数学代写|傅里叶分析代写Fourier analysis代考|MAT180

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

数学代写|傅里叶分析代写Fourier analysis代考|Differentiability

A function $f$ is differentiable at a point $x$ if and only if
$$\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$$
exists. If $f$ is differentiable at every point in a given interval $(\alpha, \beta)$, then $f$ is said to be differentiable on the interval $(\alpha, \beta)$ or, if we want to be very explicit, differentiable everywhere on $(\alpha, \beta)$.

Observe that, if a function is differentiable at a point or on some interval, then that function must also be continuous at that point or on that interval. On the other hand, there are many continuous functions which are not everywhere differentiable. It is also worth recalling the geometric significance of differentiability and the above limit; namely, that the statement ” $f$ is differentiable at $x$ ” is equivalent to the statement “the graph of $f$ has a single well-defined tangent at $x$.” Moreover, the limit in expression (3.3) gives the slope of this tangent line.

?-Exercise 3.5: Verify that $|x|$ is continuous, but not differentiable, at $x=0$.
Derivatives
For each point $x$ at which $f$ is differentiable, the derivative of $f$ at $x$, denoted by $f^{\prime}(x)$, is the number given by the limit in expression (3.3),
$$f^{\prime}(x)=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} .$$
Suppose $f$ is differentiable at all but a finite number (possibly zero) of points in each finite subinterval of $(\alpha, \beta)$. Then formula (3.4) also defines another function on $(\alpha, \beta)$, called, naturally, the derivative of $f$ on $(\alpha, \beta)$ and commonly denoted by $f^{\prime}$ (or $d f / d x$ or $d f / d t$ or …). Notice that the derivative of a function can exist on an interval even though the function is not differentiable everywhere on that interval. In fact, as our next example shows, it is possible for the derivative to be continuous (after removing the trivial discontinuities) even though the function, itself, has a nontrivial discontinuity.
Example 3.5: The step function,
$$\operatorname{step}(x)=\left{\begin{array}{ll} 0 & \text { if } \quad x<0 \ 1 & \text { if } 0<x \end{array},\right.$$
is clearly differentiable everywhere on $(-\infty, \infty)$ except at the point $x=0$ where the step function has a nontrivial jump discontinuity. It should also be clear that
$$\operatorname{step}^{\prime}(x)=\left{\begin{array}{ll} 0 & \text { if } \quad x<0 \ 0 & \text { if } 0<x \end{array} .\right.$$
The discontinuity at $x=0$ is a trivial one. Removing this discontinuity gives
$$\text { step }^{\prime}=0 \text {, }$$
which is continuous on the entire real line even though the step function is not differentiable on the real line.

数学代写|傅里叶分析代写Fourier analysis代考|Smoothness Smooth Functions

To be smooth over an interval $(\alpha, \beta)$, a function $f$ must satisfy two conditions:

1. $f$ must be differentiable (and, hence, continuous) everywhere on $(\alpha, \beta)$, and
2. $f^{\prime}$ must also be a continuous function on $(\alpha, \beta)$.
Example 3.7: The function $|x|$ is not smooth on any interval containing the origin since, as was seen in exercise $3.5,|x|$ is not differentiable at $x=0$.

Example 3.8: Even though the derivative of the step function is continuous on the real line (after removing the trivial discontinuity, see example 3.5), the step function, itself, is not smooth on any interval containing the origin because it has a jump discontinuity at $x=0$.
The graph of a smooth, real-valued function looks like a smoothly curving line. Typically, the graphs of nonsmooth functions contain nontrivial discontinuities (as with the step function at $x=0$ ) or else have sharp corners (as with $|x|$ at $x=0$ ).

From the definition it is clear that a smooth function is differentiable. And, if you were to test a random sampling of known differentiable functions, it may appear as if all differentiable functions are smooth. This, however, is not true. There are differentiable functions which are not smooth (see exercise 3.17 on page 36 ).
Uniform Smoothness
Let $(\alpha, \beta)$ be a finite interval. A function $f$ is uniformly smooth on $(\alpha, \beta)$ if and only if

1. $f$ is smooth on $(\alpha, \beta)$, and
2. both $f$ and $f^{\prime}$ are uniformly continuous on $(\alpha, \beta)$.
(This also defines uniform smoothness for a function on an infinite interval, provided the definition of uniform continuity is the alternative definition given in lemma 3.3 – with the word “finite” replaced by “infinite”)

Example 3.9: Consider the function $f(x)=x^{1 / 2}$ over the interval $(0,1)$. Both $f$ and its derivative, $f^{\prime}(x)=\frac{1}{2} x^{-1 / 2}$, are clearly continuous everywhere on $(0,1)$. In fact, $f$ is uniformly continuous on $(0,1)$ (You verify this!). But
$$\lim {x \rightarrow 0^{+}} f^{\prime}(x)=\lim {x \rightarrow 0^{+}} \frac{1}{2} x^{-1 / 2}=\infty .$$
So $f^{\prime}$ is not uniformly continuous on $(0,1)$, and hence, $f$ is not uniformly smooth on the interval $(0,1)$.

