| 1. |
- Ameur, Yacin, et al.
(författare)
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Interpolation classes and matrix monotone functions
- 2007
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Ingår i: Journal of operator theory. - 0379-4024 .- 1841-7744. ; 57:2, s. 409-427
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Tidskriftsartikel (refereegranskat)abstract
- An interpolation function of order n is a positive function -/+ on (0, infinity) such that vertical bar vertical bar -/+ (A)(1/2) T -/+ (A)-(1/2) vertical bar vertical bar <= max(vertical bar vertical bar T vertical bar vertical bar, vertical bar A(1/2)TA(-1/2) vertical bar vertical bar) for all n x ii matrices T and A such that A is positive definite. By a theorem of Donoghue, the class C-n of interpolation functions of order n coincides with the class of functions -/+ such that for each n-subset S = {lambda i}(n)(i=1)of (0,infinity) there exists a positive Pick function h on (0, co) interpolating -/+ at S. This note comprises a study of the classes C-n and their relations to matrix monotone functions of finite order. We also consider interpolation functions on general unital C*-algebras.
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| 2. |
- Araaya, Tsehaye, 1962-
(författare)
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The Symmetric Meixner-Pollaczek polynomials
- 2003
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by pn(λ)(x) instead of the standard notation pn(λ) (x/2, π/2), where λ > 0. The limiting case of these sequences of polynomials pn(0) (x) =limλ→0 pn(λ)(x), is obtained, and is shown to be an orthogonal sequence in the strip, S = {z ∈ ℂ : −1≤ℭ (z)≤1}.From the point of view of Umbral Calculus, this sequence has a special property that makes it unique in the Symmetric Meixner-Pollaczek class of polynomials: it is of convolution type. A convolution type sequence of polynomials has a unique associated operator called a delta operator. Such an operator is found for pn(0) (x), and its integral representation is developed. A convolution type sequence of polynomials may have associated Sheffer sequences of polynomials. The set of associated Sheffer sequences of the sequence pn(0)(x) is obtained, and is foundto be ℙ = {{pn(λ) (x)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied.The polynomials {pn(λ) (x)}∞n=0, λ < 0, are not orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard’s three term recurrence relation condition. For every λ ≤ 0, an associated nonstandard inner product is defined with respect to which pn(λ)(x) is orthogonal. Finally, the connection and linearization problems for the Symmetric Meixner-Pollaczek polynomials are solved. In solving the connection problem the convolution property of the polynomials is exploited, which in turn helps to solve the general linearization problem.
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| 3. |
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| 4. |
- Balkan, Andrew, et al.
(författare)
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Hardy spaces for the strip
- 2007
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 333:1, s. 347-364
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we shall study Hardy spaces of analytic functions in a strip S. Our main result is on one hand an intrinsic characterization of the spaces and on the second that polynomials are dense. We also present an orthogonal (in H-2(S)) basis of polynomials.
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| 5. |
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| 6. |
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| 7. |
- Ekstig, Kerstin, et al.
(författare)
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Sonja Lyttkens
- 2015
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Ingår i: Upsala Nya Tidning. - : Upsala nya tidning. - 1104-0173. ; 125:12, s. B13-B13
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Tidskriftsartikel (populärvet., debatt m.m.)abstract
- Sonja Lyttkens was born in 1919 and died in 2014. She was the second woman to get a PhD in mathematics in Sweden, and the first to become a university lecturer.
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| 8. |
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| 9. |
- Guo, Qi, 1957-
(författare)
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Minkowski Measure of Asymmetry and Minkowski Distance for Convex Bodies
- 2004
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of four papers about the Minkowski measure of asymmetry and the Minkowski (or Banach-Mazur) distance for convex bodies.We relate these two quantities by giving estimates for the Minkowski distance in terms of the Minkowski measure. We also investigate some properties of the Minkowski measure, in particular a stability estimate is given. More specifically, let C and D be n-dimensional convex bodies. Denote by As(C) and As(D) the Minkowski measures of asymmetry of C and D resp. and by d(C,D) the Minkowski distance between C and D.In Paper I, by using a linearisation method for affine spaces and affine maps and using a generalisation of a lemma of D.R. Lewis, we proved that d(C,D) < n(As(C) + As(D))/2 for all convex bodies C,D.In Paper II, by first proving some general existence theorems for a class of volume-increasing affine maps, we obtain the estimate that under the same conditions as in paper I, d(C,D) < (n-1) min(As(C),As(D)) + n.In Paper III we consider the Minkowski measure itself. We determine the Minkowski measures for convex hulls of sets of the form conv(C,p) where C is a convex set with known measure of asymmetry and p is a point outside C.In Paper IV, we focus on estimating the deviation of a convex body C from the simplex S if the Minkowski measure of C is close to the maximum value n (known to be attained only for the simplex). We prove that if As(C) > n - ε for 0 < ε < 1/δ where δ = 8(n+1), then d(C,S) < 1 + 8(n+1) ε .
