scholarly journals $\ell ^2$-Linear independence for the system of integer translates of a square integrable function

2012 ◽  
Vol 141 (3) ◽  
pp. 937-941 ◽  
Author(s):  
Sandra Saliani
Keyword(s):  
Integrable Function ◽  
Linear Independence ◽  
Integer Translates ◽  
Square Integrable Function

scholarly journals On linear independence for integer translates of a finite number of functions

1993 ◽  
Vol 36 (1) ◽  
pp. 69-85 ◽  
Author(s):  
Rong-Qing Jia ◽  
Charles A. Micchelli
Keyword(s):  
Finite Number ◽  
Linear Combination ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Basis Functions ◽  
Partial Difference ◽  
Integer Translates ◽  
First Case ◽  
Compactly Supported Functions ◽  
Necessary And Sufficient

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

scholarly journals On The Convolution of a Box Spline With a Compactly Supported Distribution: Linear Independence for the Integer Translates

1991 ◽  
Vol 43 (1) ◽  
pp. 19-33 ◽  
Author(s):  
Charles K. Chui ◽  
Amos Ron
Keyword(s):  
Laplace Transform ◽  
Critical Points ◽  
Linear Independence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compactly Supported ◽  
Integer Translates ◽  
Box Spline ◽  
Finite Set ◽  
Necessary And Sufficient

AbstractThe problem of linear independence of the integer translates of μ * B, where μ is a compactly supported distribution and B is an exponential box spline, is considered in this paper. The main result relates the linear independence issue with the distribution of the zeros of the Fourier-Laplace transform, of μ on certain linear manifolds associated with B. The proof of our result makes an essential use of the necessary and sufficient condition derived in [12]. Several applications to specific situations are discussed. Particularly, it is shown that if the support of μ is small enough then linear independence is guaranteed provided that does not vanish at a certain finite set of critical points associated with B. Also, the results here provide a new proof of the linear independence condition for the translates of B itself.

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,∞).

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