On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series

For the trigonometric series $$\sum_{k=0}^{\infty}a_k\sum_{l\in kV \setminus (k-1)V}e^{i(l, x)}, \quad a_k\rightarrow 0,\quad k\rightarrow \infty,$$ given on $[-\pi, \pi)^m$, where $V$ is some polyhedron in $R^m$, we prove that the inequality $$\int\limits_{T^m}\left|\sum^{\infty}_{k=0} a_k \sum_{l\...

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Datum:	2008
Hauptverfasser:	Ivashchuk, O. V., Zaderei, P. V., Pelagenko, E. N., Іващук, О. В., Задерей, П. В., Пелагенко, Є. Н.
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Sprache:	Ukrainisch Englisch
Veröffentlicht:	Institute of Mathematics, NAS of Ukraine 2008
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Ukrains’kyi Matematychnyi Zhurnal

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author	Ivashchuk, O. V. Zaderei, P. V. Pelagenko, E. N. Іващук, О. В. Задерей, П. В. Пелагенко, Є. Н.
author_facet	Ivashchuk, O. V. Zaderei, P. V. Pelagenko, E. N. Іващук, О. В. Задерей, П. В. Пелагенко, Є. Н.
author_sort	Ivashchuk, O. V.
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
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datestamp_date	2020-03-18T19:47:45Z
description	For the trigonometric series $$\sum_{k=0}^{\infty}a_k\sum_{l\in kV \setminus (k-1)V}e^{i(l, x)}, \quad a_k\rightarrow 0,\quad k\rightarrow \infty,$$ given on $[-\pi, \pi)^m$, where $V$ is some polyhedron in $R^m$, we prove that the inequality $$\int\limits_{T^m}\left\|\sum^{\infty}_{k=0} a_k \sum_{l\in kV\setminus(k-1)V}e^{i(l, x)} \right\| dx \leq C \sum^{\infty}_{k=0} (k+1) \|\Delta A_k\|,$$ holds if the coefficients $a_k$ satisfy the following conditions of the Sidon - Telyakovskii type: $$A_k\rightarrow\infty,\quad \|\Delta a_k\| \leq A_k, \quad \forall k \geq 0, \quad \sum^{\infty}_{k=0}(k+1) \|\Delta A_k\|
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fulltext	UDK 517.518.4 P. V. Zaderej, E. N. Pelahenko, O. V. Yvawuk (Kyev. nac. un-t texnolohyj y dyzajna) OB USLOVYQX TYPA SYDONA – TELQKOVSKOHO YNTEHRYRUEMOSTY KRATNÁX TRYHONOMETRYÇESKYX RQDOV For the trigonometric series a ek k i l x l kV k V= ∞ ∈ − ∑ ∑ 0 1 ( , ) \ ( ) , ak → 0, k → ∞ , given on [ , )− π π m , where V is some polyhedron in Rm , we prove that the inequality a e dx C k Ak k i l x l kV k VT k km = ∞ ∈ − = ∞ ∑ ∑∫ ∑≤ + 0 1 0 1( , ) \ ( ) ( ) ∆ , holds if the coefficients ak satisfy the following conditions of the Sidon – Telyakovskii type: Ak → 0, ∆a Ak k≤ ∀ ≥k 0, ( )k Ak k + < ∞ = ∞ ∑ 1 0 ∆ . Pokazano, wo dlq tryhonometryçnyx rqdiv vyhlqdu a ek k i l x l kV k V= ∞ ∈ − ∑ ∑ 0 1 ( , ) \ ( ) , ak → 0 , k → ∞ , wo zadani na [ , )− π π m , de V — deqkyj poliedr u Rm , vykonu[t\sq nerivnist\ a e dx C k Ak k i l x l kV k VT k km = ∞ ∈ − = ∞ ∑ ∑∫ ∑≤ + 