# Further Properties of Quantum Spline Spaces

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- (i)
- S is a polynomial of degree up to n on each interval $[{t}_{i-1},{t}_{i})$, $i=1,2,\dots ,m$.
- (ii)
- S is quantum continuous of order $n-1$ at the knots.

## 2. Preliminaries

**Definition**

**1.**

**Theorem**

**1.**

## 3. Properties of $\mathbf{q}$-B-Splines

**Property**

**1.**

**Property**

**2.**

**Property**

**3.**

**Property**

**4.**

**Property**

**5.**

**Remark**

**1.**

**Remark**

**2.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Proposition**

**1.**

**Proof.**

## 4. The Quantum Spline Space ${\mathcal{S}}_{m}^{n,q}$

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Remark**

**3.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The graphs of ${N}_{0,2}(t;q)$ with the knot sequence $(1,2,3,4)$ and $q=2$, $q=1.5$, $q=1.1$ respectively, whose first order quantum derivatives agree at knots.

**Figure 2.**q-B-spline curves with several q values, the knot sequence $\left\{0,\frac{1}{10},\frac{1}{5},\frac{3}{10},\frac{2}{5},\frac{1}{2},\frac{3}{5},\frac{7}{10},\frac{4}{5},\frac{9}{10},1\right\}$ and the control points $\left\{0,-\frac{1}{6},\frac{1}{3},-\frac{1}{2},\frac{2}{3},-\frac{5}{6},1\right\}$.

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Budakçı, G.; Oruç, H.
Further Properties of Quantum Spline Spaces. *Mathematics* **2020**, *8*, 1770.
https://doi.org/10.3390/math8101770

**AMA Style**

Budakçı G, Oruç H.
Further Properties of Quantum Spline Spaces. *Mathematics*. 2020; 8(10):1770.
https://doi.org/10.3390/math8101770

**Chicago/Turabian Style**

Budakçı, Gülter, and Halil Oruç.
2020. "Further Properties of Quantum Spline Spaces" *Mathematics* 8, no. 10: 1770.
https://doi.org/10.3390/math8101770