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Jan-Fredrik Olsen

Publications and source records attributed to Jan-Fredrik Olsen.

At least 19 recordsLinked to original sources
arXiv2026

Diverse yet consistent: How mathematicians position computational thinking across research and teaching

Recent research in mathematics education points to an "epistemic clash" when programming and computational thinking (CT) are leveraged alongside more established forms of mathematical thinking (MT). The emergence of generative AI emphasises the need to understand the mechanisms shaping relations between CT and MT. We address this need by analysing interviews with 15 mathematicians on their use of computations across their teaching and research activities. The interviews were conducted at a critical site with a history of integrating computations across its science and mathematics programs for more than 20 years. Drawing on Cultural Historical Activity Theory and Communities of Practice theory, we consider MT and CT as methodologies grounded in practice. We identify three perspectives shaping how mathematicians position CT: mathematical theory considered as a source of control, computations as a source of pragmatic reach, and real-world impact as a source of legitimacy. This three-perspectives model explains why mathematicians who emphasise real-world impact are most likely to carry programming into teaching, whereas those who position theoretical mathematics as authoritative are least likely to do so. Mathematicians working on numerical algorithms occupy an uneasy intermediate position. Our findings suggest that the perceived clash between MT and CT is not purely epistemic, but also ontological, as it depends on how computations are positioned within the goal of doing mathematics. For mathematics education, this implies that perceived meaningful integration with CT is mediated by context, and that more extensive use can be stabilised by leveraging authentic learning goals external to mathematics.

math.HO↗
arXiv2026

How Disciplinary Norms Influence Mathematicians' Views of Programming in Undergraduate Mathematics

Programming is deeply embedded in contemporary mathematical practice, yet its epistemic status in university mathematics teaching remains contested. Little is known about how mathematicians themselves understand the legitimacy of programming in their professional work, and how these views shape their teaching. We address this gap through semi-structured interviews with 15 mathematicians at a Northern European university with over two decades of systematic integration of programming across STEM subjects. Drawing on Cultural-Historical Activity Theory and Communities of Practice, we examine how mathematicians articulate the role of programming across research and teaching. We identify four epistemic archetypes - classical pure, classical applied, computational applied, and computational pure - each expressing coherent norms governing legitimate use of programming. Across archetypes, light-touch programming (e.g., numerical exploration) was widely used in research but largely invisible in teaching, where legitimacy was tied to more "substantive" integration. We argue that this gap reflects epistemic continuity across practices combined with teaching's focus on established analytic outcomes, which reduces the epistemic visibility of computational work. Given how mathematicians articulate legitimate uses of programming, our findings suggest that integration is most widely accepted when substantive programming is taught in dedicated courses, while light epistemic use - such as numerical exploration - is used to support learning in traditional theoretical courses. More extensive computational work is generally viewed as fitting naturally within specialised numerical or computational mathematics courses.

math.HO↗
arXiv2024

A sharp higher order Sobolev embedding

We obtain sharp embeddings from the Sobolev space $W^{k,2}_0(-1,1)$ into the space $L^1(-1,1)$ and determine the extremal functions. This improves on a previous estimate of the sharp constants of these embeddings due to Kalyabin.

math.FA↗
arXiv2022

Existence of Uncertainty Minimizers for the Continuous Wavelet Transform

Continuous wavelet design is the endeavor to construct mother wavelets with desirable properties for the continuous wavelet transform (CWT). One class of methods for choosing a mother wavelet involves minimizing a functional, called the wavelet uncertainty functional. Recently, two new wavelet uncertainty functionals were derived from theoretical foundations. In both approaches, the uncertainty of a mother wavelet describes its concentration, or accuracy, as a time-scale probe. While an uncertainty minimizing mother wavelet can be proven to have desirable localization properties, the existence of such a minimizer was never studied. In this paper, we prove the existence of minimizers for the two uncertainty functionals.

math.FA↗
arXiv2021

IKT$^ω$ and Łukasiewicz-models

In this note, we show that the first-order logic IK$^ω$ is sound with regard to the models obtained from continuum-valued Łukasiewicz-models for first-order languages by treating the quantifiers as infinitary strong disjunction/conjunction rather than infinitary weak disjunction/conjunction. Moreover, we show that these models cannot be used to provide a new consistency proof for the theory of truth IKT$^ω$ obtained by expanding IK$^ω$ with transparent truth because the models are inconsistent with transparent truth. Finally, we show that whether or not this inconsistency can be reproduced in the sequent calculus for IKT$^ω$ depends on how vacuous quantification is treated.

math.LO↗
arXiv2020

An operator theoretic approach to the Prime number theorem

In this short note, we establish an operator theoretic version of the Wiener-Ikehara tauberian theorem, and point out how this leads to a new proof of the Prime number theorem that should be accessible to anyone with a basic knowledge of operator theory.

math.FA↗
arXiv2017

Fatou and brother Riesz theorems in the infinite-dimensional polydisc

We study the boundary behavior of functions in the Hardy spaces on the infinite dimensional polydisk. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz-Zygmund type theorems for radial convergence. As a consequence one obtains easy new proofs of the brothers F. and M. Riesz theorems in infinite dimension. Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.