0 1 0 1( , ) \ ( ) ( ) ∆ , qkwo koefici[nty ak zadovol\nqgt\ umovy typu Sidona – Telqkovs\koho Ak → 0 , ∆a Ak k≤ ∀ ≥k 0 , ( )k Ak k + < ∞ = ∞ ∑ 1 0 ∆ . Pust\ V — zamknut¥j ohranyçenn¥j polyπdr v R m s verßynamy v toçkax s ra- cyonal\n¥my koordynatamy, zvezdn¥j otnosytel\no naçala koordynat, qvlqg- wehosq eho vnutrennej toçkoj, y takoj, çto prodolΩenye lgboj eho hrany ne proxodyt çerez naçalo koordynat; nV = { / }:x R x n Vm∈ ∈ — homotet V. Mno- Ωestvo polyπdrov s ukazann¥my svojstvamy oboznaçym çerez W. Pust\ Z m — celoçyslennaq reßetka v R m . Oboznaçym çerez L Tm 1( ) prostranstvo opredelenn¥x na T m 2π -peryodyçe- skyx yntehryruem¥x funkcyj f ( x ) s normoj f 1 = f x dx T m ( )∫ < ∞ , © P. V. ZADEREJ, E. N. PELAHENKO, O. V. YVAWUK, 2008 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5 579 580 P. V. ZADEREJ, E. N. PELAHENKO, O. V. YVAWUK hde x = ( x1 , … , xm ) y T m = [ – π , π ) m = [ – π , π ) × … × m �� [ – π , π ) . V dannoj rabote yssleduetsq sxodymost\ kratn¥x tryhonometryçeskyx rq- dov vyda a a ek k i l x l kV k V 0 1 1 + = ∞ ∈ − ∑ ∑ ( , ) \ ( ) , (1) hde x ∈ T m , l ∈ Z m , ( l, x ) = l x l xm m1 1 + … + , koπffycyent¥ kotor¥x stremqtsq k nulg pry k → ∞ y udovletvorqgt uslovyqm typa Sydona – Telqkovskoho: suwestvugt çysla Ak takye, çto Ak → 0, k → ∞ , dlq vsex k ≥ 0 ∆ak ≤ Ak , (2) ( )k Ak k + = ∞ ∑ 1 0 ∆ < ∞ . (3) Pryvedem nekotor¥e rezul\tat¥, otnosqwyesq k rassmatryvaemoj zadaçe. V rabote [1] (teoremaGA) ustanovleno, çto esly koπffycyent¥ rqda a a kxk k 0 12 + = ∞ ∑ cos (4) moΩno predstavyt\ v vyde ak = p m m m k i i k m = ∞ = ∑ ∑α , k = 1, 2, … , (5) hde αi ≤ 1 y pmm= ∞∑ 1 < ∞ , to rqd (4) qvlqetsq rqdom Fur\e. S. A. Telqkovskyj v rabote [2] (teoremaG1) prydal uslovyqm (5) bolee udob- n¥j vyd, a ymenno, pokazal, çto ony πkvyvalentn¥ uslovyqm: ak → 0, k → ∞ , y suwestvugt takye çysla Ak , çto Ak ↓ 0, ∆ak ≤ Ak ∀ k ≥ 0, Ak k= ∞ ∑ 0 < ∞ . (6) Pry πtom poluçena ocenka a a kx dxk k 0 10 2 + = ∞ ∑∫ cos π ≤ C Ak k= ∞ ∑ 0 . Zdes\ y dalee çerez C budem oboznaçat\ absolgtn¥e poloΩytel\n¥e konstan- t¥, vozmoΩno, razn¥e v razn¥x formulax. Krome toho, v [2] (teoremaG2) pokazano, çto dlq rqda a kxk k sin = ∞ ∑ 1 , (7) koπffycyent¥ kotoroho udovletvorqgt uslovyqm (6), ravnomerno otnosytel\- no p = 1, 2, … spravedlyva ocenka ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5 OB USLOVYQX TYPA SYDONA – TELQKOVSKOHO YNTEHRYRUEMOSTY … 581 a kx dxk k p sin = ∞ + ∑∫ 1 1 π π = a k O Ak k p k k= = ∞ ∑ ∑+    1 0 . Pry πtom (7) qvlqetsq rqdom Fur\e v tom y tol\ko v tom sluçae, kohda a k k k= ∞ ∑ 1 < ∞ . Uslovyq (6) naz¥vagt uslovyqmy Sydona – Telqkovskoho. G. L. Nosenko [3] obobwyl rezul\tat¥ rabot¥ [2] na sluçaj dvojn¥x rqdov (yz kosynusov, synusov y kosynusov, a takΩe synusov). Sxodymost\ dvojn¥x tryhonometryçeskyx rqdov v [3] ponymaetsq v sm¥sle Prynshejma. Obßyrnaq byblyohrafyq, posvqwennaq yntehryruemosty kratn¥x tryhono- metryçeskyx rqdov, ymeetsq v [4]. V rabote O. Y. Kuznecovoj [5] (teoremaG2.1) dokazano, çto rqd (1), koπffy- cyent¥ kotoroho udovletvorqgt uslovyqm Sydona – Telqkovskoho (6), sxodyt- sq poçty vsgdu na T m k nekotoroj funkcyy f L Tm∈ 1( ) , qvlqetsq ee rqdom Fur\e y spravedlyva ocenka a a e dxk k i l x l kV k VT m 0 1 1 + = ∞ ∈ − ∑ ∑∫ ( , ) \ ( ) ≤ C Ak k= ∞ ∑ 0 . Cel\g rabot¥ qvlqetsq dokazatel\stvo sledugwej teorem¥. Teorema. Pust\ koπffycyent¥ ak rqda (1) stremqtsq k nulg pry k → → ∞ y udovletvorqgt uslovyqm (2), (3). Tohda rqd (1) sxodytsq poçty vsgdu na T m k nekotoroj funkcyy f L Tm∈ 1( ) , qvlqetsq ee rqdom Fur\e y spraved- lyva ocenka a a e dxk k i l x l kV k VT m 0 1 1 + = ∞ ∈ − ∑ ∑∫ ( , ) \ ( ) ≤ C k Ak k ( )+ = ∞ ∑ 1 0 ∆ , (8) a ravenstvo lim ( ) n nf S f →∞ − 1 = 0, hde S f xn( , ) = a a ek k i l x l kV k V 0 1 1 + = ∞ ∈ − ∑ ∑ ( , ) \ ( ) — posledovatel\nost\ çastn¥x summ rqda (1), v¥polnqetsq tohda y tol\ko tohda, kohda lim ln n n ma n →∞ = 0. (9) Zameçanye. V teoreme posledovatel\nost\ { },A kk ≥ 0 stremytsq k nulg ne obqzatel\no monotonno v otlyçye ot uslovyj (6) y, krome toho, ukazano neob- xodymoe y dostatoçnoe uslovye sxodymosty v srednem kratn¥x tryhonometry- çeskyx rqdov vyda (1). Pry dokazatel\stve budem yspol\zovat\ pryem, predloΩenn¥j S. A. Telq- kovskym v [6]. ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5 582 P. V. ZADEREJ, E. N. PELAHENKO, O. V. YVAWUK PokaΩem snaçala, çto rqd Akk= ∞∑ 0 sxodytsq. Poskol\ku rqd ∆Ajj k= ∞∑ sxodytsq k Ak , to Ak = ∆Aj j k= ∞ ∑ . (10) Sledovatel\no, yspol\zuq preobrazovanye Abelq y uslovye (3), naxodym Ak k = ∞ ∑ 0 = ( )k Ak k + = ∞ ∑ 1 0 ∆ ≤ ( )k Ak k + = ∞ ∑ 1 0 ∆ < ∞ . (11) Yz (2) y (11) sleduet, çto ∆ak k= ∞ ∑ 0 < ∞ . (12) Zafyksyruem natural\noe n y k koneçnoj summe a a ek k n i l x l kV k V 0 1 1 + = ∈ − ∑ ∑ ( , ) \ ( ) prymenym preobrazovanye Abelq ( ):a− =1 0 : a a ek k n i l x l kV k V 0 1 1 + = ∈ − ∑ ∑ ( , ) \ ( ) = ∆a D x a D xk kV k n n nV( ) ( ) = − ∑ + 0 1 , hde D xkV ( ) : = ei l x l kV ( , ) ∈ ∑ — qdra Dyryxle, sootvetstvugwye mnoΩestvu V. Poskol\ku posledovatel\nost\ { }( ),D x nnV ≥ 0 ohranyçena poçty vsgdu na T m [5] (lemmaG2.4) y an → 0 pry n → ∞ , dlq poçty vsex x ∈ T m ymeet mesto ravenstvo a ek k i l x l kV k V= ∞ ∈ − ∑ ∑ 0 1 ( , ) \ ( ) = ∆a D xk kV k ( ) = ∞ ∑ 0 , (13) yz kotoroho, v sylu sxodymosty rqda (12), sleduet sxodymost\ rqda (1) poçty vsgdu na T m . Oboznaçym summu rqda (1) çerez f ( x ) . Perejdem teper\ k dokazatel\stvu neravenstva (8) y, sledovatel\no, summy- ruemosty funkcyy f ( x ) . Zametym, çto dlq vsex k ≥ 0 αk : = ∆a A k k ≤ 1, poçty dlq vsex x ∈ T m y dlq lgboho k ≥ 0 αk kVD x( ) ≤ Cx < ∞ , (14) hde konstanta Cx