### 数学代写|勒贝格积分代写Lebesgue Integration代考|MAT00013H

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|勒贝格积分代写Lebesgue Integration代考|The Five Big Questions

Fourier’s method for expanding an arbitrary function $F$ defined on $[-\pi, \pi]$ into a trigonometric series is to use integration to calculate coefficients:
\begin{aligned} & a_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \cos (k x) d x \quad(k \geq 0), \ & b_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \sin (k x) d x \quad(k \geq 1) . \end{aligned}
The Fourier expansion is then given by
$$F(x)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right] .$$
The heuristic argument for the validity of this procedure is that if $F$ really can be expanded in a series of the form given in Equation (1.3), then
\begin{aligned} \int_{-\pi}^\pi & F(x) \cos (n x) d x \ = & \int_{-\pi}^\pi\left(\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right]\right) \cos (n x) d x \ = & \int_{-\pi}^\pi \frac{a_0}{2} \cos (n x) d x+\sum_{k=1}^{\infty} \int_{-\pi}^\pi a_k \cos (k x) \cos (n x) d x \ & +\sum_{k=1}^{\infty} \int_{-\pi}^\pi b_k \sin (k x) \cos (n x) d x \end{aligned}
Since $n$ and $k$ are integers, all of the integrals are zero except for the one involving $a_n$. These integrals are easily evaluated:
$$\int_{-\pi}^\pi F(x) \cos (n x) d x=\pi a_n .$$

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Cauchy and Riemann Integrals

Fourier and Cauchy were among the first to fully realize the inadequacy of defining integration as the inverse process of differentiation. It is too restrictive. Fourier wanted to apply his methods to arbitrary functions. Not all functions have antiderivatives that can be expressed in terms of standard functions. Fourier tried defining the definite integral of a nonnegative function as the area between the graph of the function and the $x$-axis, but that begs the question of what we mean by area. Cauchy embraced Leibniz’s understanding as a limit of products, and he found a way to avoid infinitesimals.

To define $\int_a^b f(x) d x$, Cauchy worked with finite approximating sums. Given a partition of $[a, b]$ : $\left(a=x_0<x_1<\cdots<x_n=b\right)$, we consider
$$\sum_{k=1}^n f\left(x_{k-1}\right)\left(x_k-x_{k-1}\right) .$$
If we can force all of these approximating sums to be as close to each as other as we wish simply by limiting the size of the difference between consecutive values in the partition, then these summations have a limiting value that is designated as the value of the definite integral, and the function $f$ is said to be integrable over $[a, b]$.

Equipped with this definition, Cauchy succeeded in proving that any continuous or piecewise continuous function is integrable. The class of functions to which Fourier’s analysis could be applied was suddenly greatly expanded.

When Riemann turned to the study of trigonometric series, he wanted to know the limits of Cauchy’s approach to integration. Was there an easy test that could be used to determine whether or not a function could be integrated? Cauchy had chosen to evaluate the function at the left-hand endpoint of the interval simply for convenience. As Riemann thought about how far this definition could be pushed, he realized that his analysis would be simpler if the definition were stated in a slightly more complicated but essentially equivalent manner. Given a partition of $[a, b]$ : $\left(a=x_0<x_1<\cdots<x_n=b\right)$, we assign a tag to each interval, a number $x_j^$ contained in that interval, and consider all sums of the form $$\sum_{k=1}^n f\left(x_k^\right)\left(x_k-x_{k-1}\right) \text {. }$$

# 勒贝格积分代考

## 数学代写|勒贝格积分代写Lebesgue Integration代考|The Five Big Questions

$$a_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \cos (k x) d x \quad(k \geq 0), \quad b_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \sin (k x) d x \quad(k \geq 1) .$$

$$F(x)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right] .$$

$$\int_{-\pi}^\pi F(x) \cos (n x) d x=\int_{-\pi}^\pi\left(\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right]\right) \cos (n x) d x=\int_{-\pi}^\pi \frac{a_0}{2}$$

$$\int_{-\pi}^\pi F(x) \cos (n x) d x=\pi a_n .$$

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Cauchy and Riemann Integrals

$$\sum_{k=1}^n f\left(x_{k-1}\right)\left(x_k-x_{k-1}\right) .$$

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