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  • 1.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    A generalization of a theorem of G. Freud on the differentiability of functions1985In: Acta Scientarum Mathematicarum, ISSN 0001-6969, Vol. 49, no 1-4, p. 271-281Article in journal (Refereed)
  • 2.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    A note on capacity and Hausdorff measure in homogeneous spaces1997In: Potential analysis, Vol. 6Article in journal (Refereed)
  • 3.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A note on Gram-Schmidt's algorithm for a general angle2009In: Normat, ISSN 0801-3500, Vol. 57, no 4, p. 173-179Article in journal (Refereed)
    Abstract [en]

    The Gram-Schmidt algorithm produces a pairwise ortogonal set of vectors from a linearly independent set in a general vector space with an inner product. We give an algorithm that produces vectors that have pairwise the same angle, in all cases where this is possible.

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  • 4.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    A note on the Carleman condition for determinacy of moment problems1987In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 25, no 2, p. 289-294Article in journal (Refereed)
  • 5.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    A note on the harmonic derivative2004In: Real Analysis Exchange, ISSN 0147-1937, Vol. 30, p. 11-21Article in journal (Refereed)
  • 6.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Bernstein's analyticity theorem for binary differences1999In: Mathematische Annalen, ISSN 0025-5831 (paper), 1432-1807 (e), Vol. 315, no 2, p. 251-261Article in journal (Refereed)
  • 7.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Bernstein's analyticity theorem for quantum differences2007In: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 57, no 1, p. 67-73Article in journal (Refereed)
    Abstract [en]

    We consider real valued functions f defined on a subinterval I of the positive real axis and prove that if all of f's quantum differences are nonnegative then f has a power series representation on I. Further, if the quantum differences have fixed sign on I then f is analytic on I.

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  • 8.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Bessel potentials and extension of continuous functione on compact sets1975In: Arkiv för matematik, ISSN 0004-2080 (p), 1871-2487 (e), Vol. 13, no 2, p. 263-271Article in journal (Refereed)
    Abstract [en]

    We characterize the compact sets K in the n-dimensional Euclidean space with capacity zero relative to a certain kernel as exactly those sets for which every continous function on K has an extension to a continuous potential in the full space. A special case in the Bessel kernel and the related capacity.

  • 9.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Beurling's analyticity theorem for quantum differences2008In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 127, no 3, p. 369-380Article in journal (Refereed)
    Abstract [en]

    A theorem of Beurling states that if f satisfies , n = 1, 2,..., for some 0 < ρ < 2, on a real interval I, then f is analytic in a rhombus containing I. We study the corresponding problem for the quantum differences Δ n f (q, x), q > 1, n = 1, 2,..., for functions defined on (0, ∞) and prove quantitative and qualitative analogues of Beurling’s result. We also characterize the analyticity of f on subintervals of (0, ∞) in q-analytic terms.

  • 10.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Capacities of compact sets in linear subspaces of R^n1978In: Pacific Journal of Mathematics, ISSN 0030-8730, Vol. 78, no 1, p. 261-266Article in journal (Refereed)
    Abstract [en]

    We characterize the exceptional sets for Besov spaces in the n-dimensional Euclidean space by an extension property for continuous functions and prove an inequality between Bessel and Besov capacities.

  • 11.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Continuous functions and Riesz type potentials in homogeneous spaces2015In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 3, p. 495-511Article in journal (Refereed)
    Abstract [en]

    We develop a potential theory for a Riesz type kernel in a homogeneous space and characterize the compact sets K with capacity zero as the sets K for which every continous function f on K is the restriction to K of a continuous potential Uσfk of an absolutely continuous measure σ f supported in an arbitrarily small neighbourhood of K. The measure σ f can be choosen as a suitable restriction of a single measure σ that only depends on the set K and the kernel k.

  • 12.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Explaining Kummer's test2015Manuscript (preprint) (Other academic)
    Abstract [en]

    Mathematical theorems and their proofs are often described as being beautyful, natural or explanatory. This paper treats the last case  and discusses explanatory versus non explanatory proofs using  a convergence theorem for positive series due to Kummer, called Kummer's test. This example was studied by Pringsheim already in the beginning in the last century and has recently been discussed by Hafner and Mancuso in relation to Steiner's criteria for an explanatory proof. We provide a mathematical analysis of Kummer's test and give a proof that we claim is explanatory in this sense. Besides this we answer a number of questions that are frequently asked about the test.

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  • 13.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Local and global weighted norm inequalities for thr sharp function and the Hardy-Littlewood maximal function1993In: Ricerche di Matamatica, Vol. 42, no 2Article in journal (Refereed)
  • 14.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Nonlinear potential theory in Lebesque spaces with mixed norm1988In: Potential Theory, Prague, 1987 / [ed] J. Kral, J. Lukes, I. Netuka, J. Vesely, New York: Plenum Publishing Corporation , 1988, p. 325-331Conference paper (Refereed)
    Abstract [en]

    The classical potential theory for the Riesz kernel in the d-dimentional Euclidean space has been generalized in many different ways. We are here concerned with the L^p-potential theory which appeared around 1970 in papers by V.G. Maz'ya, V. P, Havin and N. G. Mayers. In this paper we extend that theory to the so called Lebesgue spaces with mixed norm.

  • 15.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On Lp-differentiability and difference properties of functions1982In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 74, no 2, p. 153-168Article in journal (Refereed)
  • 16.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC)2019In: Probability and Mathematical Statistics, ISSN 0208-4147Article in journal (Refereed)
    Abstract [en]

    Let Y be a standard Gamma(k) distributed random variable (rv), k > 0, and let X be an independent positive rv. If X has a hyperbolically monotone density of order k (HMk), then Y ·X and Y/X are generalized gamma convolutions (GGC). This extends work by Roynette et al. and Behme and Bondesson. The same conclusion holds with Y replaced by a finite sum of independent gamma variables with sum of shape parameters at most k. Both results are applied to subclasses of GGC.