傅里叶分析代写

数学代写|傅里叶分析代写Fourier analysis代考|Differentiability

$$\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$$

?-练习3.5:验证 $|x|$ 是连续的，但不可微的，at $x=0$．

$$f^{\prime}(x)=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} .$$

$$\operatorname{step}(x)=\left{\begin{array}{ll} 0 & \text { if } \quad x<0 \ 1 & \text { if } 0<x \end{array},\right.$$

$$\operatorname{step}^{\prime}(x)=\left{\begin{array}{ll} 0 & \text { if } \quad x<0 \ 0 & \text { if } 0<x \end{array} .\right.$$

$$\text { step }^{\prime}=0 \text {, }$$

数学代写|傅里叶分析代写Fourier analysis代考|Smoothness Smooth Functions

$f$ 必须在$(\alpha, \beta)$上处处可微(因此是连续的)，并且

$f^{\prime}$ 也必须是$(\alpha, \beta)$上的连续函数。

$f$ 是平滑的$(\alpha, \beta)$，和

$f$和$f^{\prime}$在$(\alpha, \beta)$上都是一致连续的。
(这也定义了函数在无限区间上的一致平滑性，前提是一致连续性的定义是引理3.3中给出的替代定义——用“有限”一词代替“无限”)

$$\lim {x \rightarrow 0^{+}} f^{\prime}(x)=\lim {x \rightarrow 0^{+}} \frac{1}{2} x^{-1 / 2}=\infty .$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

数学代写|傅里叶分析代写Fourier analysis代考|Math290

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

数学代写|傅里叶分析代写Fourier analysis代考|Classifying Functions Based on Continuity Continuous Functions

A function $f$ is continuous on an interval $(\alpha, \beta)$ if and only if it is continuous at each point in the interval. Remember that, if any finite subinterval of $(\alpha, \beta)$ contains a finite (but not infinite ${ }^4$ ) number of trivial discontinuities, then all trivial discontinuities are automatically assumed to have been removed.

Example 3.3: The function from example 3.1,
$$f(x)=\frac{\sin (2 \pi x)}{\sin (\pi x)},$$
is continuous on the real line.
Even though a function is continuous on a given interval, it might still be rather poorly behaved near an endpoint of the interval. For example, even though the function $1 / x$ is continuous on the finite interval $(0,1)$, it is not bounded. Instead, it “blows up” around $x=0$. To exclude such functions from discussion when $(\alpha, \beta)$ is a finite interval, we will impose the condition of “uniform continuity”, as defined in the next paragraph.

Let $(\alpha, \beta)$ be a finite interval. The function $f$ is uniformly continuous on $(\alpha, \beta)$ if, in addition to being continuous on $(\alpha, \beta)$, its one-sided limits at the endpoints,
$$\lim {x \rightarrow \alpha^{+}} f(x) \quad \text { and } \quad \lim {x \rightarrow \beta^{-}} f(x) \quad,$$
both exist.

?-Exercise 3.2: Why is $(x-1)^{-1}$ not uniformly continuous on $(0,1)$ ?
Let us observe that, if $f$ is continuous on any interval $(\alpha, \beta)$, finite or infinite, and if $\alpha<a<b<\beta$, then $f$ is continuous over the finite subinterval $(a, b)$. Moreover, since $f$ is continuous at $a$ and $b$, the one-sided limits
$$\lim {x \rightarrow a^{+}} f(x) \quad \text { and } \quad \lim {x \rightarrow b^{-}} f(x)$$
both exist. Thus, $f$ is uniformly continuous over $(a, b)$. This fact is significant enough to be recorded in a lemma for future reference.

数学代写|傅里叶分析代写Fourier analysis代考|Discontinuous Functions

Fourier analysis would be of very limited value if it only dealt with continuous functions. Still, we won’t be able to deal with every possible discontinuous function. We will have to restrict our attention to discontinuous functions we can reasonably handle. Typically, the minimal continuity requirement that we can conveniently get away with is “piecewise continuity” over the interval of interest. Occasionally the requirements can be weakened so that we can deal with some functions that are merely “continuous over some partitioning of the interval”.
Because it is the more important, we will describe “piecewise continuity” first.
Let $f$ be a function defined on an interval $(\alpha, \beta)$. If $(\alpha, \beta)$ is a finite interval, then we will say $f$ is piecewise continuous on $(\alpha, \beta)$ if and only if all of the following three statements hold:

1. $f$ has at most a finite number (possibly zero) of discontinuities on $(\alpha, \beta)$.
2. All of the (nontrivial) discontinuities of $f$ on $(\alpha, \beta)$ are jump discontinuities.
3. Both $\lim {x \rightarrow \alpha^{+}} f(x)$ and $\lim {x \rightarrow \beta^{-}} f(x)$ exist (as finite numbers).
If, on the other hand, $(\alpha, \beta)$ is an infinite interval, then $f$ will be referred to as piecewise continuous on $(\alpha, \beta)$ if and only if it is piecewise continuous on each finite subinterval of $(\alpha, \beta)$.

