scholarly journals Geometrical and Spectral Properties of the Orthogonal Projections of the Identity

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Luis González ◽  
Antonio Suárez ◽  
Dolores García
Keyword(s):  
Orthogonal Projection ◽  
Best Approximation ◽  
Spectral Properties ◽  
Positive Definite ◽  
Identity Matrix ◽  
Arbitrary Matrix ◽  
Orthogonal Projections ◽  
The Best Approximation ◽  
Special Case ◽  
New Inequalities

We analyze the best approximation (in the Frobenius sense) to the identity matrix in an arbitrary matrix subspace ( nonsingular, being any fixed subspace of ). Some new geometrical and spectral properties of the orthogonal projection are derived. In particular, new inequalities for the trace and for the eigenvalues of matrix are presented for the special case that is symmetric and positive definite.

HILBERT–SCHMIDT OPERATORS AND FRAMES — CLASSIFICATION, BEST APPROXIMATION BY MULTIPLIERS AND ALGORITHMS

2008 ◽  
Vol 06 (02) ◽  
pp. 315-330 ◽  
Author(s):  
PETER BALAZS
Keyword(s):  
Hilbert Spaces ◽  
Best Approximation ◽  
Inner Product ◽  
Riesz Bases ◽  
Wavelet Frames ◽  
Arbitrary Matrix ◽  
The Best Approximation ◽  
Rank One ◽  
Arbitrary Operator ◽  
Bessel Sequences

In this paper we deal with the theory of Hilbert–Schmidt operators, when the usual choice of orthonormal basis, on the associated Hilbert spaces, is replaced by frames. We More precisely, we provide a necessary and sufficient condition for an operator to be Hilbert–Schmidt, based on its action on the elements of a frame (i.e. an operator T is [Formula: see text] if and only if the sum of the squared norms of T applied on the elements of the frame is finite). Also, we construct Bessel sequences, frames and Riesz bases of [Formula: see text] operators using tensor products of the same sequences in the associated Hilbert spaces. We state how the [Formula: see text] inner product of an arbitrary operator and a rank one operator can be calculated in an efficient way; and we use this result to provide a numerically efficient algorithm to find the best approximation, in the Hilbert–Schmidt sense, of an arbitrary matrix, by a so-called frame multiplier (i.e. an operator which act diagonally on the frame analysis coefficients). Finally, we give some simple examples using Gabor and wavelet frames, introducing in this way wavelet multipliers.

Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices

2004 ◽  
Vol 56 (4) ◽  
pp. 776-793 ◽  
Author(s):  
Yongdo Lim
Keyword(s):  
Best Approximation ◽  
Positive Definite ◽  
Explicit Calculation ◽  
Hadamard Manifold ◽  
Minimal Distance ◽  
Positive Definite Matrices ◽  
Geometric Means ◽  
Geodesic Submanifolds ◽  
The Best Approximation ◽  
Schur Complements

AbstractWe explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold Sym(n, ℝ)++ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold Sym(p, ℝ)++ × Sym(q, ℝ)++ block diagonally embedded in Sym(n, ℝ)++ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when p ≤ 2 or q ≤ 2.

Intracranial volume-pressure relationship in man

1982 ◽  
Vol 56 (4) ◽  
pp. 524-528 ◽  
Author(s):  
Joseph Th. J. Tans ◽  
Dick C. J. Poortvliet
Keyword(s):  
Best Approximation ◽  
Continuous Monitoring ◽  
Fluid Pressure ◽  
Constant Term ◽  
Volume Index ◽  
Intracranial Volume ◽  
Content Type ◽  
The Best Approximation ◽  
Ventricular Fluid Pressure ◽  
Fine Print

✓ The pressure-volume index (PVI) was determined in 40 patients who underwent continuous monitoring of ventricular fluid pressure. The PVI value was calculated using different mathematical models. From the differences between these values, it is concluded that a monoexponential relationship with a constant term provides the best approximation of the PVI.

scholarly journals On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices

2021 ◽  
Author(s):  
Mahamet Koïta ◽  
Stanislas Kupin ◽  
Sergey Naboko ◽  
Belco Touré
Keyword(s):  
Spectral Analysis ◽  
Continuous Function ◽  
Bergman Space ◽  
Orthogonal Projection ◽  
Spectral Properties ◽  
Toeplitz Operators ◽  
Closed Subspace ◽  
Jacobi Matrices ◽  
Banded Matrices ◽  
Eigen Values

Abstract Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0, \end{align*}$$where $z=re^{i\theta }$ and $\varphi _1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

scholarly journals The best approximation and composition with inner functions.

1995 ◽  
Vol 42 (2) ◽  
pp. 367-378 ◽  
Author(s):  
M. Mateljević ◽  
M. Pavlović
Keyword(s):  
Best Approximation ◽  
Inner Functions ◽  
The Best Approximation
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