scholarly journals On the weakly-* dense subsets in $L^{\infty}(\Omega)$

2021 ◽  
Vol 16 ◽  
pp. 149
Author(s):  
P.I. Kogut ◽  
T.N. Rudyanova
Keyword(s):  
Weak Convergence ◽  
Density Property ◽  
Smooth Functions ◽  
Compactly Supported

In this paper we study the density property of the compactly supported smooth functions in the space $L^{\infty}(\Omega)$. We show that this set is dense with respect to the weak-* convergence in variable spaces.

scholarly journals On density of compactly supported smooth functions in fractional Sobolev spaces

2021 ◽  
Author(s):  
Bartłomiej Dyda ◽  
Michał Kijaczko
Keyword(s):  
Sobolev Space ◽  
Hardy Inequality ◽  
Sufficient Conditions ◽  
Fractional Sobolev Spaces ◽  
Density Property ◽  
Smooth Functions ◽  
Fractional Sobolev Space ◽  
Compactly Supported ◽  
Bounded Set ◽  
Compactly Supported Functions

AbstractWe describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $$W^{s,p}(\Omega )$$ W s , p ( Ω ) for an open, bounded set $$\Omega \subset \mathbb {R}^{d}$$ Ω ⊂ R d . The density property is closely related to the lower and upper Assouad codimension of the boundary of $$\Omega$$ Ω . We also describe explicitly the closure of $$C_{c}^{\infty }(\Omega )$$ C c ∞ ( Ω ) in $$W^{s,p}(\Omega )$$ W s , p ( Ω ) under some mild assumptions about the geometry of $$\Omega$$ Ω . Finally, we prove a variant of a fractional order Hardy inequality.

scholarly journals Compactly Supported Curvelet-Type Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Kenneth N. Rasmussen ◽  
Morten Nielsen
Keyword(s):  
Sparse Representation ◽  
Compact Support ◽  
Type Systems ◽  
Linear Combinations ◽  
Smooth Functions ◽  
Constructive Description ◽  
Compactly Supported ◽  
Single Function ◽  
Type Spaces ◽  
Finite Linear

We study a flexible method for constructing curvelet-type frames. These curvelet-type systems have the same sparse representation properties as curvelets for appropriate classes of smooth functions, and the flexibility of the method allows us to give a constructive description of how to construct curvelet-type systems with a prescribed nature such as compact support in direct space. The method consists of using the machinery of almost diagonal matrices to show that a system of curvelet molecules which is sufficiently close to curvelets constitutes a frame for curvelet-type spaces. Such a system of curvelet molecules can then be constructed using finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay.

scholarly journals A Convenient Notion of Compact Set for Generalized Functions

2017 ◽  
Vol 61 (1) ◽  
pp. 57-92
Author(s):  
Paolo Giordano ◽  
Michael Kunzinger
Keyword(s):  
Function Spaces ◽  
Generalized Functions ◽  
Distribution Theory ◽  
Compact Set ◽  
Compact Sets ◽  
Smooth Functions ◽  
Test Function ◽  
Compactly Supported ◽  
Nonlinear Generalized Functions

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.

Compactly supported analytic indices for Lie groupoids

2009 ◽  
Vol 4 (2) ◽  
pp. 223-262 ◽  
Author(s):  
Paulo Carrillo Rouse
Keyword(s):  
Index Theorem ◽  
Lie Groupoids ◽  
Smooth Functions ◽  
Lie Groupoid ◽  
Convolution Algebra ◽  
Compactly Supported ◽  
Potential Applications ◽  
Analytic Index ◽  
Image Position ◽  
Symbol Class

AbstractFor any Lie groupoid we construct an analytic index morphism taking values in a modified K-theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in [CR06]. This allows us in particular to prove a more primitive version of the Connes-Skandalis longitudinal index theorem for foliations, that is, an index theorem taking values in a group which pairs with cyclic cocycles. As another application, for D a -PDO elliptic operator with associated index ind we prove that the pairingwith τ a bounded continuous cyclic cocycle, only depends on the principal symbol class [σ(D)]∈K0. The result is completely general for étale groupoids. We discuss some potential applications to the Novikov conjecture.

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