It is important to realize that a piecewise continuous function is not simply “continuous over pieces of $(\alpha, \beta)$ “. To see this, let $(\alpha, \beta)$ be a finite interval, and let $x_1, x_2, \ldots, x_N$ be the points in $(\alpha, \beta)-$ indexed so that $x_1<x_2<\cdots<x_N-$ at which a given piecewise continuous function $f$ is discontinuous. These points partition $(\alpha, \beta)$ into a finite number of subintervals
with $f$ being continuous over each of these subintervals. But the second and third parts of the definition also ensure that
$$\lim {x \rightarrow \alpha^{+}} f(x), \lim {x \rightarrow x_1^{-}} f(x), \lim {x \rightarrow x_1^{+}} f(x), \lim {x \rightarrow x_2^{-}} f(x) \quad, \quad \ldots \quad, \lim _{x \rightarrow \beta^{-}} f(x)$$
all exist (and are finite). Thus, not only is $f$ continuous on each of the above subintervals, it is uniformly continuous on each of the above subintervals. ${ }^5$

傅里叶分析代写

数学代写|傅里叶分析代写Fourier analysis代考|Classifying Functions Based on Continuity Continuous Functions

$$f(x)=\frac{\sin (2 \pi x)}{\sin (\pi x)},$$

$$\lim {x \rightarrow \alpha^{+}} f(x) \quad \text { and } \quad \lim {x \rightarrow \beta^{-}} f(x) \quad,$$

-练习3.2:为什么$(x-1)^{-1}$在$(0,1)$上不是均匀连续的?

$$\lim {x \rightarrow a^{+}} f(x) \quad \text { and } \quad \lim {x \rightarrow b^{-}} f(x)$$

数学代写|傅里叶分析代写Fourier analysis代考|Discontinuous Functions

$f$ 在$(\alpha, \beta)$上最多有有限个不连续点(可能为零)。

$(\alpha, \beta)$上$f$的所有(非平凡)不连续都是跳变不连续。

$\lim {x \rightarrow \alpha^{+}} f(x)$和$\lim {x \rightarrow \beta^{-}} f(x)$都存在(作为有限的数字)。

$$\lim {x \rightarrow \alpha^{+}} f(x), \lim {x \rightarrow x_1^{-}} f(x), \lim {x \rightarrow x_1^{+}} f(x), \lim {x \rightarrow x_2^{-}} f(x) \quad, \quad \ldots \quad, \lim _{x \rightarrow \beta^{-}} f(x)$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

数学代写|傅里叶分析代写Fourier analysis代考|MA3266

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

数学代写|傅里叶分析代写Fourier analysis代考|Trivial Discontinuities

The function $f$ has a trivial discontinuity (also called a removable discontinuity) at $x_0$ if the limit of $f(x)$ does exist as $x$ approaches $x_0$ but, for some reason, either this limit does not equal $f\left(x_0\right)$ or $f\left(x_0\right)$ does not even exist according to the definition given for the function. A classic example is the sinc (pronounced “sink”) function on $(-\infty, \infty)$. It is given by the formula ${ }^2$
$$\operatorname{sinc}(x)=\frac{\sin (x)}{x} .$$
While this formula is indeterminate at $x=0$, we see that, using L’Hôpital’s rule,
$$\lim {x \rightarrow 0} \frac{\sin (x)}{x}=\lim {x \rightarrow 0} \frac{\frac{d}{d x} \sin (x)}{\frac{d}{d x} x}=\lim _{x \rightarrow 0} \frac{\cos (x)}{1}=1 .$$
But recall our discussion in the previous chapter. As far as we are concerned, the value of a function at a single point is irrelevant, and (re)defining the formula for it at any single point (or any finite number of points on any finite interval) does not change that function. This means we can “remove” the discontinuity in the sinc function by appropriately (re)defining $\operatorname{sinc}(x)$ to be 1 when $x=0$,
$$\operatorname{sinc}(x)=\left{\begin{array}{cl} \frac{\sin (x)}{x} & \text { if } \quad x \neq 0 \ 1 & \text { if } \quad x=0 \end{array} .\right.$$

Likewise, any other function $f$ with a trivial discontinuity at some point $x_0$ can have that discontinuity removed by (re)defining $f\left(x_0\right)$ to be $\lim _{x \rightarrow x_0} f(x)$. Since redefining a function’s formula at isolated points does not change the function as far as we are concerned, let us agree that, if any function is initially defined or otherwise described with a finite number of trivial discontinuities on any finite interval, then those trivial discontinuities are automatically assumed to be removed.

