On the functional Hodrick–Prescott filter with non-compact operators

AbstractWe study a version of the functional Hodrick–Prescott filter in the case when the associated operator is not necessarily compact but merely closed and densely defined with closed range. We show that the associated optimal smoothing operator preserves the structure obtained in the compact case when the underlying distribution of the data is Gaussian.

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A functional Hodrick–Prescott filter

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2015-0111 ◽

2017 ◽

Vol 25 (2) ◽

Cited By ~ 1

Author(s):

Boualem Djehiche ◽

Hiba Nassar

Keyword(s):

Hilbert Space ◽

Linear Operator ◽

Functional Data ◽

Separable Hilbert Space ◽

Underlying Distribution ◽

Smoothing Operator ◽

Infinite Dimensional ◽

Optimal Smoothing ◽

Functional Version

AbstractWe propose a functional version of the Hodrick–Prescott filter for functional data which take values in an infinite-dimensional separable Hilbert space. We further characterize the associated optimal smoothing operator when the associated linear operator is compact and the underlying distribution of the data is Gaussian.

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The essential approximate pseudospectrum and related results

Filomat ◽

10.2298/fil1806139a ◽

2018 ◽

Vol 32 (6) ◽

pp. 2139-2151 ◽

Cited By ~ 1

Author(s):

Aymen Ammar ◽

Aref Jeribi ◽

Kamel Mahfoudhi

Keyword(s):

Banach Space ◽

Compact Operators ◽

Linear Operators ◽

The Stability ◽

Densely Defined

In this paper, we introduce and study the essential approximate pseudospectrum of closed, densely defined linear operators in the Banach space. We begin by the definition and we investigate the characterization, the stability by means of quasi-compact operators and some properties of these pseudospectrum.

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Characterizing C-algebras of compact operators by generic categorical properties of Hilbert C-modules

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is008001031jkt035 ◽

2008 ◽

Vol 2 (3) ◽

pp. 453-462 ◽

Cited By ~ 7

Author(s):

Michael Frank

Keyword(s):

Hilbert Spaces ◽

Adjoint Operator ◽

Compact Operators ◽

Linear Operators ◽

Closed Graph ◽

Graph Theorem ◽

Closed Graph Theorem ◽

Closed Range ◽

C Algebras ◽

Categorical Properties

AbstractC*-algebras A of compact operators are characterized as those C*-algebras of coefficients of Hilbert C*-modules for which (i) every bounded A-linear operator between two Hilbert A-modules possesses an adjoint operator, (ii) the kernels of all bounded A-linear operators between Hilbert A-modules are orthogonal summands, (iii) the images of all bounded A-linear operators with closed range between Hilbert A-modules are orthogonal summands, and (iv) for every Hilbert A-module every Hilbert A-submodule is a topological summand. Thus, the theory of Hilbert C*-modules over C*-algebras of compact operators has similarities with the theory of Hilbert spaces. In passing, we obtain a general closed graph theorem for bounded module operators on arbitrary Hilbert C*-modules.

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On the Optimal Smoothing of Time Series

Behaviormetrika ◽

10.2333/bhmk.8.9_47 ◽

1981 ◽

Vol 8 (9) ◽

pp. 47-56

Author(s):

Hisao Miyano

Keyword(s):

Time Series ◽

Optimal Smoothing

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An optimal smoothing approach for trajectory reconstruction in planetary exploration

2003 European Control Conference (ECC) ◽

10.23919/ecc.2003.7086541 ◽

2003 ◽

Author(s):

Amit Brandes

Keyword(s):

Planetary Exploration ◽

Trajectory Reconstruction ◽

Optimal Smoothing

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The space of compact operators contains c 0 when a noncompact operator is suitably factorized

Czechoslovak Mathematical Journal ◽

10.1023/a:1022489103715 ◽

2000 ◽

Vol 50 (1) ◽

pp. 75-82

Author(s):

G. Emmanuele ◽

K. John

Keyword(s):

Compact Operators ◽

Space Of Compact Operators

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An asymptotic analysis for a generalized Cahn–Hilliard system with fractional operators

Journal of Evolution Equations ◽

10.1007/s00028-021-00706-1 ◽

2021 ◽

Author(s):

Pierluigi Colli ◽

Gianni Gilardi ◽

Jürgen Sprekels

Keyword(s):

Additional Term ◽

Linear Operators ◽

Phase Variable ◽

Logarithmic Potentials ◽

Relaxation Problem ◽

System Equations ◽

Indicator Functions ◽

Regularity Results ◽

Compact Resolvents ◽

Densely Defined

AbstractIn the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” (Colli et al. in Atti Accad Naz Lincei Rend Lincei Mat Appl 30:437–478, 2019), the same authors have studied viscous and nonviscous Cahn–Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers $$A^{2r}$$ A 2 r and $$B^{2\sigma }$$ B 2 σ (in the spectral sense) of general linear operators A and B, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space $$L^2(\Omega )$$ L 2 ( Ω ) , for some bounded and smooth domain $$\Omega \subset {{\mathbb {R}}}^3$$ Ω ⊂ R 3 , and have compact resolvents. Existence, uniqueness, and regularity results have been proved in the quoted paper. Here, in the case of the viscous system, we analyze the asymptotic behavior of the solution as the parameter $$\sigma $$ σ appearing in the operator $$B^{2\sigma }$$ B 2 σ decreasingly tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of B appears.

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