Integral representation of even positive-definite functions of one variable
We obtain an integral representation of even positive-definite functions of one variable for which the kernel $[k_1(x + y) + k_2 (x − y)]$ is positive definite.
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| Date: | 2010 |
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| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508854269247488 |
|---|---|
| author | Lopotko, O. V. Лопотко, О. В. |
| author_facet | Lopotko, O. V. Лопотко, О. В. |
| author_sort | Lopotko, O. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:03Z |
| description | We obtain an integral representation of even positive-definite functions of one variable for which the kernel $[k_1(x + y) + k_2 (x − y)]$ is positive definite. |
| first_indexed | 2026-03-24T02:31:49Z |
| format | Article |
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UDK 517.9
O. V. Lopotko (Nac. lisotexn. un-t Ukra]ny, L\viv)
INTEHRAL|NE ZOBRAÛENNQ
PARNYX DODATNO VYZNAÇENYX FUNKCIJ
ODNI{} ZMINNO}
We obtain the integral representation of even positive definite functions of one variable such that the
kernel k x y k x y1 2( ) ( )+ + −[ ] is positively defined.
Poluçeno yntehral\noe predstavlenye çetn¥x poloΩytel\no opredelenn¥x funkcyj odnoj
peremennoj, dlq kotor¥x qdro k x y k x y1 2( ) ( )+ + −[ ] poloΩytel\no opredeleno.
U praci [1] M.-H.-Krejn zastosuvav metod sprqmovanyx funkcionaliv dlq oder-
Ωannq intehral\nyx zobraΩen\ dodatno vyznaçenyx qder K x y( , ) , x , y R∈ 1
.
G.-M.-Berezans\kyj [2] zaproponuvav metod oderΩannq intehral\nyx zobraΩen\
dlq dodatno vyznaçenyx qder K x y( , ) , x, y R∈ 1
, za dopomohog vlasnyx funk-
cij dyferencial\nyx operatoriv. Cej metod polqha[ u vvedenni za qdrom
K x( , y) , x, y R∈ 1
, hil\bertovoho prostoru i pobudovi rozvynennq za uzahal\-
nenymy vlasnymy vektoramy samosprqΩenyx operatoriv, qki rozhlqdagt\sq u
c\omu prostori; vidpovidna rivnist\ Parsevalq da[ potribne zobraΩennq. U da-
nij roboti pobudovano intehral\ne zobraΩennq dlq parnyx dodatno vyznaçenyx
funkcij odni[] zminno]. Dovedena teorema [ uzahal\nennqm teorem-3.18, 3.19 [3,
s. 697 – 699].
Oznaçennq. Paru parnyx dijsnyx neperervnyx funkcij k x1( ) , k x2( ) , x ∈
∈ R1
, budemo nazyvaty parno dodatno vyznaçenymy (p.-d. v .), qkwo dlq do-
vil\no] finitno] funkci] u x( ) ∈ C R0
1∞( ) vykonu[t\sq nerivnist\
k x y k x y u y u x dxdy
RR
1 2
11
0( ) ( ) ( ) ( )+ + −[ ] ≥∫∫ . (1)
Tobto neperervne qdro K x y( , ) = k x y k x y1 2( ) ( )+ + −[ ] ma[ buty dodatno
vyznaçenym.
Teorema. KoΩna para p. d. v. funkcij k x1( ) , k x2( ) , x R∈ 1
, dopuska[ zob-
raΩennq
k x k
x
d
x
d
R R
1 2 10
1
2
1
21 1
( ) ( )
cos
( )
cos
+ =
+
+
−
∫ ∫
λ
ρ λ
λ
λ
ρρ λ2( ) , (2)
k k x
x
d
x
d
R R
1 2 10
1
2
1
21 1
( ) ( )
cos
( )
cos
+ =
+
−
−
∫ ∫
λ
ρ λ
λ
λ
ρρ λ2( ) , (3)
de dρ λ1( ) , dρ λ2( ) — borelivs\ki nevid’[mni miry; qkwo k x1( ) ≤ ceNx2
i
k x2( ) ≤ ceNx2
, c, N > 0 dlq vsix x R∈ 1
, to miry u (2) i (3) vyznaçagt\sq
odnoznaçno. Navpaky, funkci] vyhlqdu (2), (3) [ parog p.-d. v. funkcij.
Dovedennq. Za funkciqmy k x1( ) , k x2( ) vvedemo kvaziskalqrnyj dobutok u
prostori L R dx2
1,( ) :
© O. V. LOPOTKO, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2 281
282 O. V. LOPOTKO
u K x y u y x dxdyH
RR
k
, ( , ) ( ) ( )v v= ∫∫
11
, u, v ∈ ( )∞C R0
1
.
Pislq provedennq faktoryzaci] j popovnennq oderΩymo hil\bertovyj pros-
tir- H k .