数学代写|傅里叶分析代写Fourier analysis代考|Jump Discontinuities

The function $f$ has a jump discontinuity at $x_0$ if the left- and right-hand limits of the function at $x_0$,
$$\lim {x \rightarrow x_0^{-}} f(x) \quad \text { and } \quad \lim {x \rightarrow x_0^{+}} f(x) \quad,$$
both exist but are not equal (see figure 3.2). The jump in $f$ at $x_0$ is the difference
$$j_0=\lim {x \rightarrow x_0^{+}} f(x)-\lim {x \rightarrow x_0^{-}} f(x) .$$
Clearly, such a function cannot be made continuous by (re)defining the function at the jump discontinuity. We could, for reasons of aesthetics (again, see figure 3.2), (re)define the value of a function at a jump discontinuity to be the midpoint of the jump,
$$f\left(x_0\right)=\frac{1}{2}\left[\lim {x \rightarrow x_0^{+}} f(x)+\lim {x \rightarrow x_0^{-}} f(x)\right],$$

but this will not appreciably simplify the mathematics of interest to us. Since this is the case and since we have already agreed that the value of a function at a single point is irrelevant, we will simply not worry about the value of a function at a jump. And if the value of a function is accidentally specified at a jump, we will feel free to ignore that specification.

Example 3.2 (the step function): One of the simplest examples of a function with a jump discontinuity is the unit step function
$$\operatorname{step}(x)=\left{\begin{array}{ll} 0 & \text { if } \quad x<0 \ 1 & \text { if } 0<x \end{array} .\right.$$
Note that step $=u=h$ where $u$ and $h$ are the functions from example 2.2. ${ }^3$

傅里叶分析代写

数学代写|傅里叶分析代写Fourier analysis代考|Trivial Discontinuities

$$\operatorname{sinc}(x)=\frac{\sin (x)}{x} .$$

$$\lim {x \rightarrow 0} \frac{\sin (x)}{x}=\lim {x \rightarrow 0} \frac{\frac{d}{d x} \sin (x)}{\frac{d}{d x} x}=\lim _{x \rightarrow 0} \frac{\cos (x)}{1}=1 .$$

$$\operatorname{sinc}(x)=\left{\begin{array}{cl} \frac{\sin (x)}{x} & \text { if } \quad x \neq 0 \ 1 & \text { if } \quad x=0 \end{array} .\right.$$

数学代写|傅里叶分析代写Fourier analysis代考|Jump Discontinuities

$$\lim {x \rightarrow x_0^{-}} f(x) \quad \text { and } \quad \lim {x \rightarrow x_0^{+}} f(x) \quad,$$

$$j_0=\lim {x \rightarrow x_0^{+}} f(x)-\lim {x \rightarrow x_0^{-}} f(x) .$$

$$f\left(x_0\right)=\frac{1}{2}\left[\lim {x \rightarrow x_0^{+}} f(x)+\lim {x \rightarrow x_0^{-}} f(x)\right],$$

$$\operatorname{step}(x)=\left{\begin{array}{ll} 0 & \text { if } \quad x<0 \ 1 & \text { if } 0<x \end{array} .\right.$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

数学代写|傅里叶分析代写Fourier analysis代考|MATH4303

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

数学代写|傅里叶分析代写Fourier analysis代考|Integral Formulas

We will be using a number of integral formulas in our work, most of which should be well known from basic calculus. For example, you surely recall that no one really calculates an integral via Riemann sums. Instead, we use the fact that, as long as $f$ is uniformly smooth on a finite interval $(\alpha, \beta)$,
$$\int_\alpha^\beta f^{\prime}(x) d x=” f(\beta)-f(\alpha) ” .$$
Notice the quotes around the right-hand side of this equation. As written, this formula assumes $f$ is continuous at the endpoints of $(\alpha, \beta)$. Often, though, we will be dealing with functions that have jump discontinuities at the endpoints of the intervals over which we are integrating. In these cases, the correct formula is actually
$$\int_\alpha^\beta f^{\prime}(x) d x=\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x) .$$
For convenience, this will often be written as
$$\int_\alpha^\beta f^{\prime}(x) d x=\left.f(x)\right|\alpha ^\beta,$$ where it is understood that $$\left.f(x)\right|\alpha ^\beta=\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x) .$$
Because we will often be integrating functions that are not smooth, let us state and verify the following slight generalization of the above:
Theorem 4.1
Let $f$ be continuous and piecewise smooth on the finite interval $(\alpha, \beta)$. Then
$$\int_\alpha^\beta f^{\prime}(x) d x=\left.f(x)\right|_\alpha ^\beta$$

PROOF (partial): First of all, if $f^{\prime}$ has no discontinuities, then $f$ is uniformly smooth on $(\alpha, \beta)$ and, from elementary calculus, we know equation (4.7) holds.

