It is shown that every regular Krein-Feller eigenvalue problem can be transformed to a semidefinite Sturm-Liouville problem introduced by Atkinson. This makes it possible to transfer results between the corresponding theories. In particular, Prüfer angle methods become available for Krein-Feller problems.

This paper analyzes the generalization error of two-layer neural networks for computing ground state Schrödinger operator on add-dimensional hyperc...

Krein space operator-theoretic methods are used to prove that the operator (sgnx)-j-j is similar to a selfadjoint operator in the Hubert space ¿2(R) • Let L be a symmetric ordinary differential expression. Spectral properties of the operators associated with the weighted eigenvalue problem Lu = Xwu have been studied extensively. When w is positive, this problem leads to a selfadjoint problem in...

We consider the Krein realization of the Hilbert space for a massless scalar field in 1+1 dimensions. We find convergence criteria and the completion of the space of test functions S with the topology induced by the Krein scalar product. Finally, we show that the interpretation for the Fourier components as probability amplitudes for the momentum operator is lost in this case.

Whitney's extension problem asks the following: Given a compact set E?Rn and function f:E?R, how can we tell whether there exists F?Cm(Rn) such that F=f on E? A 2006 theorem of Charles Fefferman [6] answers this question in its full generality. In paper, establish version adapted for variants Whitney problem, including nonnegative extensions smooth selection problems. Among other things, genera...

We give a representation of the Gamma subordinator as a Krein functional of Brownian motion, using the known representations for stable subordinators and Esscher transforms. In particular, we have obtained Krein representations of the subordinators which govern the two parameter Poisson-Dirichlet family of distributions [23].

Abstract John’s fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ $\mathbb{R}^{d}$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids positions (affine images) of another $L$. Another, more recent direction consider logarithmically concave functions on instead bodies: we designate some spec...

The classical concept of Q-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized Q-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein ...