The $*$-algebra of unbounded GLT: construction and theoretical foundations

Abstract

In the present paper, we are concerned with the study of matrix-sequences arising from the discretization of PDEs and FDEs on domains $Ω\subset \mathbb{R}^d$ with finite measure. When $Ω$ is either a hypercube or a bounded domain, the theory of Generalized Locally Toeplitz (GLT) sequences and of reduced GLT sequences cover the spectral analysis of the matrix-sequences derived from the approximation of the continuous problem. This work aims to extend the machinery and tools of the GLT apparatus to the case of unbounded domains with finite measure. For any unbounded domain $Ω\subset \mathbb{R}^d$ with finite measure, we define a new class of sequences, which we call unbounded GLT, and study their spectral properties.

Figures (7)

• Figure 1: Eigenvalues distribution of $B_{\mathbf n,t}$ for different values of $h$ together with the sampling of $\mathrm{f}(x,y,\theta_1,\theta_2)=a(x,y)(4-2\cos \theta_1-2\cos \theta_2)$, $a(x,y)= (10 + x^2 + 2y^2+ \sin^2(x+y)) / ( 1 + x^2 + y^2)$ over $\Omega_t \times [-\pi,\pi]^2$, $t=2$.
• Figure 2: Eigenvalues distribution of $B_{\mathbf n,t}$ for different values of $h$ together with the sampling of $\mathrm{f}(x,y,\theta_1,\theta_2)=a(x,y)(4-2\cos \theta_1-2\cos \theta_2)$, $a(x,y)= (10 + x^2 + 2y^2+ \sin^2(x+y)) / ( 1 + x^2 + y^2)$ over $\Omega_t \times [-\pi,\pi]^2$, $t=4$.
• Figure 3: Eigenvalues distribution of $B_{\mathbf n,t}$ for different values of $h$ together with the sampling of $\mathrm{f}(x,y,\theta_1,\theta_2)=a(x,y)(4-2\cos \theta_1-2\cos \theta_2)$, $a(x,y)= (10 + x^2 + 2y^2+ \sin^2(x+y)) / ( 1 + x^2 + y^2)$ over $\Omega_t \times [-\pi,\pi]^2$, $t=8$.
• Figure 4: Eigenvalues distribution of $E_{\mathbf n,\Omega_t,\Omega}(B_{\mathbf n,t})$ for different values of $h$ together with the sampling of $\mathrm{f}(x,y,\theta_1,\theta_2)=a(x,y)(4-2\cos \theta_1-2\cos \theta_2)$, $a(x,y)= (10 + x^2 + 2y^2+ \sin^2(x+y)) / ( 1 + x^2 + y^2)$ over $\Omega \times [-\pi,\pi]^2$, $t=2$.
• Figure 5: Eigenvalues distribution of $E_{\mathbf n,\Omega_t,\Omega}(B_{\mathbf n,t})$ for different values of $h$ together with the sampling of $\mathrm{f}(x,y,\theta_1,\theta_2)=a(x,y)(4-2\cos \theta_1-2\cos \theta_2)$, $a(x,y)= (10 + x^2 + 2y^2+ \sin^2(x+y)) / ( 1 + x^2 + y^2)$ over $\Omega \times [-\pi,\pi]^2$, $t=4$.

Theorems & Definitions (74)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5: s.v. and s.u. matrix sequences
• Remark 2.6
• Proposition 2.7
• Lemma 2.8