Sample Numbers and Optimal Lagrange Interpolation of Sobolev Spaces Wr1

2021 ◽  
Vol 42 (4) ◽  
pp. 519-528
Author(s):  
Guiqiao Xu ◽  
Zehong Liu ◽  
Hui Wang
Keyword(s):  
Sobolev Spaces ◽  
Lagrange Interpolation

scholarly journals A note on optimal Hermite interpolation in Sobolev spaces

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Guiqiao Xu ◽  
Xiaochen Yu
Keyword(s):  
Weight Function ◽  
Sobolev Spaces ◽  
Lagrange Interpolation ◽  
Hermite Interpolation ◽  
Weighted Spaces ◽  
Interpolation Algorithms ◽  
Interpolation Systems

AbstractThis paper investigates the optimal Hermite interpolation of Sobolev spaces $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] , $n\in \mathbb{N}$ n ∈ N in space $L_{\infty }[a,b]$ L ∞ [ a , b ] and weighted spaces $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p< \infty $ 1 ≤ p < ∞ with ω a continuous-integrable weight function in $(a,b)$ ( a , b ) when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in $L_{\infty }$ L ∞ (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ) are optimal for $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] in $L_{\infty }[a,b]$ L ∞ [ a , b ] (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

scholarly journals Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Thomas Eiter ◽  
Mads Kyed
Keyword(s):  
Rigid Body ◽  
Sobolev Spaces ◽  
Periodic Motion ◽  
Viscous Incompressible Fluid ◽  
Body Motion ◽  
Translational Velocity ◽  
Ill Posed ◽  
Homogeneous Sobolev Spaces ◽  
The Mean ◽  
Time Periodic

AbstractThe equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

scholarly journals The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
André Guerra ◽  
Lukas Koch ◽  
Sauli Lindberg
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Spaces ◽  
Regularity Of Solutions ◽  
Jacobian Equation ◽  
L Log L

AbstractWe study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $$\det D u =f$$ det D u = f , where f is integrable and bounded away from zero. In particular, we take $$f\in L^p$$ f ∈ L p , where $$p>1$$ p > 1 , or in $$L\log L$$ L log L . We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.

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