# Fourier Series for Singular Measures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

#### 1.1. Effective Sequences

**Definition**

**3.**

## 2. Main Results

**Theorem**

**1.**

**Theorem**

**2.**

**Lemma**

**1.**

**Proof.**

**Definition 4**(Stationary Sequences).

**Theorem 3**(Kwapień and Mycielski).

**Proof**

**of Theorem 1.**

**Remark**

**1.**

**Lemma**

**2.**

**Proof.**

**Remark**

**2.**

**Definition**

**5.**

**Corollary**

**1.**

**Proof.**

#### 2.1. Non-Uniqueness of Fourier Coefficients

**Theorem**

**4.**

**Proof.**

**Remark**

**3.**

#### 2.2. Connection to the Normalized Cauchy Transform

**Proposition**

**1.**

**Proof.**

## 3. A Shannon Sampling Formula

**Theorem**

**5.**

**Proof.**

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Herr, J.E.; Weber, E.S.
Fourier Series for Singular Measures. *Axioms* **2017**, *6*, 7.
https://doi.org/10.3390/axioms6020007

**AMA Style**

Herr JE, Weber ES.
Fourier Series for Singular Measures. *Axioms*. 2017; 6(2):7.
https://doi.org/10.3390/axioms6020007

**Chicago/Turabian Style**

Herr, John E., and Eric S. Weber.
2017. "Fourier Series for Singular Measures" *Axioms* 6, no. 2: 7.
https://doi.org/10.3390/axioms6020007