6 research outputs found
ON BESSEL-RIESZ OPERATORS
We consider a class of conv olution operator denoted ϕα W obtained by convolution with a generalized function expressible in terms of the Bessel function on first kind γ J with argument the distribution ( ) P ± i0 . We study some elementary properties of the operator ϕα W like the semigroup property ϕ = ϕ α β α+β W W W ; and ( +m2 ) α α−2 W ϕ = W for α > 2 where ( +m2 ) is the Klein-Gordon ultrahyperbolic operator. Moreover we prove that the operator ϕα W may be consider as a negative power of the Klein-Gordon operat
On the Inversion of Bessel Ultrahyperbolic Kernel of Marcel Riesz
We define the Bessel ultrahyperbolic Marcel Riesz operator on the function f by Uαf=RαB*f, where RαB is Bessel ultrahyperbolic kernel of Marcel Riesz, α…C, the symbol * designates as the convolution, and f∈S, S is the Schwartz space of functions. Our purpose in this paper is to obtain the operator Eα=Uα-1 such that, if Uαf=φ, then Eαφ=f
Learning Theory and Approximation
The main goal of this workshop – the third one of this type at the MFO – has been to blend mathematical results from statistical learning theory and approximation theory to strengthen both disciplines and use synergistic effects to work on current research questions. Learning theory aims at modeling unknown function relations and data structures from samples in an automatic manner. Approximation theory is naturally used for the advancement and closely connected to the further development of learning theory, in particular for the exploration of new useful algorithms, and for the theoretical understanding of existing methods. Conversely, the study of learning theory also gives rise to interesting theoretical problems for approximation theory such as the approximation and sparse representation of functions or the construction of rich kernel reproducing Hilbert spaces on general metric spaces. This workshop has concentrated on the following recent topics: Pitchfork bifurcation of dynamical systems arising from mathematical foundations of cell development; regularized kernel based learning in the Big Data situation; deep learning; convergence rates of learning and online learning algorithms; numerical refinement algorithms to learning; statistical robustness of regularized kernel based learning
Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis
The positivity of the energy in relativistic quantum mechanics implies that
wave functions can be continued analytically to the forward tube T in complex
spacetime. For Klein-Gordon particles, we interpret T as an extended (8D)
classical phase space containing all 6D classical phase spaces as symplectic
submanifolds. The evaluation maps of wave functions on T are
relativistic coherent states reducing to the Gaussian coherent states in the
nonrelativistic limit. It is known that no covariant probability interpretation
exists for Klein-Gordon particles in real spacetime because the time component
of the conserved "probability current" can attain negative values even for
positive-energy solutions. We show that this problem is solved very naturally
in complex spacetime, where is interpreted as a probability
density on all 6D phase spaces in T which, when integrated over the "momentum"
variables y, gives a conserved spacetime probability current whose time
component is a positive regularization of the usual one. Similar results are
obtained for Dirac particles, where the evaluation maps are spinor-valued
relativistic coherent states. For free quantized Klein-Gordon and Dirac fields,
the above formalism extends to n-particle/antiparticle coherent states whose
scalar products are Wightman functions. The 2-point function plays the role of
a reproducing kernel for the one-particle and antiparticle subspaces.Comment: 252 pages, no figures. Originally published as a book by
North-Holland, 1990. Reviewed by Robert Hermann in Bulletin of the AMS Vol.
28 #1, January 1993, pp. 130-132; see http://wavelets.co