Nexaj teper u prostori L R dx2
1,( ) di[ minimal\nyj operator A, qkyj vidpo-
vida[ vyrazu L = −
d
dx
2
2 . Cej operator dopuska[ prodovΩennq osnawennq — v
qkosti D moΩna vzqty prostir C R0
1∞( ) , topolohizovanyj naleΩnym çynom.
ZvuΩennq A∗
na D zbiha[t\sq z vidobraΩennqm u L u→ +
, u C R∈ ( )∞
0
1
. Todi
umova komutuvannq k x y( , ) i A ekvivalentna ermitovosti zvuΩennq A∗
na D
u prostori H k , tobto rivnosti
L u u L+ +=, ,v v , u, v ∈ ( )∞C R0
1
. (4)
Dlq hladkoho dodatno vyznaçenoho qdra k x y( , ) rivnist\ (4) vykonu[t\sq.
Dlq dovil\noho dodatno vyznaçenoho qdra k x y( , ) rivnist\ (4) takoΩ vyko-
nu[t\sq. Dijsno,
L u k x y k x y
y
u y x d
RR
+ = + + −[ ] ∂
∂∫∫, ( ) ( ) ( ) ( )v v1 2
2
2
11
xxdy =
= k x y
y
u y x dxdy
RR
1
2
2
11
( ) ( ) ( )+
∂
∂∫∫ v + k x y
y
u y x dxdy
RR
2
2
2
11
( ) ( ) ( )−
∂
∂∫∫ v =
= k y
y
u y x dy x dx
RR
1
2
2
11
( ) ( ) ( )
∂
∂
−
∫∫ v +
+ k y
y
u y x dy x dx
RR
2
2
2
11
( ) ( ) ( )
∂
∂
+
∫∫ v =
= k y
x
u y x dy x dx
RR
1
2
2
11
( ) ( ) ( )
∂
∂
−
∫∫ v +
+ k y
x
u y x dy x dx
RR
2
2
2
11
( ) ( ) ( )
∂
∂
+
∫∫ v =
= k y
x
u y x x dx dy
R R
1
2
2
1 1
( ) ( ) ( )∫ ∫
∂
∂
−
v +
+ k y
x
u y x x dx dy
R R
2
2
2
1 1
( ) ( ) ( )∫ ∫
∂
∂
+
v =
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
INTEHRAL|NE ZOBRAÛENNQ PARNYX DODATNO VYZNAÇENYX FUNKCIJ … 283
= k y u y x
x
x dx dy
R R
1
2
2
1 1
( ) ( ) ( )∫ ∫ −
∂
∂
v + k y u y x
x
x dxdy
R R
2
2
2
1 1
( ) ( ) ( )∫ ∫ +
∂
∂
v =
= k x y k x y u y
x
x dxdy
RR
1 2
2
2
11
( ) ( ) ( ) ( )+ + −[ ] ∂
∂∫∫ v = u L, +v .
Takym çynom, K x y( , ) komutu[ z −
d
dx
2
2 .
Teper dlq qdra K x y( , ) moΩna zastosuvaty teoremu 3.9 [3, s. 669], qkwo za
fundamental\nu systemu rozv’qzkiv rivnqnnq −
d u
dx
2
2 + λ u = 0 vzqty χ λ0( ; )x =
= cos λ x ; χ λ1( ; )x =
sin λ
λ
x
, i oderΩaty take zobraΩennq:
k x y k x y1 2( ) ( )+ + −[ ] = cos cos ( )λ λ σ λx y d
R1
00∫ +
+ cos
sin
( )λ
λ
λ
σ λx
y
d
R1
01∫ +
sin
cos ( )
λ
λ
λ σ λ
x
y d
R
10
1
∫ +
+
sin sin
( )
λ λ
λ
σ λ
x y
d
R
11
1
∫ . (5)
Vykonavßy u (5) zaminu x na – x, y na – y, oderΩymo zobraΩennq
k x y k x y1 2( ) ( )+ + −[ ] = cos cos ( )λ λ σ λx y d
R1
00∫ +
+
sin sin
( )
λ λ
λ
σ λ
x y
d
R
11
1
∫ . (6)
Dali, poklavßy u (6) y = x, znajdemo
k x k( ) ( )2 0+[ ] = cos ( )2
00
1
λ σ λx d
R
∫ +
sin
( )
2
11
1
λ
λ
σ λ
x
d
R
∫ ,
abo
k x k1 2 0( ) ( )+ =
1
2
1
1
+
∫
cos
( )
λ
ρ λ
x
d
R
+
1
2
2
1
−
∫
cos
( )
λ
λ
ρ λ
x
d
R
,
tobto zobraΩennq (2).
