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Temirkhanova, Ainur
##### Publications (6 of 6) Show all publications
Temirkhanova, A. (2015). Estimates for Discrete Hardy-type Operators in Weighted Sequence Spaces (ed.). (Doctoral dissertation). Paper presented at . Luleå tekniska universitet
Open this publication in new window or tab >>Estimates for Discrete Hardy-type Operators in Weighted Sequence Spaces
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This PhD thesis consists of an introduction and eight papers, which deal with questions of the validity of some new discrete Hardy type inequalities in weighted spaces of sequences and on the cone of non-negative monotone sequences, and their applications. In the introduction we give an overview of the area that serves as a frame for the rest of the thesis. In particular, a short description of the development of Hardy type inequalities is given. In Paper 1 we find necessary and sufficient conditions on weighted sequences $\omega_i$, $i=1, 2,...,n-1$, $u$ and $v$, for which the operator $$(S_{n}f)_i=\sum\limits_{k_1=1}^i\omega_{1,k_1}\cdots\sum\limits_{k_{n-1}=1}^{k_{n-2}} \omega_{n-1,k_{n-1}}\sum\limits_{j=1}^{k_{n-1}}f_j,~~ i\geq 1,~~~~~(1)$$ is bounded from $l_{p,v}$ to $l_{q,u}$ for $1<p\leq q<\infty$. In Paper 2 we prove a new discrete Hardy-type inequality $$\|Af\|_{q,u}\leq C\|f\|_{p,v},~~~~1<p\leq q<\infty,~~~~~~~~~~~(2)$$ where the matrix operator $A$ is defined by $\left(Af\right)_i:=\sum\limits_{j=1}^ia_{i,j}f_j,$ ~$a_{i, j}\geq 0$, where the entries $a_{i, j}$ satisfy less restrictive additional conditions than studied before. Moreover, we study the problem of compactness for the operator $A$, and also the dual result is stated, proved and discussed. In Paper 3 we derive the necessary and sufficient conditions for inequality (2) to hold for the case $1<q<p<\infty$. In Papers 4 and 5 we obtain criteria for the validity of the inequality (2) for slightly more general classes of matrix operators $A$ defined by $\left(Af\right)_j:=\sum\limits_{i=j}^\infty a_{i,j}f_i,$ ~$a_{i, j}\geq 0$, when $1<p, q<\infty$. Moreover, we study the problem of compactness for the operator $A$ for the case $1<p\leq q<\infty$, also the dual result is established and here we also give some applications of the main results. In Paper 6 we state boundedness for the operator of multiple summation with weights (1) in weighted sequence spaces for the case $1<q<p<\infty$. Paper 7 deals with new Hardy-type inequalities restricted to the cones of monotone sequences. The case $1<q<p<\infty$ is considered. Also some applications related to H\"{o}lder's summation method are pointed out. In Paper 8 we obtain necessary and sufficient conditions for which three-weighted Hardy type inequalities hold.

##### Place, publisher, year, edition, pages
Luleå tekniska universitet, 2015
##### Series
Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, ISSN 1402-1544
##### National Category
Mathematical Analysis
Mathematics
##### Note

Godkänd; 2015; 20151021 (aintem); Nedanstående person kommer att disputera för avläggande av teknologie doktorsexamen. Namn: Ainur Temirkhanova Ämne: Matematik/Mathematics Avhandling: Estimates for Discrete Hardy-type Operators in Weighted Sequence Spaces Opponent: Professor Mikhail Goldman, Dept of Nonlinear Analysis and Optimization, Peoples’Friendship University of Russia, Moscow, Russia. Ordförande: Professor Peter Wall, Avd för matematiska vetenskaper, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet, Luleå. Tid: Tisdag 8 december kl 10.00 Plats: E246, Luleå tekniska universitet

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-11-24Bibliographically approved
Temirkhanova, A. (2009). Some new boundedness and compactness results for discrete Hardy type operators with kernels (ed.). (Licentiate dissertation). Paper presented at . Luleå: Luleå tekniska universitet
Open this publication in new window or tab >>Some new boundedness and compactness results for discrete Hardy type operators with kernels
2009 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis consists of an introduction and three papers, which deal with some new discrete Hardy type inequalities. In the introduction we give an overview of the area that serves as a frame for the rest of the thesis. In particular, a short description of the development of Hardy type inequalities is given.In Paper 1 we prove a new discrete Hardy-type inequality $$\|Af\|_{q,u}\leq C\|f\|_{p,v},~~~~1$$where the matrix operator $A$ is defined by $\left(Af\right)_i:=\sum\limits_{j=1}^ia_{i,j}f_j,$ ~$a_{i, j}\geq 0$, where the entries $a_{i, j}$ satisfies less restrictive additional conditions than studied before. Moreover, we study the problem of compactness for the operator $A$, and also the dual result is stated, proved and discussed.In Paper 2 we derive the necessary and sufficient conditions for inequality (1) to hold for the case $1 In Paper 3 we consider an operator of multiple summation with weights in weighted sequence spaces, which cover a wide class of matrix operators and we state, prove and discuss both boundedness and compactness forthis operator, for the case$1

##### Place, publisher, year, edition, pages
Luleå: Luleå tekniska universitet, 2009. p. 6
##### Series
Licentiate thesis / Luleå University of Technology, ISSN 1402-1757
##### National Category
Mathematical Analysis
Mathematics
##### Note

Godkänd; 2009; 20090423 (aintem); LICENTIATSEMINARIUM Ämnesområde: Matematik/Mathematics Examinator: Professor Lars-Erik Persson, Luleå tekniska universitet Tid: Torsdag den 4 juni 2009 kl 13.00 Plats: D 2214, Luleå tekniska universitet

