# 数学代写|实分析作业代写Real analysis代考|Fourier Series

## 数学代写|实分析作业代写Real analysis代考|Fourier Series

In this chapter, we consider the problem of expressing a real-valued periodic function of period $2 \pi$ in terms of a trigonometric series
$$\frac{1}{2} a_0+\sum_{n=1}^{\infty}\left(a_n \cos n x+b_n \sin n x\right),$$
where the $a_n$ and $b_n$ are real numbers. As we will see, such series afford much greater generality in the type of functions that can be represented as opposed to Taylor series. The study of trigonometric series has its origins in the monumental work of Joseph Fourier (1768-1830) on heat conduction in solids. His 1807 presentation to the French Academy introduced a whole new subject area in mathematics while at the same time providing very useful techniques for solving physical problems.

Fourier’s work is the source of all modern methods in mathematical physics involving boundary value problems and has been a source of new ideas in mathematical analysis for the past two centuries. To see how greatly mathematics has been influenced by the studies of Fourier one only needs to look at the two volume work Trigonometric Series by A. Zygmund (Cambridge University Press, 1968). In addition to trigonometric series, Fourier’s original method of separation of variables leads very naturally to the study of orthogonal functions and the representation of functions in terms of a series of orthogonal functions. All of these have many applications in mathematical physics and engineering.

Fourier initially claimed and tried to show, with no success, that the Fourier series expansion of a function actually represented the function. Although his claim is false, in view of the eighteenth century concept of a function this was not an unrealistic expectation. Fourier’s claim had an immediate impact on nineteenth century mathematics. It caused mathematicians to reconsider the definition of “function.” The question of what type of function has a Fourier series expansion also led Riemann to the development of the theory of the integral and the notion of an integrable function. The first substantial progress on the convergence of a Fourier series to its defining function is due to Dirichlet in 1829 . Instead of trying to prove like Fourier that the Fourier series always converged to its defining function, Dirichlet considered the more restrictive problem of trying to find sufficient conditions on the function $f$ for which the Fourier series converges pointwise to the function.

## 数学代写|实分析作业代写Real analysis代考|Orthogonal Functions

In this section, we provide a brief introduction to the topic of orthogonal functions and the question of representing a function by means of a series of orthogonal functions. Although these topics have their origins in the study of partial differential equations and boundary value problems ${ }^1$, they are closely related to concepts normally encountered in the study of vector spaces.
If $X$ is a vector space over $\mathbb{R}$, a function $\langle\rangle:, X \times X \rightarrow \mathbb{R}$ is an inner product on $X$ if
(a) $\langle\mathrm{x}, \mathrm{x}\rangle \geq 0$ for all $\mathrm{x} \in X$,
(b) $\langle\mathbf{x}, \mathbf{x}\rangle=0$ if and only if $\mathbf{x}=\mathbf{0}$,
(c) $\langle\mathbf{x}, \mathbf{y}\rangle=\langle\mathbf{y}, \mathbf{x}\rangle$ for all $\mathbf{x}, \mathbf{y} \in X$, and
(d) $\langle a \mathbf{x}+b \mathbf{y}, \mathbf{z}\rangle=a\langle\mathbf{x}, \mathbf{z}\rangle+b\langle\mathbf{y}, \mathbf{z}\rangle$ for all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in X$ and $a, b \in \mathbb{R}$.
In $\mathbb{R}^n$, the usual inner product is given by
$$\langle\mathbf{a}, \mathbf{b}\rangle=\sum_{j=1}^n a_j b_j$$
for $\mathbf{a}=\left(a_1, \ldots, a_n\right)$ and $\mathbf{b}=\left(b_1, \ldots, b_n\right)$ in $\mathbb{R}^n$. If $\langle$,$\rangle is an inner product on X$, then two non-zero vectors $\mathbf{x}, \mathbf{y} \in X$ are orthogonal if $\langle\mathbf{x}, \mathbf{y}\rangle=0$. The term “orthogonal” is synonymous with “perpendicular” and comes from geometric considerations in $\mathbb{R}^n$. Two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^n$ are orthogonal if and only if they are mutually perpendicular; that is, the angle $\theta$ between the two vectors $\mathbf{a}$ and $\mathbf{b}$ is $\pi / 2$ or $90^{\circ}$ (see Exercise 9 , Section 7.4).

In the study of analysis we typically encounter vector spaces whose elements are functions. For example, in previous sections we have shown that the space $\ell^2$ of square summable sequences as well as the space $\mathcal{C}[a, b]$ of continuous real-valued functions on $[a, b]$ are vector spaces over $\mathbb{R}$. With the usual rules of addition and scalar multiplication, $\mathcal{R}[a, b]$, the set of Riemann integrable functions on $[a, b]$, is also a vector space over $\mathbb{R}$. If for $f, g \in \mathcal{R}[a, b]$ we define
$$\langle f, g\rangle=\int_a^b f(x) g(x) d x,$$
then it is easily shown that $\langle$,$\rangle satisfies (a), (c), and (d) of the definition of$ an inner product. It however does not satisfy (b).

# 实分析代考

## 数学代写|实分析作业代写Real analysis代考|Fourier Series

$$\frac{1}{2} a_0+\sum_{n=1}^{\infty}\left(a_n \cos n x+b_n \sin n x\right)$$

## 数学代写|实分析作业代写Real analysis代考|Orthogonal Functions

(一) $\langle\mathrm{x}, \mathrm{x}\rangle \geq 0$ 对全部 $\mathrm{x} \in X_{\text {， }}$
$($ 二) $\langle\mathbf{x}, \mathbf{x}\rangle=0$ 当且仅当 $\mathbf{x}=\mathbf{0}$ ，
(三) $\langle\mathbf{x}, \mathbf{y}\rangle=\langle\mathbf{y}, \mathbf{x}\rangle$ 对全部 $\mathbf{x}, \mathbf{y} \in X$ ，和
(d) $\langle a \mathbf{x}+b \mathbf{y}, \mathbf{z}\rangle=a\langle\mathbf{x}, \mathbf{z}\rangle+b\langle\mathbf{y}, \mathbf{z}\rangle$ 对全部 $\mathbf{x}, \mathbf{y}, \mathbf{z} \in X$ 和 $a, b \in \mathbb{R}$.

$$\langle\mathbf{a}, \mathbf{b}\rangle=\sum_{j=1}^n a_j b_j$$

$$\langle f, g\rangle=\int_a^b f(x) g(x) d x$$

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