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Register a new userGeneralized solutions of variational problems and applications
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Ultrafunctions are a particular class of generalized functions defined on a hyperreal field $mathbb{R}^{*}supsetmathbb{R}$ that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach.
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Splines come in a variety of flavors that can be characterized in terms of some differential operator L. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, one can extend the traditional notion of total variation by considering more general operators than the derivative. This leads us to the definition of the generalized Beppo-Levi space M, which is further identified as the direct sum of two Banach spaces. We then prove that the minimization of the generalized total variation (gTV) over M, subject to some arbitrary (convex) consistency constraints on the linear measurements of the signal, admits nonuniform L-spline solutions with fewer knots than the number of measurements. This shows that non-uniform splines are universal solutions of continuous-domain linear inverse problems with LASSO, L1, or TV-like regularization constraints. Remarkably, the spline-type is fully determined by the choice of L and does not depend on the actual nature of the measurements.
Let $Sinmathcal{M}_d(mathbb{C})^+$ be a positive semidefinite $dtimes d$ complex matrix and let $mathbf a=(a_i)_{iinmathbb{I}_k}in mathbb{R}_{>0}^k$, indexed by $mathbb{I}_k={1,ldots,k}$, be a $k$-tuple of positive numbers. Let $mathbb T_{d}(mathbf a )$ denote the set of families $mathcal G={g_i}_{iinmathbb{I}_k}in (mathbb{C}^d)^k$ such that $|g_i|^2=a_i$, for $iinmathbb{I}_k$; thus, $mathbb T_{d}(mathbf a )$ is the product of spheres in $mathbb{C}^d$ endowed with the product metric. For a strictly convex unitarily invariant norm $N$ in $mathcal{M}_d(mathbb{C})$, we consider the generalized frame operator distance function $Theta_{( N , , , S, , , mathbf a)}$ defined on $mathbb T_{d}(mathbf a )$, given by $$ Theta_{( N , , , S, , , mathbf a)}(mathcal G) =N(S-S_{mathcal G }) quad text{where} quad S_{mathcal G}=sum_{iinmathbb{I}_k} g_i,g_i^*inmathcal{M}_d(mathbb{C})^+,. $$ In this paper we determine the geometrical and spectral structure of local minimizers $mathcal G_0inmathbb T_{d}(mathbf a )$ of $Theta_{( N , , , S, , , mathbf a)}$. In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of $N$.
In this work we continue our research on nonharmonic analysis of boundary value problems as initiated in our recent paper (IMRN 2016). There, we assumed that the eigenfunctions of the model operator on which the construction is based do not have zeros. In this paper we have weakened this condition extending the applicability of the developed pseudo-differential analysis. Also, we do not assume that the underlying set is bounded.
In this paper we present the groundwork for an It^o/Malliavin stochastic calculus and Hidas white noise analysis in the context of a supersymmentry with Z3-graded algebras. To this end we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes in this context.
It is shown that generalized trigonometric functions and generalized hyperbolic functions can be transformed from each other. As an application of this transformation, a number of properties for one immediately lead to the corresponding properties for the other. In this way, Mitrinovi{c}-Adamovi{c}-type inequalities, multiple-angle formulas, and double-angle formulas for both can be produced.
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