math.CV↗
arXiv2017

Balian-Low type theorems in finite dimensions

We formulate and prove finite dimensional analogs for the classical Balian-Low theorem, and for a quantitative Balian-Low type theorem that, in the case of the real line, we obtained in a previous work. Moreover, we show that these results imply their counter-parts on the real line.

math.CA↗
arXiv2015

Zeros of random functions generated with de Branges kernels

We study the point process given by the set of real zeros of random sums of orthonormal bases of reproducing kernels of de Branges spaces. Examples of these kernels are the cardinal sine, Airy and Bessel kernels. We find an explicit formula for the first intensity function in terms of the phase of the Hermite-Biehler function. We prove that the first intensity of the point process completely characterizes the underlying de Branges space. This result is a real version of the so called Calabi rigidity for GAFs proved by M. Sodin.

math.CA↗
arXiv2014

On a sharp estimate for Hankel operators and Putnam's inequality

We obtain a sharp norm estimate for Hankel operators with anti-analytic symbol for weighted Bergman spaces. For the classical Bergman space, the estimate improves the corresponding classical Putnam inequality for commutators of Toeplitz operators with analytic symbol by a factor of $1/2$, answering a recent conjecture by Bell, Ferguson and Lundberg. As an application, this yields a new proof of the de St. Venant inequality, which relates the torsional rigidity of a domain with its area.

math.FA↗
arXiv2012

Fourier multipliers for Hardy spaces of Dirichlet series

We obtain new results on Fourier multipliers for Dirichlet-Hardy spaces. As a consequence, we establish a Littlewood-Paley type inequality which yields a simple proof that the Dirichlet monomials form a Schauder basis for p>1.

math.CV↗
arXiv2012

A quantitative Balian-Low theorem

We study functions generating Gabor Riesz bases on the integer lattice. The classical Balian-Low theorem restricts the simultaneous time and frequency localization of such functions. We obtain a quantitative estimate that extends both this result and other related theorems.

math.CA↗
arXiv2011

Gap probabilities for the cardinal sine

We study the zero set of random analytic functions generated by a sum of the cardinal sine functions that form an orthogonal basis for the Paley-Wiener space. As a model case, we consider real-valued Gaussian coefficients. It is shown that the asymptotic probability that there is no zero in a bounded interval decays exponentially as a function of the length.

math.CV↗
arXiv2011

Sampling and interpolation in de Branges spaces with doubling phase

The de Branges spaces of entire functions generalise the classical Paley-Wiener space of square summable bandlimited functions. Specifically, the square norm is computed on the real line with respect to weights given by the values of certain entire functions. For the Paley-Wiener space, this can be chosen to be an exponential function where the phase increases linearly. As our main result, we establish a natural geometric characterisation, in terms of densities, for real sampling and interpolating sequences in the case when the derivative of the phase function merely gives a doubling measure on the real line. Moreover, a consequence of this doubling condition, is that the spaces we consider are one component model spaces. A novelty of our work is the application to de Branges spaces of techniques developed by Marco, Massaneda and Ortega-Cerdá for Fock spaces satisfying a doubling condition analogue to ours.

math.CA↗
arXiv2010

Local properties of Hilbert spaces of Dirichlet series

We show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the Hilbert space of Dirichlet series defined using these as weights. This extends results recently obtained describing the local behavior of Dirichlet series with square summable coefficients in terms of local integrability, boundary behavior, Carleson measures and interpolating sequences. As these spaces can be identified with functions spaces on the infinite-dimensional polydisk, this gives new results on the Dirichlet and Bergman spaces on the infinite dimensional polydisk, as well as the scale of Besov-Sobolev spaces containing the Drury-Arveson space on the infinite dimensional unit ball. We use both techniques from the theory of sampling in Paley-Wiener spaces, and classical results from analytic number theory.

math.CV↗
arXiv2010

From exact systems to Riesz bases in the Balian-Low theorem

We look at the time-frequency localisation of generators of lattice Gabor systems. For a generator of a Riesz basis, this localisation is described by the classical Balian-Low theorem. We establish Balian-Low type theorems for complete and minimal Gabor systems with a frame-type approximation property. These results describe how the best possible localisation of a generator is limited by the degree of control over the coefficients in approximations given by the system, and provide a continuous transition between the classical Balian-Low conditions and the corresponding conditions for generators of complete and minimal systems. Moreover, this holds for the non-symmetric generalisations of these theorems as well.

math.CA↗
arXiv2009

Modified zeta functions as kernels of integral operators

The modified zeta functions $\sum_{n \in K} n^{-s}$, where $K \subset \N$, converge absolutely for $\Re s > 1/2$. These generalise the Riemann zeta function which is known to have a meromorphic continuation to all of $\C$ with a single pole at $s=1$. Our main result is a characterisation of the modified zeta functions that have pole-like behaviour at this point. This behaviour is defined by considering the modified zeta functions as kernels of certain integral operators on the spaces $L^2(I)$ for symmetric and bounded intervals $I \subset \R$. We also consider the special case when the set $K \subset \N$ is assumed to have arithmetic structure. In particular, we look at local $L^p$ integrability properties of the modified zeta functions on the abscissa $\Re s=1$ for $p \in [1,\infty]$.

math.CA↗