If $f^{\prime}$ has only one discontinuity in $(\alpha, \beta)$, say, at $x=x_0$, then $f$ is uniformly smooth on $\left(\alpha, x_0\right)$ and $\left(x_0, \beta\right)$. Thus,
\begin{aligned} \int_\alpha^\beta f^{\prime}(x) d x & =\int_\alpha^{x_0} f^{\prime}(x) d x+\int_{x_0}^\beta f^{\prime}(x) d x \ & =\left[\lim {x \rightarrow x_0^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x)\right]+\left[\lim {x \rightarrow \beta-} f(x)-\lim {x \rightarrow x_0^{+}} f(x)\right] \ & =\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x)+\lim {x \rightarrow x_0^{-}} f(x)-\lim {x \rightarrow x_0^{+}} f(x) . \end{aligned}

数学代写|傅里叶分析代写Fourier analysis代考|Infinite Series (Summations)

For mathematicians (and others indoctrinated by mathematicians – like you), an infinite series is simply any expression that looks like the summation of an infinite number of things. For example, you should recognize
$$\sum_{k=1}^{\infty} \frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\cdots$$
(with the “…” denoting “continue the obvious pattern”) as the famous harmonic series.
In Fourier analysis we must deal with infinite series of numbers, infinite series of functions, and, ultimately, infinite series of generalized functions. Here, we will review some basic facts concerning infinite series of numbers. Later, as the need arises, we’ll extend our discussions to include those other infinite series.
Basic Facts
Let $c_0, c_1, c_2, \ldots$ be any sequence of numbers, and consider the infinite series with these numbers as its terms,
$$\sum_{k=0}^{\infty} c_k=c_0+c_1+c_2+\cdots .$$
Here the index, $k$, started at 0 . In practice, it can start at any convenient integer $M$. For any integer $N$ with $N \geq 0$ (or, more generally, with $N \geq M$ ), the $N^{\text {th }}$ partial sum $S_N$ is simply the value obtained by adding all the terms up to and including $c_N$,
$$S_N=\sum_{k=0}^N c_k=c_0+c_1+c_2+\cdots+c_N .$$
The sum (or value) of the infinite series, which is also denoted by $\sum_{k=0}^{\infty} c_k$, is the value we get by taking the limit of the $N^{\text {th }}$ partial sums as $N \rightarrow \infty$,
$$\sum_{k=0}^{\infty} c_k=\lim {N \rightarrow \infty} S_N=\lim {N \rightarrow \infty} \sum_{k=0}^N c_k .$$

傅里叶分析代写

数学代写|傅里叶分析代写Fourier analysis代考|Integral Formulas

$$\int_\alpha^\beta f^{\prime}(x) d x=” f(\beta)-f(\alpha) ” .$$

$$\int_\alpha^\beta f^{\prime}(x) d x=\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x) .$$

$$\int_\alpha^\beta f^{\prime}(x) d x=\left.f(x)\right|\alpha ^\beta,$$其中，$$\left.f(x)\right|\alpha ^\beta=\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x) .$$

$$\int_\alpha^\beta f^{\prime}(x) d x=\left.f(x)\right|_\alpha ^\beta$$

\begin{aligned} \int_\alpha^\beta f^{\prime}(x) d x & =\int_\alpha^{x_0} f^{\prime}(x) d x+\int_{x_0}^\beta f^{\prime}(x) d x \ & =\left[\lim {x \rightarrow x_0^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x)\right]+\left[\lim {x \rightarrow \beta-} f(x)-\lim {x \rightarrow x_0^{+}} f(x)\right] \ & =\lim {x \rightarrow \beta^{-}} f(x)-\lim {x \rightarrow \alpha^{+}} f(x)+\lim {x \rightarrow x_0^{-}} f(x)-\lim {x \rightarrow x_0^{+}} f(x) . \end{aligned}

数学代写|傅里叶分析代写Fourier analysis代考|Infinite Series (Summations)

$$\sum_{k=1}^{\infty} \frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\cdots$$
(“…”表示“继续明显的模式”)作为著名的和声级数。

$$\sum_{k=0}^{\infty} c_k=c_0+c_1+c_2+\cdots .$$

$$S_N=\sum_{k=0}^N c_k=c_0+c_1+c_2+\cdots+c_N .$$

$$\sum_{k=0}^{\infty} c_k=\lim {N \rightarrow \infty} S_N=\lim {N \rightarrow \infty} \sum_{k=0}^N c_k .$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

数学代写|傅里叶分析代写Fourier analysis代考|MATH668

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

数学代写|傅里叶分析代写Fourier analysis代考|Some Useful Inequalities

At various points in our work we will need to determine “suitable upper bounds” for various numerical expressions. At some of these points, the inequalities discussed below will be invaluable.

Two basic inequalities will be identified. You are probably well acquainted with the first one, the triangle inequality, though you may not have given it a name before. You may not be as well acquainted with the second one, the Schwarz inequality. It is somewhat more subtle than the triangle inequality and will require a formal proof. Both, it should be mentioned, are fundamental inequalities in analysis and have applications and generalizations beyond the simple formulas discussed in this section.
The Triangle Inequality
Let $A$ and $B$ be any two real numbers. If you just consider how values of $|A|,|B|,|A+B|$, and $|A|+|B|$ depend on the signs of $A$ and $B$, then you should realize that
$$|A+B| \leq|A|+|B|$$
This inequality is called the triangle inequality. The reason for its name is explained in chapter 6 (see page 58), where it is also shown that this inequality holds when $A$ and $B$ are complex numbers as well.