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| 10. |
- Janson, Svante, 1955-, et al.
(författare)
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Higher moments of Banach space valued random variables
- 2015
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Ingår i: Memoirs of the American Mathematical Society. - : American Mathematical Society (AMS). - 0065-9266 .- 1947-6221. ; 238:1127, s. 1-110
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Tidskriftsartikel (refereegranskat)abstract
- We define the k:th moment of a Banach space valued random variable as the expectation of its k: th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general. Several chapters are devoted to results in special Banach spaces, including Hilbert spaces, C(K) and D[0,1]. The latter space is non-separable, which complicates the arguments, and we prove various preliminary results on e.g. measurability in D[0,1] that we need. One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix.
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| 11. |
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| 12. |
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| 13. |
- Kaijser, Sten, et al.
(författare)
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Asymmetry of Some Convex Bodies
- 2002
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Ingår i: DiscreteComput. Geom.. ; 27, s. 239-247
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Tidskriftsartikel (refereegranskat)
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| 14. |
- Kaijser, Sten
(författare)
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Biography of Matts Essén.
- 2004
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Ingår i: Complex Var. Theory Appl.. ; :49, s. 7-9
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Tidskriftsartikel (populärvet., debatt m.m.)
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| 15. |
- Kaijser, Sten, et al.
(författare)
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Hardy-type inequalities via convexity
- 2005
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Ingår i: Mathematical Inequalities & Applications. - : Element d.o.o.. - 1331-4343 .- 1848-9966. ; 8:3, s. 403-417
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Tidskriftsartikel (refereegranskat)abstract
- A recently discovered Hardy-Pólya type inequality described by a convex function is considered and further developed both in weighted and unweighted cases. Also some corresponding multidimensional and reversed inequalities are pointed out. In particular, some new multidimensional Hardy and Pólya-Knopp type inequalities and some new integral inequalities with general integral operators (without additional restrictions on the kernel) are derived
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| 16. |
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| 17. |
- Kaijser, Sten
(författare)
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Interpolation of Banach algebras and open sets
- 2001
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Ingår i: INTEGRAL EQUATIONS AND OPERATOR THEORY. - : BIRKHAUSER VERLAG AG. - 0378-620X. ; 41:2, s. 189-222
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Tidskriftsartikel (refereegranskat)abstract
- In this article we shall consider interpolation of commutative Banach algebras. Our main result is a description of the Gelfand space of maximal ideals for the interpolated Banach algebras. A case of special interest is the interpolation of algebras of bo
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| 18. |
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| 19. |
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| 20. |
- Kaijser, Sten, et al.
(författare)
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Lp-Boundedness of Two Singular Integral Operators of Convolution Type
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056
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Bokkapitel (refereegranskat)abstract
- We investigate boundedness properties of two singular integral operators defined on Lp-spaces (1 < p < ∞) on the real line, both as convolution operators on Lp(R) and on the spaces Lp(w), where w(x) = 1/2cosh πx/2. It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for p = 2 and weak boundedness for p = 1, and then using interpolation to obtain boundedness for 1 < p ≤ 2. To obtain boundedness also for 2 ≤ p < ∞, we use duality in the translation invariant case, while the weighted case is partly based on the expositions on the conjugate function operator in [7].π/2
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| 21. |
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| 22. |
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| 23. |
- Kaijser, Sten, et al.
(författare)
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On Carleman's and Knopp's inequalities
- 2002
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Ingår i: Journal of Approximation Theory. - : Elsevier BV. - 0021-9045 .- 1096-0430. ; 117:1, s. 140-151
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Tidskriftsartikel (refereegranskat)abstract
- A sharpened version of Carleman's inequality is proved. This result unifies and generalizes some recent results of this type. Also the “ordinary” sum that serves as the upper bound is replaced by the corresponding Cesaro sum. Moreover, a Carleman-type inequality with a more general measure is proved and this result may also be seen as a generalization of a continuous variant of Carleman's inequality, which is usually referred to as Knopp's inequality. A new elementary proof of (Carleman–)Knopp's inequality and a new inequality of Hardy–Knopp type is pointed out
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| 24. |
- Kaijser, Sten, et al.