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-11-24Bibliographically approved
Temirkhanova, A. (2009). Weighted inequalities for a certain class of matrix operators: the case q (ed.). Paper presented at . Luleå: Department of Mathematics, Luleå University of Technology
Open this publication in new window or tab >>Weighted inequalities for a certain class of matrix operators: the case q
##### Place, publisher, year, edition, pages
Luleå: Department of Mathematics, Luleå University of Technology, 2009. p. 14
##### Series
Gula serien, ISSN 1400-4003 ; 2009:07
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-23190 (URN)5f08c080-8323-483e-b549-e919f3b56ced (Local ID)5f08c080-8323-483e-b549-e919f3b56ced (Archive number)5f08c080-8323-483e-b549-e919f3b56ced (OAI)
##### Note

Godkänd; 2009; 20120416 (andbra)

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-11-24Bibliographically approved
Oinarov, R. & Temirkhanova, A. (2008). Boundedness and compactness of a class of matrix operators in weighted sequence spaces (ed.). Journal of Mathematical Inequalities, 2(4), 555-570
Open this publication in new window or tab >>Boundedness and compactness of a class of matrix operators in weighted sequence spaces
2008 (English)In: Journal of Mathematical Inequalities, ISSN 1846-579X, E-ISSN 1848-9575, Vol. 2, no 4, p. 555-570Article in journal (Refereed) Published
##### Abstract [en]

Characterisations of bounded and compact multiple weighted summation operators from weighted ℓ p into weighted ℓ q spaces are established. Let 1<p,q<∞ , let ℓ p denote the space of all p -summable real sequences, let (ω i,k ) ∞ k=1 for i=1,2,…,n−1 , u=(u i ) ∞ i=1 and v=(v i ) ∞ i=1 be nonnegative sequences, and let ℓ p,v be the space of all sequences f=(f i ) ∞ i=1 such that fv=(f i v i ) ∞ i=1 ∈ℓ p , endowed with the natural norm ∥⋅∥ ℓ p,v defined by (∑ ∞ i=1 |f i v i | p ) 1/p . The n -tuple summation operator S n is defined by (S n f) i =∑ k 1 =1 i ω 1,k 1 ∑ k 2 =1 k 1 ω 2,k 2 ∑ k 3 =1 k 2 ω 3,k 3 ⋯∑ k n−1 =1 k n−2 ω n−1,k n−1 ∑ j=1 k n−1 f j . A necessary and sufficient condition is established for the inequality ∥S n f∥ ℓ q,u ≤C∥f∥ ℓ p,u to hold in the case 1<p≤q<∞ , for all sequences f∈ℓ q,u , where C is an absolute constant. This condition immediately yields a necessary and sufficient condition for S n to be a bounded operator from ℓ q,u into ℓ p,v . This result is a generalisation of a known result by K. F. Andersen and H. P. Heinig in the case n=1 when the operator S n reduces to a discrete Hardy operator of the form (S 1 f) i =∑ i j=1 f j . Finally, a necessary and sufficient condition is established for S n to be a compact operator from ℓ q,u into ℓ p,v when 1<p≤q<∞ . It should be noted that if n=2 then S 2 f can be expressed as a special matrix transformation of the form (Af) i =∑ i j=1 a ij f j .

##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-13733 (URN)d02eff0e-e315-4793-a1bb-9461d958962f (Local ID)d02eff0e-e315-4793-a1bb-9461d958962f (Archive number)d02eff0e-e315-4793-a1bb-9461d958962f (OAI)
##### Note

Upprättat; 2008; 20130627 (andbra)

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-11-24Bibliographically approved
Oinarov, R., Persson, L.-E. & Temirkhanova, A. (2008). Boundedness and compactness of a class of matrix operators: the case p (ed.). Paper presented at . Luleå: Department of Mathematics, Luleå University of Technology
Open this publication in new window or tab >>Boundedness and compactness of a class of matrix operators: the case p
##### Place, publisher, year, edition, pages
Luleå: Department of Mathematics, Luleå University of Technology, 2008. p. 17
##### Series
Gula serien, ISSN 1400-4003 ; 2008:05
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-22785 (URN)446de0ea-5b8c-47dc-98e3-23e8dae59fd1 (Local ID)446de0ea-5b8c-47dc-98e3-23e8dae59fd1 (Archive number)446de0ea-5b8c-47dc-98e3-23e8dae59fd1 (OAI)
##### Note

Godkänd; 2008; 20120425 (andbra)

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-11-24Bibliographically approved
Oinarov, R. & Temirkhanova, A. (2008). Boundedness and compactness of the operator of multiply summation with weights in weighted spaces of the sequences (ed.). Paper presented at . Luleå: Department of Mathematics, Luleå University of Technology
Open this publication in new window or tab >>Boundedness and compactness of the operator of multiply summation with weights in weighted spaces of the sequences
##### Place, publisher, year, edition, pages
Luleå: Department of Mathematics, Luleå University of Technology, 2008. p. 21
##### Series
Gula serien, ISSN 1400-4003 ; 2008:06
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-23665 (URN)7e680045-060c-4172-9cc3-2cc92e542b29 (Local ID)7e680045-060c-4172-9cc3-2cc92e542b29 (Archive number)7e680045-060c-4172-9cc3-2cc92e542b29 (OAI)
##### Note

Godkänd; 2008; 20120425 (andbra)

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-11-24Bibliographically approved

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