Qkwo u (6) poklademo y = – x, to otryma[mo
k k x1 20 2( ) ( )+ = cos ( )2
00
1
λ σ λx d
R
∫ –
sin
( )
2
11
1
λ
λ
σ λ
x
d
R
∫ ,
abo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
284 O. V. LOPOTKO
k k x1 20( ) ( )+ =
1
2
1
1
+
∫
cos
( )
λ
ρ λ
x
d
R
–
1
2
2
1
−
∫
cos
( )
λ
λ
ρ λ
x
d
R
,
tobto zobraΩennq (3).
Odnoznaçnist\ miry u (2), (3) vyplyva[ iz teoremy 4.3 [2, s. 708 – 710].
Ostann[ tverdΩennq teoremy dovodyt\sq takym çynom. Iz (2) znaxodymo
k x y k1 2 0( ) ( )+ + =
1
2
1
1
+ +
∫
cos ( )
( )
λ
ρ λ
x y
d
R
+
+
1
2
2
1
− +
∫
cos ( )
( )
λ
λ
ρ λ
x y
d
R
. (7)
Iz (3) ma[mo
k k x y1 20( ) ( )+ − =
1
2
1
1
+ −
∫
cos ( )
( )
λ
ρ λ
x y
d
R
–
–
1
2
2
1
− −
∫
cos ( )
( )
λ
λ
ρ λ
x y
d
R
. (8)
Oskil\ky z (6) vyplyva[ rivnist\
k k1 20 0( ) ( )+ = d
R
ρ λ1
1
( )∫ ,
to, dodavßy (7) do (8), oderΩymo
k x y k x y1 2( ) ( )+ + − = cos cos ( )λ λ ρ λx y d
R1
1∫ +
+
sin sin
( )
λ λ
λ
ρ λ
x y
d
R
2
1
∫ . (9)
Za dopomohog rivnosti (9) perevirq[mo umovu (1).
Teoremu dovedeno.
U vypadku, koly u nerivnosti (1) k x1 0( ) = , oderΩymo zobraΩennq z teore-
my-3.11 [3] dlq parnyx funkcij.
U vypadku, koly k x2 0( ) = , oderΩymo zobraΩennq z teoremy-3.17 [3] dlq
parnyx funkcij.
ZauvaΩennq. 1. Qkwo k x1( ) = k x2( ) = ( / ) ( )1 2 k x , to oderΩymo zobraΩen-
nq (3.87) iz [3, s. 697].
2. Qkwo k x1( ) = ( / ) ( )1 2 k x , k x2( ) = ( / ) ( )−1 2 k x ta k( )0 0= , to budemo ma-
ty zobraΩennq (3.92) iz [3, s. 699].
1. Krejn M. H. Ob odnom obwem metode razloΩenyq poloΩytel\no opredelenn¥x qder na
πlementarn¥e proyzvedenyq // Dokl. AN SSSR. – 1946. – 53, # 1. – S. 3 – 6.
2. Berezanskyj G. M. Obobwenye teorem¥ Boxnera na razloΩenyq po sobstvenn¥m funkcyqm
dyfferencyal\n¥x operatorov // Tam Ωe. – 1956. – 108, # 3. – S. 893 – 896.
3. Berezanskyj G. M. RazloΩenye po sobstvenn¥m funkcyqm samosoprqΩenn¥x operatorov.
– Kyev: Nauk. dumka, 1965. – 798 s.
OderΩano 01.07.08,
pislq doopracgvannq — 09.10.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
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| id | umjimathkievua-article-2865 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:31:49Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f5/c3cde54797079ce6bc40004958d475f5.pdf |
| spelling | umjimathkievua-article-28652020-03-18T19:39:03Z Integral representation of even positive-definite functions of one variable Інтегральне зображення парних додатно визначених функцій однієї змінної Lopotko, O. V. Лопотко, О. В. We obtain an integral representation of even positive-definite functions of one variable for which the kernel $[k_1(x + y) + k_2 (x − y)]$ is positive definite. Получено интегральное представление четных положительно определенных функций одной переменной, для которых ядро $[k_1(x + y) + k_2 (x − y)]$ положительно определено. Institute of Mathematics, NAS of Ukraine 2010-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2865 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 2 (2010); 281 – 284 Український математичний журнал; Том 62 № 2 (2010); 281 – 284 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2865/2482 https://umj.imath.kiev.ua/index.php/umj/article/view/2865/2483 Copyright (c) 2010 Lopotko O. V. |
| spellingShingle | Lopotko, O. V. Лопотко, О. В. Integral representation of even positive-definite functions of one variable |
| title | Integral representation of even positive-definite functions of one variable |
| title_alt | Інтегральне зображення парних додатно визначених функцій однієї змінної |
| title_full | Integral representation of even positive-definite functions of one variable |
| title_fullStr | Integral representation of even positive-definite functions of one variable |
| title_full_unstemmed | Integral representation of even positive-definite functions of one variable |
| title_short | Integral representation of even positive-definite functions of one variable |
| title_sort | integral representation of even positive-definite functions of one variable |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2865 |
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