scholarly journals Orthogonal expansions of vectors in a Hilbert space for non-Gaussian measures

1982 ◽  
Vol 86 (4) ◽  
pp. 638-638
Author(s):  
Yoshiaki Okazaki
Keyword(s):  
Hilbert Space ◽  
Gaussian Measures ◽  
Orthogonal Expansions ◽  
Non Gaussian

THE EUCLIDEAN DISTORTION OF FLAT TORI

2013 ◽  
Vol 05 (02) ◽  
pp. 205-223 ◽  
Author(s):  
ISHAY HAVIV ◽  
ODED REGEV
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Upper Bound ◽  
Gaussian Function ◽  
Smoothing Parameter ◽  
Gaussian Measures ◽  
Dimensional Lattice ◽  
Flat Tori ◽  
Euclidean Distortion

We show that for every n-dimensional lattice [Formula: see text] the torus [Formula: see text] can be embedded with distortion [Formula: see text] into a Hilbert space. This improves the exponential upper bound of O(n3n/2) due to Khot and Naor (FOCS 2005, Math. Ann. 2006) and gets close to their lower bound of [Formula: see text]. We also obtain tight bounds for certain families of lattices. Our main new ingredient is an embedding that maps any point [Formula: see text] to a Gaussian function centered at u in the Hilbert space [Formula: see text]. The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine–Zolotarev bases.

p-ADIC HILBERT SPACE REPRESENTATION OF QUANTUM SYSTEMS WITH AN INFINITE NUMBER OF DEGREES OF FREEDOM

1996 ◽  
Vol 10 (13n14) ◽  
pp. 1665-1673 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
ANDREW KHRENNIKOV
Keyword(s):  
Hilbert Space ◽  
Infinite Number ◽  
Degrees Of Freedom ◽  
Quantum Systems ◽  
Space Representation ◽  
Gaussian Measures ◽  
Infinite Dimensional ◽  
Integrable Functions ◽  
Symmetric Operators ◽  
Square Integrable Functions

Gaussian measures on infinite-dimensional p-adic spaces are introduced and the corresponding L2-spaces of p-adic valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in p-adic L2-spaces. p-adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in L2-spaces with respect to a p-adic Gaussian measure.

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