# Mollification Based on Wavelets

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## Abstract

**:**

## 1. Introduction

## 2. Basis of Mollification Based on Wavelets

#### 2.1. Expansion in Orthogonal Wavelets

**Condition**

**1**

**Lemma**

**1**

**Definition**

**1**

**Definition**

**2**

**Lemma**

**2**

#### 2.2. Main Results

**Remark**

**1**

**Condition**

**2**

**Condition**

**3**

**Proposition**

**1**

**Proposition**

**2**

**Remark**

**2**

#### 2.3. Proofs of Propositions 1 and 2

**Lemma**

**3**

## 3. Mollifiers Based on Wavelets

**Requirement**

**1**

**Requirement**

**2**

**Requirement**

**3**

**Figure 1.**${\widehat{\mu}}_{1}(w)$ and ${\mu}_{1}(x)$ for the mollifiers based on the rdH-wavelets with $l=1$. The three curves for ${u}_{p}(w)$ with $p=1$, 2, and 3, are shown by thin solid, thick solid, and dashed lines, respectively.

**Figure 2.**(

**a**) ${M}_{8}{f}_{0}(x)$ for the mollifiers based on the rdH-wavelet with $l=1$, and (

**b**) those based on the scaled B-spline wavelets of order $m=1$, 2. In (

**a**), the three curves for ${u}_{p}(w)$ with $p=1$, 2, and 3, are shown by thin solid, thick solid, and dashed lines, respectively. In (

**b**), two curves almost overlap. ${f}_{0}(x)$ is also drawn both in (

**a**) and in (

**b**).

**Figure 3.**${\widehat{\mu}}_{1}(w)$ and ${\mu}_{1}(x)$ for the mollifiers based on the scaled B-spline wavelets of order $m=1$, 2, 4, ∞. The curves take greater values as m increases at $w=5$ in (

**a**), and at $x=0.2$ and $x=1.2$ in (

**b**).

**Lemma**

**4**

#### 3.1. Rapidly Decaying Harmonic Wavelets

**Lemma**

**5**

**Lemma**

**6**

**Remark**

**3**

**Remark**

**4**

#### 3.2. B-Splines

**Lemma**

**7**

**Remark**

**5**

#### 3.3. Scaled B-Splines

## 4. Numerical Computation

**Mollifier**

**1**

**Mollifier**

**2**

**Mollifier**

**3**

**Figure 4.**(

**a**) ${f}_{1}(x)$; (

**b**) ${f}_{1,\u03f5}({x}_{k})$; (

**c**) ${f}_{1}^{\prime}(x)$; (

**d**) ${\dot{f}}_{1,\u03f5}({x}_{k})$.

**Figure 5.**(

**a**), (

**c**), (

**e**): ${M}_{\nu}{f}_{1}(x)$, ${M}_{\nu}{f}_{1,\u03f5}(x)$; (

**b**), (

**d**), (

**f**): ${({M}_{\nu}{f}_{1})}^{\prime}(x)$ and ${({M}_{\nu}{f}_{1,\u03f5})}^{\prime}(x)$, for Mollifier 1. The thinner curves show ${M}_{\nu}{f}_{1}(x)$ and ${({M}_{\nu}{f}_{1})}^{\prime}(x)$.

**Figure 6.**(

**a**), (

**c**), (

**e**): ${M}_{\nu}{f}_{1}(x)$, ${M}_{\nu}{f}_{1,\u03f5}(x)$; (

**b**), (

**d**), (

**f**): ${({M}_{\nu}{f}_{1})}^{\prime}(x)$ and ${({M}_{\nu}{f}_{1,\u03f5})}^{\prime}(x)$, for Mollifier 2. The thinner curves show ${M}_{\nu}{f}_{1}(x)$ and ${({M}_{\nu}{f}_{1})}^{\prime}(x)$.

**Figure 7.**(

**a**), (

**c**), (

**e**): ${M}_{\nu}{f}_{1}(x)$, ${M}_{\nu}{f}_{1,\u03f5}(x)$; (

**b**), (

**d**), (

**f**): ${({M}_{\nu}{f}_{1})}^{\prime}(x)$ and ${({M}_{\nu}{f}_{1,\u03f5})}^{\prime}(x)$, for Mollifier 3. The thinner curves show ${M}_{\nu}{f}_{1}(x)$ and ${({M}_{\nu}{f}_{1})}^{\prime}(x)$.

**Figure 8.**(

**a**), (

**c**), (

**e**): ${M}_{\nu}{f}_{1}(x)$, ${M}_{\nu}{f}_{1,\u03f5}(x)$; (

**b**), (

**d**), (

**f**): ${({M}_{\nu}{f}_{1})}^{\prime}(x)$ and ${({M}_{\nu}{f}_{1,\u03f5})}^{\prime}(x)$, for the mollifier ${\mu}_{\nu}(x)$ based on the rdH-wavelet using $l=\frac{1}{3}$ and $u(w)={u}_{2}(w)$. The thinner curves show ${M}_{\nu}{f}_{1}(x)$ and ${({M}_{\nu}{f}_{1})}^{\prime}(x)$.

## 5. Concluding Remarks

## Acknowledgements

## Appendices

## A. Proof of Lemma 5

## B. Use of Discrete Fourier Transform (DFT)

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**MDPI and ACS Style**

Morita, T.; Sato, K.-i.
Mollification Based on Wavelets. *Axioms* **2013**, *2*, 67-84.
https://doi.org/10.3390/axioms2020067

**AMA Style**

Morita T, Sato K-i.
Mollification Based on Wavelets. *Axioms*. 2013; 2(2):67-84.
https://doi.org/10.3390/axioms2020067

**Chicago/Turabian Style**

Morita, Tohru, and Ken-ichi Sato.
2013. "Mollification Based on Wavelets" *Axioms* 2, no. 2: 67-84.
https://doi.org/10.3390/axioms2020067