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1
Content available remote

A gradient-projective basis of compactly supported wavelets in dimension n > 1

100%
Battle G.
Open Mathematics
|
2013
|
tom 11
|
nr 7
1317-1333
EN
A given set W = {W X } of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum\limits_\chi {(\int_{\mathbb{R}^n } {\nabla f \cdot } \nabla W_\chi ^* )} W_\chi $ converges to f with respect to the norm \(\left\| {\nabla ( \cdot )} \right\|_{L^2 (\mathbb{R}^n )} \) . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = {W x } of compactly supported class C 2−ɛ functions on ℝn such that [...] where X has the structure \(\chi = (\tilde \chi ,\nu )\) , ν ∈ ℤ. W is a wavelet set in the sense that the functions indexed by \(\tilde \chi \) are generated by an averaging of lattice translations with wave propagations, and there are two additional discrete parameters associated with the latter. These functions indexed by \(\tilde \chi \) are the unit-scale wavelets. The support volumes of our unit-scale wavelets are not uniformly bounded, however. While the practical value of this construction is doubtful, our motivation is theoretical. The point is that a gradient-orthonormal basis of compactly supported wavelets has never been constructed in dimension n > 1. (In one dimension the construction of such a basis is easy - just anti-differentiate the Haar functions.)
2
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On a space of entire functions rapidly decreasing on Rn and its Fourier transform

80%
Musin I. d. K.
Concrete Operators
|
2015
|
tom 2
|
nr 1
EN
A space of entire functions of several complex variables rapidly decreasing on Rn and such that their growth along iRn is majorized with the help of a family of weight functions is considered in this paper. For such space an equivalent description in terms of estimates on all of its partial derivatives as functions on Rn and a Paley-Wiener type theorem are obtained.
3
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Boehmians of type S and their Fourier transforms

80%
Bhuvaneswari R., Karunakaran V.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
|
2010
|
tom 64
|
nr 1
EN
Function spaces of type S are introduced and investigated in the literature. They are also applied to study the Cauchy problem. In this paper we shall extend the concept of these spaces to the context of Boehmian spaces and study the Fourier transform theory on these spaces. These spaces enable us to combine the theory of Fourier transform on these function spaces as well as their dual spaces.
4
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Fourier transform, oscillatory multipliers and evolution equations in rearrangement invariant function spaces

80%
Brandolini L., Colzani L.
Colloquium Mathematicum
|
1996
|
tom 71
|
nr 2
273-286
5
Content available remote

An application of the Fourier transform to optimization of continuous 2-D systems

70%
Dymkou V., Dymkov M.
International Journal of Applied Mathematics and Computer Science
|
2003
|
tom 13
|
nr 1
45-54
EN
This paper uses the theory of entire functions to study the linear quadratic optimization problem for a class of continuous 2D systems. We show that in some cases optimal control can be given by an analytical formula. A simple method is also proposed to find an approximate solution with preassigned accuracy. Some application to the 1D optimization problem is presented, too. The obtained results form a theoretical background for the design problem of optimal controllers for relevant processes.
6
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Closure of dilates of shift-invariant subspaces

70%
Soto-Bajo M.
Open Mathematics
|
2013
|
tom 11
|
nr 10
1785-1799
EN
Let V be any shift-invariant subspace of square summable functions. We prove that if for some A expansive dilation V is A-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of V, among them the origin is a point of A*-approximate continuity of the spectral function if we assume this value to be one. We present our results also in a more general setting of A-reducing spaces. We also prove that the origin is a point of A*-approximate continuity of the Fourier transform of any semiorthogonal tight frame wavelet if we assume this value to be zero.
7
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Tauberian theorems for vector-valued Fourier and Laplace transforms

61%
Chill R.
Studia Mathematica
|
1998
|
tom 128
|
nr 1
55-69
EN
Let X be a Banach space and $f ∈ L^1_loc(ℝ;X)$ be absolutely regular (i.e. integrable when divided by some polynomial). If the distributional Fourier transform of f is locally integrable then f converges to 0 at infinity in some sense to be made precise. From this result we deduce some Tauberian theorems for Fourier and Laplace transforms, which can be improved if the underlying Banach space has the analytic Radon-Nikodym property.
8
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On stabilizability of evolution systems of partial differential equations on ℝn×[0,+∞) by time-delayed feedback controlsby time-delayed feedback controls

51%
Fardigola L.
Open Mathematics
|
2003
|
tom 1
|
nr 2
141-156
EN
In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system $$\frac{{\partial w\left( {x,t} \right)}}{{\partial t}} = A(D_x )w(x,t) - A(D_x )u(x,t), x \in \mathbb{R}^n , t > h, $$ where D x=(-i∂/∂x 1,...-i∂/∂x n), A(σ) and B(σ) are polynomial matrices (m×m), det B(σ)≡0 on ℝn, w is an unknown function, u(·,t)=P(D x)w(·,t−h) is a control, h>0. Here P is an infinite differentiable matrix (m×m), and the norm of each of its derivatives does not exceed Γ(1+|σ|2)γ for some Γ, γ∈ℝ depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the Schrödinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski-Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det {Iλ-A(σ)+B(σ)P(σ)e -hλ.
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