# Stability of Spline-Type Systems in the Abelian Case

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## Abstract

**:**

## 1. Introduction

## 2. Notation and Mathematical Preliminaries

**Definition**

**1.**

**Theorem**

**1.**

- (I)
- R can be extended to $T\left({B}_{1}\right)$, i.e., it is a left inverse for T
- (II)
- T is a Banach space isomorphism from ${B}_{1}$ onto $T\left({B}_{1}\right)$; in particular there exist ${C}_{1},{C}_{2}>0$ such that:$$\begin{array}{c}\hfill {C}_{1}\parallel h{\parallel}_{{B}_{1}}\le \parallel Th{\parallel}_{{B}_{2}}\le {C}_{2}\parallel h{\parallel}_{{B}_{1}}\phantom{\rule{2.em}{0ex}}\forall h\in {B}_{1}.\end{array}$$
- (III)
- R is surjective and such that:$$\begin{array}{c}\hfill \mathrm{if}\phantom{\rule{4.pt}{0ex}}Rh=f\phantom{\rule{1.em}{0ex}}\to \phantom{\rule{1.em}{0ex}}\parallel h{\parallel}_{{B}_{2}}\le {C}_{2}\parallel f{\parallel}_{{B}_{1}}.\end{array}$$
- (IV)
- $P=T\circ R$ is a bounded projection in ${B}_{2}$ onto $T\left({B}_{1}\right)$. In particular $T\left({B}_{1}\right)$ is complementable in ${B}_{2}$, i.e., $\exists W\subset {B}_{2}$ a closed subspace such that ${B}_{2}=T\left({B}_{1}\right)\oplus W$.

**Definition**

**2.**

**Definition**

**3.**

- 1.
- There is a solid Banach space of coefficients s.t. the synthesis map is a well-defined continuous bijection.
- 2.
- The synthesis operator has a bounded left inverse.

## 3. ${L}^{p}$-Stability of Spline-Type Spaces

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**2.**

- (i)
- $\nexists \mathit{f}\left(x\right)=\left({f}_{1}\left(\alpha \right),\dots ,{f}_{R}\left(\alpha \right)\right)\in {\left({L}^{\infty}\left(H\right)\right)}^{R}$ such that ${U}_{\mathsf{\Phi},H}\mathit{f}=0$
- (ii)
- $\nexists \xi \in {H}^{\perp}$ such that the functions on ${H}^{\perp}$, ${\left\{{\widehat{\varphi}}_{j}(\xb7\xi )\right\}}_{j=1,\dots ,R}$ are linearly dependent
- (iii)
- $\exists \delta >0$ such that $\forall 1\le p\le \infty ,\forall \mathit{f}\in {\left({L}^{p}\left(H\right)\right)}^{R}$$$\begin{array}{c}\hfill \delta \parallel \mathit{f}{\parallel}_{{\left({L}^{p}\left(H\right)\right)}^{R}}\le \parallel {U}_{\mathsf{\Phi},H}\mathit{f}{\parallel}_{{L}^{p}\left(G\right)}.\end{array}$$

**Proof.**

Algorithm 1 Computation of the coefficients as in (12) | |

Precondition: Windows $\mathsf{\Phi}={\left\{{\varphi}_{r}\right\}}_{r=1}^{R}$, subgroup H | |

1: function COEFFICIENT_COMPUTATION($\mathsf{\Phi}$,H) | |

2: | |

3: for $r\leftarrow 1\mathrm{to}R$ do | |

4: | |

5: ${\widehat{\varphi}}_{r}$←$\mathcal{F}\left(\varphi \right)$ | |

6: | |

7: end for | |

8: | |

9: for $r,l\leftarrow 1\mathrm{to}R$ do | |

10: | |

11: ${G}_{r,l}$←${\mathcal{F}}^{-1}\left({\widehat{\varphi}}_{r}{\widehat{\overline{\varphi}}}_{l}\right)$ | % Convolutions in (9) |

12: | |

13: end for | |

14: | |

15: ${G}_{r,l}^{\left(s\right)}$←${G}_{r,l}\left(H\right)$ | % Subsampling to H |

16: | |

17: for $r,l\leftarrow 1\mathrm{to}R$ do | |

18: | |

19: ${\widehat{G}}_{r,l}^{\left(s\right)}$←${\mathcal{F}}_{H}\left({G}_{r,l}^{\left(s\right)}\right)$ | % PD matrix in (10) |

20: | |

21: end for | |

22: | |

23: for $x\in H$ do | |

24: | |

25: $\widehat{\mathbf{g}}\left(x\right)$←${\left({\widehat{G}}^{\left(s\right)}\left(x\right)\right)}^{-1}$ | % Wiener’s inversion in (11) |

26: | |

27: end for | |

28: | |

29: for $r,l\leftarrow 1\mathrm{to}R$ do | |

30: | |

31: ${\mathbf{g}}_{r,l}$←${\mathcal{F}}_{H}^{-1}\left({G}_{r,l}^{\left(s\right)}\right)$ | % Coefficients (12) |

32: | |

33: end for | |

34: | |

35: return $\mathbf{g}$ | |

36: | |

37: end function |

## 4. Stability for Sequence of Projections

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Lemma**

**3.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**3.**

**Example**

**1.**

**Remark**

**1.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Level | k = 4 | k = 5 | k = 6 | k = 7 | k = 8 |

Singular value | 0.4299 | 0.7830 | 1.4886 | 2.9000 | 5.7229 |

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**MDPI and ACS Style**

Onchis, D.; Zappalà, S.
Stability of Spline-Type Systems in the Abelian Case. *Symmetry* **2018**, *10*, 7.
https://doi.org/10.3390/sym10010007

**AMA Style**

Onchis D, Zappalà S.
Stability of Spline-Type Systems in the Abelian Case. *Symmetry*. 2018; 10(1):7.
https://doi.org/10.3390/sym10010007

**Chicago/Turabian Style**

Onchis, Darian, and Simone Zappalà.
2018. "Stability of Spline-Type Systems in the Abelian Case" *Symmetry* 10, no. 1: 7.
https://doi.org/10.3390/sym10010007