There are two other inequalities that we can immediately derive from the triangle inequality. The first is the obvious extension to the case where we are adding up some (finite) set of numbers $\left{A_1, A_2, A_3, \ldots, A_N\right}$. Successively applying the triangle inequality,
\begin{aligned} \left|A_1+A_2+A_3+\cdots+A_N\right| & \leq\left|A_1\right|+\left|A_2+A_3+\cdots+A_N\right| \ & \leq\left|A_1\right|+\left|A_2\right|+\left|A_3+\cdots+A_N\right| \ & \leq \cdots, \end{aligned}
we are, eventually, left with the inequality
$$\left|A_1+A_2+A_3+\cdots+A_N\right| \leq\left|A_1\right|+\left|A_2\right|+\left|A_3\right|+\cdots+\left|A_N\right|,$$
which can also be called the triangle inequality.

数学代写|傅里叶分析代写Fourier analysis代考|The Schwarz Inequality (for Finite Sums)

The Schwarz inequality is a generalization of the well-known fact that, if $\mathbf{a}$ and $\mathbf{b}$ are any two two- or three-dimensional vectors, then
$$|\mathbf{a} \cdot \mathbf{b}| \leq|\mathbf{a}||\mathbf{b}|$$
In component form, with $\mathbf{a}=\left(a_1, a_2, a_3\right)$ and $\mathbf{b}=\left(b_1, b_2, b_3\right)$, this inequality is
$$\left|\sum_{k=1}^3 a_k b_k\right| \leq\left(\sum_{k=1}^3\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^3\left|b_k\right|^2\right)^{1 / 2} .$$

Theorem 3.7 (Schwarz inequality for finite summations)
Let $N$ be any integer, and let $\left{a_1, a_2, a_3, \ldots, a_N\right}$ and $\left{b_1, b_2, b_3, \ldots, b_N\right}$ be any two sets of $N$ numbers (real or complex). Then,
$$\left|\sum_{k=1}^N a_k b_k\right| \leq\left(\sum_{k=1}^N\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^N\left|b_k\right|^2\right)^{1 / 2} .$$
PROOF: Suppose we can show
$$\sum_{k=1}^N\left|a_k\right|\left|b_k\right| \leq\left(\sum_{k=1}^N\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^N\left|b_k\right|^2\right)^{1 / 2} .$$
Then inequality (3.7) follows immediately by combining the above inequality with the triangle inequality,
$$\left|\sum_{k=1}^N a_k b_k\right| \leq \sum_{k=1}^N\left|a_k b_k\right|=\sum_{k=1}^N\left|a_k\right|\left|b_k\right| .$$
So we only need to verify that inequality (3.8) holds.
Consider, first, the trivial case where either
$$\sum_{k=1}^N\left|a_k\right|^2=0 \quad \text { or } \quad \sum_{k=1}^N\left|b_k\right|^2=0$$

傅里叶分析代写

数学代写|傅里叶分析代写Fourier analysis代考|Some Useful Inequalities

$$|A+B| \leq|A|+|B|$$

\begin{aligned} \left|A_1+A_2+A_3+\cdots+A_N\right| & \leq\left|A_1\right|+\left|A_2+A_3+\cdots+A_N\right| \ & \leq\left|A_1\right|+\left|A_2\right|+\left|A_3+\cdots+A_N\right| \ & \leq \cdots, \end{aligned}

$$\left|A_1+A_2+A_3+\cdots+A_N\right| \leq\left|A_1\right|+\left|A_2\right|+\left|A_3\right|+\cdots+\left|A_N\right|,$$

数学代写|傅里叶分析代写Fourier analysis代考|The Schwarz Inequality (for Finite Sums)

Schwarz不等式是一个众所周知的事实的推广，如果$\mathbf{a}$和$\mathbf{b}$是任意两个二维或三维向量，则
$$|\mathbf{a} \cdot \mathbf{b}| \leq|\mathbf{a}||\mathbf{b}|$$

$$\left|\sum_{k=1}^3 a_k b_k\right| \leq\left(\sum_{k=1}^3\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^3\left|b_k\right|^2\right)^{1 / 2} .$$

$$\left|\sum_{k=1}^N a_k b_k\right| \leq\left(\sum_{k=1}^N\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^N\left|b_k\right|^2\right)^{1 / 2} .$$

$$\sum_{k=1}^N\left|a_k\right|\left|b_k\right| \leq\left(\sum_{k=1}^N\left|a_k\right|^2\right)^{1 / 2}\left(\sum_{k=1}^N\left|b_k\right|^2\right)^{1 / 2} .$$

$$\left|\sum_{k=1}^N a_k b_k\right| \leq \sum_{k=1}^N\left|a_k b_k\right|=\sum_{k=1}^N\left|a_k\right|\left|b_k\right| .$$

$$\sum_{k=1}^N\left|a_k\right|^2=0 \quad \text { or } \quad \sum_{k=1}^N\left|b_k\right|^2=0$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

