Shift-Invariant Subspaces of Sobolev Spaces and Shift-Preserving Operators

Aleksandar Aksentijević (), Suzana Aleksić () and Stevan Pilipović ()
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Aleksandar Aksentijević: University of Kragujevac
Suzana Aleksić: University of Kragujevac
Stevan Pilipović: University of Novi Sad

Chapter Chapter 2 in Analysis, Approximation, Optimization: Computation and Applications, 2025, pp 23-56 from Springer

Abstract: Abstract We study the shift-preserving operator L : V s → V s $$L:V_s\to V_s$$ and the range operator R s $$R_s$$ and their relationship, where V s $$V_s$$ is a shift-invariant subspace of Sobolev space H s ( ℝ n ) $$H^s(\mathbb {R}^n)$$ , s ∈ ℝ $$s\in \mathbb {R}$$ . Using the range operator, we give a result about dual frames. For the shift-invariant space V s $$V_s$$ generated by d functions, we find conditions on L and a finite set { ϕ i : ϕ i ∈ V s , i = 1 , … , m } $$\{\phi _i: \phi _i\in V_s, i=1,\ldots ,m\}$$ so that the collection { L j ϕ i : i = 1 , … , m , j = 0 , … , d −1 } $$\{L^j\phi _i:i=1,\ldots ,m, j =0,\ldots ,d-1\}$$ is a frame generator for V s $$V_s$$ .

Date: 2025
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DOI: 10.1007/978-3-031-85743-0_2

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