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  • 17.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On Multivariate Hyperbolically Completely Monotone Densities and Their Laplace transforms2015Manuscript (preprint) (Other academic)
    Abstract [en]

    We study classes of multivariate (bivariate) random variables with hyperbolically completely monotone densities (MVHCM) and prove that these classes are closed with respect to the Laplace transform. We then use this result to define new classes of bivariate generalized gamma convolutions (BVGGC_L) that contain Bondesson's class BVGGC in the strong sense.

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  • 18.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On ordinary differentiability of Bessel potentials1984In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 44, no 3, p. 325-352Article in journal (Refereed)
  • 19.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On properties of functions with conditions on their mean oscillation over cubes1982In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 20, no 2, p. 275-291Article in journal (Refereed)
  • 20.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    On s-sets and mutual absolute continuity of measures on homogeneous spaces1997In: Manuscripta Mathematica, ISSN 0025-2611 (p), 1432-1785 (e), Vol. 94, p. 169-186Article in journal (Refereed)
  • 21.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On the almost everywhere differentiability of the metric projection on closed sets in lp(ℝn), 2 < p < ∞2018In: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 68, no 143, p. 943-951Article in journal (Refereed)
    Abstract [en]

    Let F be a closed subset of ℝn and let P(x) denote the metric projection (closest point mapping) of x ∈ ℝn onto F in lp-norm. A classical result of Asplund states that P is (Fréchet) differentiable almost everywhere (a.e.) in ℝn in the Euclidean case p = 2. We consider the case 2 < p < ∞ and prove that the ith component Pi(x) of P(x) is differentiable a.e. if Pi(x) 6= xi and satisfies Hölder condition of order 1/(p−1) if Pi(x) = xi.

  • 22.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Polar sets and capacitary potentials in homogeneous spaces2013In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, p. 771-783Article in journal (Refereed)
    Abstract [en]

    A set E in a space X is called a polar set in X, relative to a kernel k(x; y), if thereis a nonnegative measure in X such that the potential Uk(x) = ∞ precisely when x ∈ E. Polarsets have been characterized in various classical cases as G-sets (countable intersections of opensets) with capacity zero. We characterize polar sets in a homogeneous space (X; d; ) for severalclasses of kernels k(x; y), among them the Riesz -kernels and logarithmic Riesz kernels. The latercase seems to be new even in Rn.

  • 23.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The fundamental theorem of algebra2014In: Proofs from THE BOOK / [ed] Martin Aigner, Günter M. Ziegler, Berlin-Heidelberg: Springer, 2014, 5, p. 147-149Chapter in book (Refereed)
    Abstract [en]

    Everyn nonconstant polynomial with complex coefficients has at least one root in the field of complex numbers

  • 24.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Thinness in non-linear potential theory for non-isotropic Sobolev spaces1997In: Ann. Acad. Aci. Fenn., Vol. 22Article in journal (Refereed)
  • 25.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Weighted Lp-inequalities for multi-parameter Riesz type potentials and strong fractional maximal operators2007In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 337, p. 317-333Article in journal (Refereed)
    Abstract [en]

    We prove weighted L^p-inequalities  for multi-parameter Riesz type potentials, strong fractional maximal operators and their dyadic counterparts. Our proofs avoids the Good-Lambda inequalities used earlier in the R^m-case and are based on our integrated multi-parameter summation by parts lemma, that might be of independent interest.

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  • 26.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Weighted norm inequalities for Riesz potentials and fractional maximal functions in mixed norm Lebesgue spaces1990In: Studia Mathematica, ISSN 0039-3233, Vol. 97, no 3, p. 239-244Article in journal (Refereed)
    Abstract [en]

    Norm inequalities for Riesz potentials and fractional maximal functions in weighted Lebesgue spaces were proved by Muckenhoupt and Wheeden in the 1970's. We prove such inequalities in weighted mixed norm Lebesgue spaces for the full range oh indices. Our proofs make extensive use of the concept of independence of weights in the Muckenhoupt classes.

  • 27.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Wolff's inequality in multi-parameter Morrey spaces2012In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 271, no 3, p. 781-787Article in journal (Refereed)
    Abstract [en]

    We prove Wolff inequalities for multi-parameter Riesz potentials and Wolff potentials in Lebesque spaces Lp(Rd) and multi-parameter Morrey spaces Lp λ(Rd), where Rd = Rn1 × Rn2 ×···× Rnk , λ = (λ1,...,λk) and 0 < λi ≤ ni, 1 ≤ i ≤ k, in the dyadic case as well as in the non-dyadic (continuous) case.

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  • 28.
    Sjödin, Tord
    et al.
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Løvtrup, Søren
    Umeå University, Faculty of Science and Technology, Studies in Biology and Environmental Sciences.
    Biomathematics and the fate map of the amphibian blastula1990In: Rivista di Biologia - Biology forum, ISSN 0035-6050, Vol. 83, no 1, p. 81-92Article in journal (Refereed)
    Abstract [en]

    A fate map of the amphibian embryo has been drawn by a computer on the basis of two simple and well-known processes of the cell differentiation. The two processes are related in so far as they both consist of the formation of a new type of cells, here called f-cells. The first process is the spontaneous formation of the Ruffini cells along the marginal edge around the circumference of the embryo, slightly below the equator. When this occurs, they will induce the neighbouring cells located towards the animal pole to undergo the same differentiation, and the induction, which seems to be rather an activation, spreads from cell to cell. The actual shape of the map is due to the fact that the formation of the Ruffine cells occurs along a temporal gradient from the dorsal to the ventral side.

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