zavysyt ot x . Rassmotrym çastnug summu rqda v pravoj çasty ravenstva (13): ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5 OB USLOVYQX TYPA SYDONA – TELQKOVSKOHO YNTEHRYRUEMOSTY … 583 ∆a D xk kV k n ( ) = ∑ 0 = A D xk k kV k n α ( ) = ∑ 0 y v¥polnym preobrazovanye Abelq ( ):A− =1 0 ∆a D xk kV k n ( ) = ∑ 0 = ∆A D xk k n j jV j k = − = ∑ ∑ 0 1 0 α ( ) + A D xn k kV k n α ( ) = ∑ 0 : = S S1 2+ . Yz (14) sleduet, çto dlq poçty vsex x ∈ T m αk kV k n D x( ) = ∑ 0 ≤ C nx( )+ 1 . (15) Yspol\zuq neravenstvo (15) y sootnoßenye (10), dlq poçty vsex x ∈ T m pry n → ∞ poluçaem S2 = ∆A D xj j n k kV k n = ∞ = ∑ ∑ α ( ) 0 ≤ C j Ax j j n ( )+ = ∞ ∑ 1 ∆ → 0. (16) V sylu sxodymosty rqda v pravoj çasty ravenstva (13) y ocenky (16) ymeem f ( x ) = ∆A D xk k j jV j k = ∞ = ∑ ∑ 0 0 α ( ). (17) Yspol\zuq predstavlenye funkcyy (17), a takΩe ocenku [5] (lemmaG2.3) α j jV j k T D x dx m ( ) = ∑∫ 0 ≤ Ck , (18) naxodym f x dx T m ( )∫ ≤ ∆A D x dxk k j jV j k T m= ∞ = ∑ ∑∫ 0 0 α ( ) ≤ C k Ak k ( )+ = ∞ ∑ 1 0 ∆ . Ostalos\ pokazat\, çto uslovye (9) qvlqetsq kryteryem sxodymosty rqda (1) v metryke prostranstva L Tm 1( ). Prymenqq metod A. N. Kolmohorova [7] dlq dokazatel\stva sxodymosty v srednem rqda (1), poluçaem f S fn− ( ) 1 = a e dxk k n i l x l kV k VT m = + ∞ ∈ − ∑ ∑∫ 1 1 ( , ) \ ( ) = ∆a D x a D x dxk kV k n n nV T m ( ) ( ) = + ∞ +∑∫ − 1 1 . (19) PokaΩem, çto yntehral ot pervoj summ¥ v poslednem ravenstve (19) stre- mytsq k nulg pry n → ∞ . V¥polnqq preobrazovanye Abelq y yspol\zuq ocen- ku (18), naxodym ∆a D x dxk kV k nT m ( ) = + ∞ ∑∫ 1 = A D x dxk k kV k nT m α ( ) = + ∞ ∑∫ 1 ≤ ≤ ∆A D x dx A D x dxk k n j jV j k T n j jV j n Tm m= + ∞ = + = ∑ ∑∫ ∑∫+ 1 0 1 0 α α( ) ( ) ≤ ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5 584 P. V. ZADEREJ, E. N. PELAHENKO, O. V. YVAWUK ≤ C k A A nk k n n( ) ( )+ + +       = + ∞ +∑ 1 1 1 1∆ ≤ ≤ C k A A nk k n k k n ( ) ( )+ + +       = + ∞ = + ∞ ∑ ∑1 1 1 1 ∆ ∆ → 0 (20) pry n → ∞ . Na osnovanyy (20) sootnoßenye (19) moΩno zapysat\ tak: f S fn− ( ) 1 = a D x dx on nV T m + ∫ +1 1( ) ( ). Yzvestno [8], çto dlq lgboho polyπdra V suwestvugt konstant¥ C1 y C2 takye, çto C nm 1 ln ≤ D x dxnV T m ( )∫ ≤ C nm 2 ln . Teorema dokazana. V nastoqwee vremq yzvestn¥ druhye uslovyq yntehryruemosty kratn¥x try- honometryçeskyx rqdov vyda (1) [9]. Odnako poluçenn¥e v πtoj rabote uslovyq v nekotor¥x sluçaqx bolee udobn¥ v prymenenyy. Rassmotrym sledugwyj prymer. Pust\ dlq lgb¥x n, p > 0 takyx, çto n / p = O ( 1 ) , koπffycyent¥ ak = λk n( ) , hde λk n( ) = 1 0 1 1 1 0 1 , , , , , . ≤ ≤ − − − + + − ≤ ≤ ≥ +       k n p n k p n p k n k n V¥berem koπffycyent¥ Ak n( ) sledugwym obrazom: Ak n( ) : = ∆λk n( ) = 0 0 1 1 1 0 1 , , , , , . ≤ ≤ − − + − ≤ ≤ ≥ +       k n p p n p k n k n (21) PokaΩem, çto posledovatel\nost\ { }( )Ak n k= ∞ 0 udovletvorqet uslovyqm teorem¥. Dejstvytel\no, yz (21) sleduet, çto Ak n( ) → 0 pry k → ∞ y ∆λk n( ) ≤ Ak n( ) dlq vsex k ≥ 0. Proverym v¥polnenye uslovyq (3). Yspol\zuq (21), naxodym ∆Ak n( ) = 0 0 2 1 1 1 0 1 1 1 0 1 , , , , , , , , , . ≤ ≤ − − − + = − − − ≤ ≤ − + = ≥ +           k n p p k n p n p k n p k n k n Tohda rqd ( ) ( )k Ak n k + = ∞ ∑ 1 0 ∆ = 2 1 1 n p p − + + ≤ C sxodytsq. ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5 OB USLOVYQX TYPA SYDONA – TELQKOVSKOHO YNTEHRYRUEMOSTY … 585 Takym obrazom, koπffycyent¥ ak = λk n( ) udovletvorqgt uslovyqm teo- rem¥. Rassmotrym rqd (1) s koπffycyentamy ak = λk n( ) : λk n k i l x l kV k V e( ) ( , ) \ ( )= ∞ ∈ − ∑ ∑ 0 1 . (22) Prymenqq teoremu, poluçaem λk n k i l x l kV k VT e dx m ( ) ( , ) \ ( )= ∞ ∈ − ∑ ∑∫ 0 1 ≤ C. Zametym, çto rqd (22) qvlqetsq qdrom porqdka n s yndeksom n – p metoda summyrovanyq Valle Pussena: V f xn n p− ( ; ) = λk n k i l x l kV k V e( ) ( , ) \ ( )= ∞ ∈ − ∑ ∑ 0 1 . Sledstvye. Dlq lgboj funkcyy f L Tm∈ 1( ) s rqdom Fur\e vyda (1) qdra Valle Pussena V f xn n p− ( ; ) porqdka n s yndeksom n – p, hde n, p > 0, n / p = = O ( 1 ) , ohranyçen¥: V fn n p− ( ) 1 ≤ C. 1. Sidon S. Hinreichende Bedingungen für den Fourier-Charakter einer Trigonometrischen Reihe // J. London Math. Soc. – 1939. – 14, # 5. – P. 158 – 160. 2. Telqkovskyj S. A. Ob odnom dostatoçnom uslovyy Sydona yntehryruemosty tryhonometry- çeskyx rqdov // Mat. zametky. – 1973. – 14, # 3. – S.G317 – 328. 3. Nosenko G. L. Ob uslovyqx typa Sydona yntehryruemosty dvojn¥x tryhonometryçeskyx rqdov // Teoryq funkcyj y otobraΩenyj. – Kyev: Nauk. dumka, 1979. – S.G132 – 149. 4. Zaderej P. V. Ob uslovyqx yntehryruemosty kratn¥x tryhonometryçeskyx rqdov // Ukr. mat. Ωurn. – 1992. – 44, # 3. – S.G340 – 365. 5. Kuznecova O. Y. Konstant¥ Lebeha y approksymatyvn¥e svojstva lynejn¥x srednyx krat- n¥x rqdov Fur\e: Dys. … kand. fyz.-mat. nauk. – Doneck, 1985. – 115Gs. 6. Telqkovskyj S. A. Ob uslovyqx yntehryruemosty kratn¥x tryhonometryçeskyx rqdov // Tr. Mat. yn-ta AN SSSR. – 1983. – 164. – S.G180 – 188. 7. Kolmogorov A. N. Sur l’ordre be grandeur des coefficients de la serie be Fourier – Lebesgue // Bull. Acad. pol. sci. (A). – 1923. – P. 83 – 86. 8. Podkor¥tov A. N. Porqdok konstant Lebeha summ Fur\e po polyπdram // Vestn. Lenynhr. un-ta. Mat., mex., astron. – 1982. – 7. – S.G110 – 111. 9. Kuznecova O. Y. Ob odnom klasse N-mern¥x tryhonometryçeskyx rqdov // Mat. zametky. – 1998. – 63, # 3. – S.G402 – 406. Poluçeno 16.02.07 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