(författare)
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Some approximation properties and nuclear operators in spaces of analytical functions
- 2019
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Ingår i: ADVANCES IN OPERATOR THEORY. - : TUSI MATHEMATICAL RESEARCH GROUP. - 2538-225X. ; 4:1, s. 265-283
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Tidskriftsartikel (refereegranskat)abstract
- We introduce and investigate a new notion of the approximation property AP([c]), where c = (c(n)) is an arbitrary positive real sequence, tending to infinity. Also, we study the corresponding notion of [c]-nuclear operators in Banach spaces. Some characterization of the AP([c]) in terms of tensor products, as well as sufficient conditions for a Banach space to have the AP([c]), are given. We give also sufficient conditions for a positive answer to the question: When it follows from the [c]-nuclearity of an adjoint operator the nuclearity of the operator itself. Obtained results are applied then to the study of properties of nuclear operators in some spaces of analytical functions. Many examples are given.
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| 25. |
- Kikonko, Mervis
(författare)
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Qualitative and Spectral theory of some regular non-definite Sturm-Liouville problems
- 2014
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this Licentiate thesis, we study some regular non-definite Sturm-Liouville problems. In this case, the weight function takes on both positive and negative signs on a given interval [a, b]. One feature of the non-definite Sturm-Liouville problem is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist.This thesis consists of three papers (papers A-C) and an introduction to this area, which puts these papers into a more general frame.In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper.In paper B we show that the interlacing property which holds in the one turning point case does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (−1, 2). We also present some theoretical results which support the numerical results.In paper C we extend results found in the paper by Jussi Behrndt et.al, in an essential way, to a case in which the weight function vanishes identically in a subinterval of [a, b]. In particular, we present some surprising numerical results on a specific problem in which the weight function is allowed to vanish identically on a subinterval of [−1, 2]. We also give some theoretical results which support these numerical examples.
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| 26. |
- Lundin, Sverker, 1975
(författare)
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Skolans matematik : En kritisk analys av den svenska skolmatematikens förhistoria, uppkomst och utveckling
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- I argue that common beliefs regarding mathematics originate in the practices of elementary mathematics instruction rather than in science. While learning how to solve mathematical problems in school, we come to believe that mathematics has a set of properties in itself, for instance that it is useful in everyday life, even though this is not necessarily so. I call this object of belief the mathematics of schooling (skolans matematik), while the system of practices by which the belief is produced is called mathematics education (skolmatematik). I introduce a terminology inspired by psychoanalytic theory to describe the peculiar properties of the mathematics of schooling and suggest that it can be understood as a sublime object of an ideology propagated by the system of education. To substantiate this claim I give an overview of the history of Swedish mathematics education, based on curricula, textbooks, discussions in teacher magazines, and other published material – covering in general terms the eighteenth to the twenty-first century. The historical narrative moves between social factors determining the practices of mathematics education, the changing ideas about mathematics expressed in these contexts, and the interplay between external social factors and internal “ideological” meaning. My conclusion is that while elementary arithmetics is, and should be, a part of common knowledge, the mathematics of schooling is something quite different. This object is thoroughly ideological and plays a central part in society mainly by making the social effects of mathematics education – keeping children away from production while sorting them – to appear as something else, namely as most often failed attempts to give children a necessary knowledge of mathematics.