数学代写|傅里叶分析代写Fourier analysis代考|MATH382

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

数学代写|傅里叶分析代写Fourier analysis代考|Classifying Functions Based on Continuity Continuous Functions

A function $f$ is continuous on an interval $(\alpha, \beta)$ if and only if is continuous at each point in the interval. Remember that, if any finite subinterval of $(\alpha, \beta)$ contains a finite (but not infinite ${ }^4$ ) number of trivial discontinuities, then all trivial discontinuities are automatically assumed to have been removed.

$$f(x)=\frac{\sin (2 \pi x)}{\sin (\pi x)},$$
is continuous on the real line.
Even though a function is continuous on a given interval, it might still be rather poorly behaved near an endpoint of the interval. For example, even though the function $1 / x$ is continuous on the finite interval $(0,1)$, it is not bounded. Instead, it “blows up” around $x=0$. To exclude such functions from discussion when $(\alpha, \beta)$ is a finite interval, we will impose the condition of “uniform continuity”, as defined in the next paragraph.

Let $(\alpha, \beta)$ be a finite interval. The function $f$ is uniformly continuous on $(\alpha, \beta)$ if, in addition to being continuous on $(\alpha, \beta)$, its one-sided limits at the endpoints,
$$\lim {x \rightarrow \alpha^{+}} f(x) \quad \text { and } \quad \lim {x \rightarrow \beta^{-}} f(x) \quad,$$
both exist.

数学代写|傅里叶分析代写Fourier analysis代考|Discontinuous Functions

Fourier analysis would be of very limited value if it only dealt with continuous functions. Still, we won’t be able to deal with every possible discontinuous function. We will have to restrict our attention to discontinuous functions we can reasonably handle. Typically, the minimal continuity requirement that we can conveniently get away with is “piecewise continuity” over the interval of interest. Occasionally the requirements can be weakened so that we can deal with some functions that are merely “continuous over some partitioning of the interval”.
Because it is the more important, we will describe “piecewise continuity” first.
Let $f$ be a function defined on an interval $(\alpha, \beta)$. If $(\alpha, \beta)$ is a finite interval, then we will say $f$ is piecewise continuous on $(\alpha, \beta)$ if and only if all of the following three statements hold:

1. $f$ has at most a finite number (possibly zero) of discontinuities on $(\alpha, \beta)$.
2. All of the (nontrivial) discontinuities of $f$ on $(\alpha, \beta)$ are jump discontinuities.
3. Both $\lim {x \rightarrow \alpha^{+}} f(x)$ and $\lim {x \rightarrow \beta^{-}} f(x)$ exist (as finite numbers).
If, on the other hand, $(\alpha, \beta)$ is an infinite interval, then $f$ will be referred to as piecewise continuous on $(\alpha, \beta)$ if and only if it is piecewise continuous on each finite subinterval of $(\alpha, \beta)$.

It is important to realize that a piecewise continuous function is not simply “continuous over pieces of $(\alpha, \beta)$ “. To see this, let $(\alpha, \beta)$ be a finite interval, and let $x_1, x_2, \ldots, x_N$ be the points in $(\alpha, \beta)-$ indexed so that $x_1<x_2<\cdots<x_N-$ at which a given piecewise continuous function $f$ is discontinuous. These points partition $(\alpha, \beta)$ into a finite number of subintervals
$$\begin{array}{lllllll} \left.\alpha, x_1\right) & , & \left(x_1, x_2\right) & , & \left(x_2, x_3\right) & , & \ldots \end{array} \quad, \quad\left(x_N, \beta\right) \quad,$$
with $f$ being continuous over each of these subintervals. But the second and third parts of the definition also ensure that
$$\lim {x \rightarrow \alpha^{+}} f(x) \quad, \quad \lim {x \rightarrow x_1^{-}} f(x) \quad, \quad \lim {x \rightarrow x_1^{+}} f(x) \quad, \quad \lim {x \rightarrow x_2^{-}} f(x) \quad, \quad \ldots \quad, \quad \lim _{x \rightarrow \beta^{-}} f(x)$$
all exist (and are finite). Thus, not only is $f$ continuous on each of the above subintervals, it is uniformly continuous on each of the above subintervals. ${ }^5$

傅里叶分析代写

考|Classifying Functions Based on Continuity Continuous Functions

$$f(x)=\frac{\sin (2 \pi x)}{\sin (\pi x)},$$

$$\lim {x \rightarrow \alpha^{+}} f(x) \quad \text { and } \quad \lim {x \rightarrow \beta^{-}} f(x) \quad,$$

数学代写|傅里叶分析代写Fourier analysis代考|Discontinuous Functions

$f$ 在$(\alpha, \beta)$上最多有有限个不连续点(可能为零)。

$(\alpha, \beta)$上$f$的所有(非平凡)不连续都是跳变不连续。

$\lim {x \rightarrow \alpha^{+}} f(x)$和$\lim {x \rightarrow \beta^{-}} f(x)$都存在(作为有限的数字)。

$$\begin{array}{lllllll} \left.\alpha, x_1\right) & , & \left(x_1, x_2\right) & , & \left(x_2, x_3\right) & , & \ldots \end{array} \quad, \quad\left(x_N, \beta\right) \quad,$$

$$\lim {x \rightarrow \alpha^{+}} f(x) \quad, \quad \lim {x \rightarrow x_1^{-}} f(x) \quad, \quad \lim {x \rightarrow x_1^{+}} f(x) \quad, \quad \lim {x \rightarrow x_2^{-}} f(x) \quad, \quad \ldots \quad, \quad \lim _{x \rightarrow \beta^{-}} f(x)$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