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spelling	umjimathkievua-article-31772020-03-18T19:47:45Z On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series Об условиях типа Сидона - Теляковского интегрируемости кратных тригонометрических рядов Ivashchuk, O. V. Zaderei, P. V. Pelagenko, E. N. Іващук, О. В. Задерей, П. В. Пелагенко, Є. Н. For the trigonometric series $$\sum_{k=0}^{\infty}a_k\sum_{l\in kV \setminus (k-1)V}e^{i(l, x)}, \quad a_k\rightarrow 0,\quad k\rightarrow \infty,$$ given on $[-\pi, \pi)^m$, where $V$ is some polyhedron in $R^m$, we prove that the inequality $$\int\limits_{T^m}\left\|\sum^{\infty}_{k=0} a_k \sum_{l\in kV\setminus(k-1)V}e^{i(l, x)} \right\| dx \leq C \sum^{\infty}_{k=0} (k+1) \|\Delta A_k\|,$$ holds if the coefficients $a_k$ satisfy the following conditions of the Sidon - Telyakovskii type: $$A_k\rightarrow\infty,\quad \|\Delta a_k\| \leq A_k, \quad \forall k \geq 0, \quad \sum^{\infty}_{k=0}(k+1) \|\Delta A_k\| Показано, що для тригонометричних рядів вигляду $$\sum_{k=0}^{\infty}a_k\sum_{l\in kV \setminus (k-1)V}e^{i(l, x)}, \quad a_k\rightarrow 0,\quad k\rightarrow \infty,$$ що задані на $[-\pi, \pi)^m$ , де $V$ — деякии поліедр у $R^m$ , виконується нерівність $$\int\limits_{T^m}\left\|\sum^{\infty}_{k=0} a_k \sum_{l\in kV\setminus(k-1)V}e^{i(l, x)} \right\| dx \leq C \sum^{\infty}_{k=0} (k+1) \|\Delta A_k\|,$$ якщо коефіцієнти $a_k$ задовольняють умови типу Сідона - Теляковського $$A_k\rightarrow\infty,\quad \|\Delta a_k\| \leq A_k, \quad \forall k \geq 0, \quad \sum^{\infty}_{k=0}(k+1) \|\Delta A_k\| Institute of Mathematics, NAS of Ukraine 2008-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3177 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 5 (2008); 579–585 Український математичний журнал; Том 60 № 5 (2008); 579–585 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3177/3101 https://umj.imath.kiev.ua/index.php/umj/article/view/3177/3102 Copyright (c) 2008 Ivashchuk O. V.; Zaderei P. V.; Pelagenko E. N.
spellingShingle	Ivashchuk, O. V. Zaderei, P. V. Pelagenko, E. N. Іващук, О. В. Задерей, П. В. Пелагенко, Є. Н. On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series
title	On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series
title_alt	Об условиях типа Сидона - Теляковского интегрируемости кратных тригонометрических рядов
title_full	On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series
title_fullStr	On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series
title_full_unstemmed	On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series
title_short	On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series
title_sort	on sidon-telyakovskii-type conditions for the integrability of multiple trigonometric series
url	https://umj.imath.kiev.ua/index.php/umj/article/view/3177
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On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series

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