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| 27. |
- Musonda, John
(författare)
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Orthogonal Polynomials, Operators and Commutation Relations
- 2017
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Orthogonal polynomials, operators and commutation relations appear in many areas of mathematics, physics and engineering where they play a vital role. For instance, orthogonal functions in general are central to the development of Fourier series and wavelets which are essential to signal processing. In particular, as demonstrated in this thesis, orthogonal polynomials can be used to establish the L2-boundedness of singular integral operators which is a fundamental problem in harmonic analysis and a subject of extensive investigations. The Lp-convergence of Fourier series is closely related to the Lp-boundedness of singular integral operators. Many important relations in physical sciences are represented by operators satisfying various commutation relations. Such commutation relations play key roles in such areas as quantum mechanics, wavelet analysis, representation theory, spectral theory, and many others.This thesis consists of three main parts. The first part presents a new system of orthogonal polynomials, and establishes its relation to the previously studied systems in the class of Meixner–Pollaczek polynomials. Boundedness properties of two singular integral operators of convolution type are investigated in the Hilbert spaces related to the relevant orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on L2-spaces, and estimates of the norms are obtained.The second part extends the investigation of the boundedness properties of the two singular integral operators to Lp-spaces on the real line, both in the weighted and unweighted spaces. It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for L2 and weak boundedness for L1, and then using interpolation to obtain boundedness for the intermediate spaces. To obtain boundedness for the remaining spaces, duality is used in the translation invariant case, while the weighted case is partly based on the methods developed by M. Riesz in his paper of 1928 for the conjugate function operator.The third and final part derives simple and explicit formulas for reordering elements in an algebra with three generators and Lie type relations. Centralizers and centers are computed as an example of an application of the formulas.
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| 28. |
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| 29. |
- Musonda, John, 1981-
(författare)
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Reordering in Noncommutative Algebras, Orthogonal Polynomials and Operators
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The main object studied in this thesis is the multi-parametric family of unital associative complex algebras generated by the element $Q$ and the finite or infinite set $\{S_j\}_{j\in J}$ of elements satisfying the commutation relations $S_jQ=\sigma_j(Q)S_j$, where $\sigma_j$ is a polynomial for all $j\in J$. A concrete representation is given by the operators $Q_x(f)(x)=xf(x)$ and $\alpha_{\sigma_j}(f)(x)=f(\sigma_j(x))$ acting on polynomials or other suitable functions. The main goal is to reorder arbitrary elements in this family and some of its generalizations, and to study properties of operators in some representing operator algebras, including their connections to orthogonal polynomials. For $J=\{1\}$ and $\sigma(x)=x+1$, the above commutation relations reduce to the famous classical Heisenberg--Lie commutation relation $SQ-QS=S$. Reordering an element in $S$ and $Q$ means to bring it, using the commutation relation, into a form where all elements $Q$ stand either to the left or to the right. For example, $SQ^2=Q^2S+2QS+S$. In general, one can use the commutation relation $SQ-QS=S$ successively and transform for any positive integer $n$ the element $SQ^n$ into a form where all elements $Q$ stand to the left. The coefficients which appear upon reordering in this case are the binomial coefficients. General reordering formulas for arbitrary elements in noncommutative algebras defined by commutation relations are important in many research directions, open problems and applications of the algebras and their operator representations. In investigation of the structure, representation theory and applications of noncommutative algebras, an important role is played by the explicit description of suitable normal forms for noncommutative expressions or functions of generators. Further investigation of the operator representations of the commutation relations by difference type operators on Hilbert function spaces leads to interesting connections to functional analysis and orthogonal polynomials. This thesis consists of two main parts. The first part is devoted to the multi-parametric family of algebras introduced above. General reordering formulas for arbitrary elements in this family are derived, generalizing some well-known results. As an example of an application of the formulas, centralizers and centers are computed. Some operator representations of the above algebras are also described, including considering them in the context of twisted derivations. The second part of this thesis is devoted to a special representation of these algebras by difference operators associated with action by shifts on the complex plane. It is shown that there are three systems of orthogonal polynomials of the class of Meixner--Pollaczek polynomials that are connected by these operators. Boundedness properties of two singular integral operators of convolution type connected to these difference operators are investigated in the Hilbert spaces related to these systems of orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on the $L^2$-spaces and estimates of the norms are obtained. This investigation is also extended to $L^p$-spaces on the real line where it is proved again that the two operators are bounded.
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| 30. |
- Musonda, John, et al.
(författare)
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Three systems of orthogonal polynomials and L2-boundedness of associated operators
- 2018
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 459:1, s. 464-475
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we describe three systems of orthogonal polynomials belonging to the class of Meixner-Pollaczek polynomials, and establish some useful connections between them in terms of three basic operators that are related to them. Furthermore, we investigate boundedness properties of two other operators, both as convolution operators in the translation invariant case where we use Fourier transforms and for the weights related to the relevant orthogonal polynomials. We consider only the most important but also simplest case of L-2-spaces. However, in subsequent papers, we intend to extend the study to L-p-spaces (1 < p < infinity).
