On the properties of the integer translates of a square integrable function

2010 ◽  
pp. 233-249 ◽  
Author(s):  
Eugenio Hernández ◽  
Hrvoje Šikić ◽  
Guido Weiss ◽  
Edward Wilson
Keyword(s):  
Integrable Function ◽  
Integer Translates ◽  
Square Integrable Function

scholarly journals What is the Wigner Function Closest to a Given Square Integrable Function?

2018 ◽  
Vol 50 (5) ◽  
pp. 5161-5197 ◽  
Author(s):  
J. S. Ben-Benjamin ◽  
L. Cohen ◽  
N. C. Dias ◽  
P. Loughlin ◽  
J. N. Prata
Keyword(s):  
Wigner Function ◽  
Integrable Function ◽  
Square Integrable Function

scholarly journals LOG-TANGENT INTEGRALS AND THE RIEMANN ZETA FUNCTION

2019 ◽  
Vol 24 (3) ◽  
pp. 404-421
Author(s):  
Lahoucine Elaissaoui ◽  
Zine El-Abidine Guennoun
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Integrable Function ◽  
Harmonic Series ◽  
Algebraic Properties ◽  
Tangent Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Square Integrable Function ◽  
Positive Integers

We show that integrals involving the log-tangent function, with respect to any square-integrable function on (0,π/2), can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive integers are discussed. Furthermore, we show among other things, that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and its values depend on the Dirichlet series ζh(s) :=∑n≥1hnn−s−8, where hn=∑nk=1(2k−1)−1.

On dichotomy of Riesz products

1979 ◽  
Vol 85 (1) ◽  
pp. 79-89 ◽  
Author(s):  
G. Ritter
Keyword(s):  
Integrable Function ◽  
Abelian Groups ◽  
Major Step ◽  
Riesz Product ◽  
Major Interest ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Riesz Products ◽  
Square Integrable Function ◽  
Continuous Measures

Background. Riesz products are very useful for the construction of singular measures on compact, Abelian groups. Under some circumstances, two Riesz products are either equivalent or singular in the measure-theoretic sense. Exact knowledge of these circumstances has been of major interest ever since the 1930s, when Riesz's famous example (8) was recognized as a fertile source of examples of singular continuous measures. Zygmund(11) showed that any Riesz product over a Hadamard dissociate subset of ℕ is either a square integrable function or singular with respect to Lebesgue measure. Hewitt–Zuckerman(4) generalized these products to all compact, Abelian groups, introducing the notion of a dissociate subset. They extended Zygmund's result in certain cases. The next major step was taken by Brown–Moran(3) and Peyrière(6), (7), who showed that two Riesz productsare mutually singular ifThe author (9) has improved another result of Brown–Moran (3) by showing that µa and µb are equivalent if

scholarly journals Bounded andL2-solutions to a second order nonlinear differential equation with a square integrable forcing term

1999 ◽  
Vol 22 (3) ◽  
pp. 569-571 ◽  
Author(s):  
Allan Kroopnick
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
Second Order ◽  
Forcing Term ◽  
Order Nonlinear Differential Equation ◽  
Square Integrable Function

This paper presents two theorems concerning the nonlinear differential equationx″+c(t)f(x)x′+a(t,x)=e(t), wheree(t)is a continuous square-integrable function. The first theorem gives sufficient conditions when all the solutions of this equation are bounded while the second theorem discusses when all the solutions are inL2[0,∞).

scholarly journals White noise approach to Gaussian random fields

1990 ◽  
Vol 119 ◽  
pp. 93-106 ◽  
Author(s):  
Ke-Seung Lee
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Integrable Function ◽  
Smooth Boundary ◽  
Noise Analysis ◽  
Variational Calculus ◽  
Main Tool ◽  
Gaussian Random Fields ◽  
Dimensional Euclidean Space ◽  
Square Integrable Function

The purpose of this paper is to investigate way of dependency of Gaussian random fields X(D) indexed by a domain D in d-dimensional Euclidean space Rd. Our main tool is variational calculus, where the boundary of a domain varies and deforms and we appeal to the white noise analysis. We therefore assume that X(D) is expressed white noise integral of the form(0.1) X(D) = X(D, W)=∫D F(D, u)W(u)du,where W is the Rd-parameter white noise and the kernel F(D, u) is a square integrable function over Rd, and where D is a bounded domain with smooth boundary.

scholarly journals POSITIVE PERTURBATIONS OF SELF-ADJOINT SCHRÖDINGER OPERATORS ON RIEMANNIAN MANIFOLDS

2005 ◽  
Vol 02 (04) ◽  
pp. 543-552
Author(s):  
OGNJEN MILATOVIC
Keyword(s):  
Riemannian Manifold ◽  
Differential Expression ◽  
Riemannian Manifolds ◽  
Integrable Function ◽  
Schrödinger Operators ◽  
Sufficient Conditions ◽  
Schrodinger Operators ◽  
Square Integrable Function

We consider a Schrödinger differential expression L0 = ΔM + V0 on a Riemannian manifold (M,g) with metric g, where ΔM is the scalar Laplacian on M and V0 is a real-valued locally square integrable function on M. We consider a perturbation L0 + V, where V is a non-negative locally square-integrable function on M, and give sufficient conditions for L0 + V to be essentially self-adjoint on [Formula: see text]. This is an extension of a result of T. Kappeler.

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