arXiv subjects
Publications and source records attributed to Steen Pedersen.
Let $\Q=[0,1)^d$ denote the unit cube in $d$-dimensional Euclidean space \Rd and let \T be a discrete subset of \Rd. We show that the exponentials $e_t(x):=exp(i2πtx)$, $t\in\T$ form an othonormal basis for $L^2(\Q)$ if and only if the translates $\Q+t$, $t\in\T$ form a tiling of \Rd.
Let $D$ be a bounded domain in ${\Bbb R}^n$ whose boundary has a Minkowski dimension $α<n$. Suppose that $E_Λ= {\{e^{2 πi x \cdot λ}\}}_{λ\in Λ}$, $Λ$ an infinite discrete subset of ${\Bbb R}^n$, is a frame of exponentials for $L^2(D)$, with frame constants $A,B$, $A \leq B$. Then if $$ R \ge C{(\frac{{B|\partial D|}_α}{A|D|} )}^ {\frac{1}{n-α}},$$ where $C$ depends only on the ambient dimension $n$ and ${|\partial D|}_α$ denotes the Minkowski content, then every cube of sidelength $R$ contains at least one element of $Λ$. We give examples that illustrate the extent to which our estimates are sharp.
We consider the problem of finding commuting self-adjoint extensions of the partial derivatives {(1/i)(\partial/\partial x_j):j=1,...,d} with domain C_c^\infty(Ω) where the self-adjointness is defined relative to L^2(Ω), and Ωis a given open subset of R^d. The measure on Ωis Lebesgue measure on R^d restricted to Ω. The problem originates with I.E. Segal and B. Fuglede, and is difficult in general. In this paper, we provide a representation-theoretic answer in the special case when Ω=I\timesΩ_2 and I is an open interval. We then apply the results to the case when Ωis a d-cube, I^d, and we describe possible subsets Λof R^d such that {e^(i2πλ\dot x) restricted to I^d:λ\inΛ} is an orthonormal basis in L^2(I^d).
Production of heat shock proteins are induced when a living cell is exposed to a rise in temperature. The heat shock response of protein DnaK synthesis in E.coli for temperature shifts from temperature T to T plus 7 degrees, respectively to T minus 7 degrees is measured as function of the initial temperature T. We observe a reversed heat shock at low T. The magnitude of the shock increases when one increase the distance to the temperature $T_0 \approx 23^o$, thereby mimicking the non monotous stability of proteins at low temperature. Further we found that the variation of the heat shock with T quantitatively follows the thermodynamic stability of proteins with temperature. This suggest that stability related to hot as well as cold unfolding of proteins is directly implemented in the biological control of protein folding. We demonstrate that such an implementation is possible in a minimalistic chemical network.
We show that certain iteration systems lead to fractal measures admitting exact orthogonal harmonic analysis.
We consider self-similar measures $μ$ with support in the interval $0\leq x\leq 1$ which have the analytic functions $\left\{e^{i2πnx}:n=0,1,2,... \right\} $ span a dense subspace in $L^{2}(μ) $. Depending on the fractal dimension of $μ$, we identify subsets $P\subset \mathbb{N}_{0}=\{0,1,2,... \} $ such that the functions $\{e_{n}:n\in P\} $ form an orthonormal basis for $L^{2}(μ) $. We also give a higher-dimensional affine construction leading to self-similar measures $μ$ with support in $\mathbb{R}^ν$. It is obtained from a given expansive $ν$-by-$ν$ matrix and a finite set of translation vectors, and we show that the corresponding $L^{2}(μ) $ has an orthonormal basis of exponentials $e^{i2πλ\cdot x}$, indexed by vectors $λ$ in $\mathbb{R}^ν$, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.
We describe a class of measurable subsets $Ω$ in $\br^d$ such that $L^2(Ω)$ has an orthogonal basis of frequencies $e_λ(x)=e^{i2πλ\cdot x}(x\inΩ)$ indexed by $λ\inΛ\subset\br^d$. We show that such spectral pairs $(Ω,Λ)$ have a self-similarity which may be used to generate associated fractal measures $μ$ with Cantor set support. The Hilbert space $L^2(μ)$ does not have a total set of orthogonal frequencies, but a harmonic analysis of $μ$ may be built instead from a natural representation of the Cuntz C$^*$- algebra which is constructed from a pair of lattices supporting the given spectral pair $(Ω,Λ)$. We show conversely that such a pair may be reconstructed from a certain Cuntz-representation given to act on $L^2(μ)$.