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| 31. |
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| 32. |
- Prytz, Johan, 1975-
(författare)
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Speaking of Geometry : A study of geometry textbooks and literature on geometry instruction for elementary and lower secondary levels in Sweden, 1905-1962, with a special focus on professional debates
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This dissertation deals with geometry instruction in Sweden in the period 1905-1962. The purpose is to investigate textbooks and other literature used by teachers in elementary schools (ES) and lower secondary schools (LSS) – Folkskolan and Realskolan – connection to geometry instruction. Special attention is given to debates about why a course should be taught and how the content should be communicated.In the period 1905-1962, the Swedish school system changed greatly. Moreover, in this period mathematics instruction was reformed in several countries and geometry was a major issue; especially, classical geometry based on the axiomatic method. However, we do not really know how mathematics instruction changed in Sweden. Moreover, in the very few works where the history of mathematics instruction in Sweden is mentioned, the time before 1950 is often described in terms of “traditional”, “static” and “isolation”.In this dissertation, I show that geometry instruction in Sweden did change in the period 1905-1962: geometry instruction in LSS was debated; the axiomatic method and spatial intuition were major issues. Textbooks for LSS not following Euclid were produced also, but the axiomatic method was kept. By 1930, these alternative textbooks were the most popular.Also the textbooks in ES changed. In the debate about geometry instruction in ES, visualizability was a central concept.Nonetheless, some features did not change. Throughout the period, the rationale for keeping axiomatic geometry in LSS was to provide training in reasoning. An important aspect of the debate on geometry instruction in LSS is that the axiomatic method was the dominating issue; other issues, e.g. heuristics, were not discussed. I argue that a discussion on heuristics would have been relevant considering the final exams in the LSS; in order to succeed, it was more important to be a skilled problem solver than a master of proof.
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| 33. |
- Sigstam, Kibret, 1959-
(författare)
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Optimization and Estimation of Solutions of Riccati Equations
- 2004
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of three papers on topics related to optimization and estimation of solutions of Riccati equations. We are concerned with the initial value problemf'+f² =r², f(0)=0, (*)and we want to optimiseF(T)= ∫0T f(t) dtwhen r is allowed to vary over the set R(φ ) of all equimeasurable rearrangements of a decreasing function φ and its convex hull CR(φ). In the second paper we give a new proof of a lemma of Essén giving lower and upper bounds for the solution to the above equation, when r is increasing. We also generalize the lemma to a more general equation.It was proved by Essén that the infimum of F(T) over R(φ) and RC(φ) is attained by the solution f of (*) associated to the increasing rearrangement of an element in R(φ). The supremum of F(T) over RC(φ) is obtained for the solution associated to a decreasing function p, though not necessarily the decreasing rearrangement φ, of an element in R(φ). By changing the perspective we determine the function p that solves the supremum problem.
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| 34. |
- Silvestrov, Sergei, et al.
(författare)
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Interpolation classes, operator and matrix monotone functions
- 2006
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Ingår i: Proceedings of the Estonian Academy of Sciences. Physics & Mathematics. - 1406-0086. ; 55:3, s. 141-145
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Tidskriftsartikel (refereegranskat)abstract
- Connections between interpolation spaces, Pick functions, and matrix monotone functions are investigated. Characterizations, inclusion results, open problems, and conjectures on these function classes and their interrelations are presented.
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| 35. |
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| 36. |
- Sunehag, Peter, 1976-
(författare)
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Interpolation of Subcouples, New Results and Applications
- 2003
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Suppose that X and Y are Banach couples and suppose that there is a bounded linear couple map Q from Y to X which has the property that Q restricted to the endpoint spaces is injective and the images of the endpointspaces of Y are closed in the endpoint spaces of X, then we say that Y is a subcouple of X. If F is an interpolation functor we want to know how F(Y) is related to F(X). In particular we want to know for which F it holds that Q is an injection that maps F(Y) onto a closed subspace of F(X). In recent years interest has been paid to subcouples of finite codimension and in particular to subcouples of codimension one. We will in this thesis present an interpolation theory for subcouples of codimension one and then generalize it to finite codimension. Our theory will include both a larger class of couples and a larger class of interpolation functors than earlier results.The interpolation method that will be considered is the regular real method. Our general theory will imply older results by Kalton, Ivanov and Löfström. We will use the theory to answer questions about Hardy-type inequalities that were raised by Krugljak, Maligranda and Persson in 1999 and our new theory will also answer a question concerning interpolation of Banach algebras.
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