
«
-
-
-»
2011—2012 .;
2)
2.3.18
«
-
-
--
»
2012 .
Ȼɢɛɥɢɨɝɪɚɮɢɱɟɫɤɢɣ ɫɩɢɫɨɤ
1.
2. - / . . , . . ,
. . , . . . . : - , 2010. 336 .
3. / . . ,
. . , . . , . . . . : - , 2009. 336 .
4. ɁɚɯɚɪɨɜɚɌ.ȼ., ɒɟɫɬɚɤɨɜɈ.ȼ. - . . : - , 2012.
158 .
5. Ʉɟɱȼ., Ɍɟɨɞɨɪɟɫɤɭɉ.
-. -. : , 1978. 518 .
ɉɨɫɬɭɩɢɥɚɜɪɟɞɚɤɰɢɸɜɚɩɪɟɥɟ 2012 ɝ.
: ɆɨɡɝɚɥɟɜɚɆɚɪɢɧɚɅɟɨɧɢɞɨɜɧɚ — , ,
, ɎȽȻɈɍ ȼɉɈ «Ɇɨɫɤɨɜɫɤɢɣ
ɝɨɫɭɞɚɪ-ɫɬɜɟɧɧɵɣ ɫɬɪɨɢɬɟɥɶɧɵɣ ɭɧɢɜɟɪɫɢɬɟɬ» (ɎȽȻɈɍ ȼɉɈ «ɆȽɋɍ»), 129337, . , -, . 26-, (499) 183-59-94-, [email protected];
Ⱥɤɢɦɨɜ ɉɚɜɟɥ Ⱥɥɟɤɫɟɟɜɢɱ — , - ,
, ɎȽȻɈɍ ȼɉɈ «Ɇɨɫɤɨɜɫɤɢɣ
ɝɨɫɭ-ɞɚɪɫɬɜɟɧɧɵɣɫɬɪɨɢɬɟɥɶɧɵɣɭɧɢɜɟɪɫɢɬɟɬ» (ɎȽȻɈɍȼɉɈ «ɆȽɋɍ»), 129337, . , -, . 26-, (499) 183-59-94-, [email protected].
: ɆɨɡɝɚɥɟɜɚɆ.Ʌ., Ⱥɤɢɦɨɜɉ.Ⱥ.
// . 2012. № 8. . 98—103.
M.L. Mozgaleva, P.A. Akimov
COMPUTATION OF CONVOLUTION OF FUNCTIONS WITHIN THE HAAR BASIS
The Wavelet analysis, that replaces the conventional Fourier analysis, is an exciting new problem-solving tool employed by mathematicians, scientists and engineers. Recent decades have witnessed intensive research in the theory of wavelets and their applications. Wavelets are math-ematical functions that divide the data into different frequency components, and examine each component with a resolution adjusted to its scale. Therefore, the solution to the boundary problem of structural mechanics within multilevel wavelet-based methods has local and global components. The researcher may assess the infl uence of various factors. High-quality design models and rea-sonable design changes can be made.
The Haar wavelet, known since 1910, is the simplest possible wavelet. Corresponding computational algorithms are quite fast and effective. The problem of computing the convolution of functions in the Haar basis, considered in this paper, arises, in particular, within the wavelet-based discrete-continual boundary element method of structural analysis. The authors present their concept of convolution of functions within the Haar basis (one-dimensional case), share their useful ideas concerning Haar functions, and derive a relevant convolution formula of Haar functions.
Key words: wavelet analysis, Haar basis, convolution, structural analysis, discrete-continual boundary element method.
References
1. Zolotov A.B., Akimov P.A., Sidorov V.N., Mozgaleva M.L. Diskretnye i diskretno-kontinual’nye realizatsii metoda granichnykh integral’nykh uravneniy [Discrete and Discrete-continual Versions of the Boundary Integral Equation Method]. Moscow, MSUCE, 2011, 368 p.
2. Zolotov A.B., Akimov P.A., Sidorov V.N., Mozgaleva M.L. Diskretno-kontinual’nye metody ra-scheta sooruzheniy [Discrete-continual Methods of Structural Analysis]. Moscow, Arhitektura-S Publ., 2010, 336 p.
3. Zolotov A.B., Akimov P.A., Sidorov V.N., Mozgaleva M.L. Chislennye i analiticheskie metody ra-scheta stroitel’nykh konstruktsiy [Numerical and Analytical Methods of Structural Analysis]. Moscow, ASV Publ., 2009, 336 p.
5. Kech V., Teodoresku P. Vvedenie v teoriyu obobshchennykh funktsiy s prilozheniyami v tekhnike
[Introduction into the Theory of Generalized Functions and Their Engineering Applications]. Moscow, Mir Publ., 1978, 518 p.
A b o u t t h e a u t h o r s: MozgalevaMarina Leonidovna — Candidate of Technical Sciences, Associ-ated Professor, Department of Computer Science and Applied Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; marina. [email protected]; +7 (499) 183-59-94;
Akimov Pavel Alekseevich — Doctor of Technical Sciences, Associate Member of the Russian Academy of Architecture and Construction Science, Professor, Department of Computer Science and Ap-plied Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; [email protected]; +7 (499) 183-59-94.