数学代写|傅里叶分析代写Fourier analysis代考|MATH382

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

数学代写|傅里叶分析代写Fourier analysis代考|Functions, Formulas, and Variables Basics

Most of the time we will be concerned with functions defined on some given interval of the real line. If no interval is explicitly stated or obviously implied by other conditions, then you may assume that the functions under consideration are defined on the entire real line.

Typically, a function $f$ is described (or defined) by stating its domain and a formula for computing the value of $f(x)$ for all “relevant values of $x$ “. (For now, “all relevant values of $x$ ” should be taken as meaning “all $x$ in the domain of the function”, though we’ll soon see that this is not always quite the case.) For our purposes, a formula for $f$ is any set of instructions for determining the value of $f(x)$ for each relevant value of $x$. Sometimes the formula will be a simple expression involving well-known functions (e.g., $(3+x)^2$ or $\sin (2 \pi x)$ ). Other times the formula may be a collection of simple formulas with each valid over a different interval. For example, the ramp function is the function on $(-\infty, \infty)$ given by the formula
$$\operatorname{ramp}(x)=\left{\begin{array}{ll} 0 & \text { if } \quad x<0 \ x & \text { if } \quad 0 \leq x \end{array} .\right.$$
We should also expect formulas involving integrals and infinite summations, such as
$$f(x)=\int_{t=0}^x 3 t^2 d t \quad \text { and } \quad g(x)=\sum_{k=1}^{\infty} \frac{1}{(1+k)^k} \sin (n \pi x)$$

数学代写|傅里叶分析代写Fourier analysis代考|A Pragmatic Approach to Domains and Function Equality

Often we must deal with functions that are not well defined at a few isolated points in the intervals of interest. Sometimes this is because the formula defining the function has ambiguities. Other times this is because of inherent discontinuities in the function. In practice, though, we are only concerned with the behavior of a function over intervals, not at isolated points. Because of this, we can take a rather pragmatic point of view concerning these functions and adopt the following convention:
Convention (irrelevance of function values at isolated points)
Let $f$ and $g$ be two functions on an interval $(a, b)$. If $f(x)=g(x)$ for all but a finite number of $x$ ‘s in $(a, b)$, then $f$ and $g$ are viewed as the same function over that interval.,4

To a great extent, this convention concerns how we use formulas to define functions. A few examples may help clarify the matter.
Example 2.1: A trivial example is given by
$$f(x)=\frac{x^2-1}{x-1}$$
which is undefined for $x=1$. In applications, however, most of us would feel justified in “simplifying” $f(x)$,
$$f(x)=\frac{x^2-1}{x-1}=\frac{(x+1)(x-1)}{x-1}=x+1,$$
and then ignoring the fact that the original formula for $f(x)$ was not defined for $x=1$. In other words, “for all practical purposes” we would agree that
$$\frac{x^2-1}{x-1}=x+1$$

傅里叶分析代写

数学代写|傅里叶分析代写Fourier analysis代考|Functions, Formulas, and Variables Basics

$$\operatorname{ramp}(x)=\left{\begin{array}{ll} 0 & \text { if } \quad x<0 \ x & \text { if } \quad 0 \leq x \end{array} .\right.$$

$$f(x)=\int_{t=0}^x 3 t^2 d t \quad \text { and } \quad g(x)=\sum_{k=1}^{\infty} \frac{1}{(1+k)^k} \sin (n \pi x)$$

数学代写|傅里叶分析代写Fourier analysis代考|A Pragmatic Approach to Domains and Function Equality

$$f(x)=\frac{x^2-1}{x-1}$$

$$f(x)=\frac{x^2-1}{x-1}=\frac{(x+1)(x-1)}{x-1}=x+1,$$

$$\frac{x^2-1}{x-1}=x+1$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

数学代写|傅里叶分析代写Fourier analysis代考|MTH710

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

数学代写|傅里叶分析代写Fourier analysis代考|Wavelets

We are concerned with orthonormal bases of $L^2(\mathbf{R})$ generated by translations and dilations of a single function such as the Haar functions we encountered in Section 5.4. The Haar functions are generated by integer translations and dyadic dilations of the single function $\chi_{\left[0, \frac{1}{2}\right)}-\chi_{\left[\frac{1}{2}, 1\right)}$. This function is not smooth, and the main question addressed in this section is whether there exist smooth analogues of the Haar functions.

Definition 5.6.1. A square integrable function $\varphi$ on $\mathbf{R}^n$ is called a wavelet if the family of functions
$$\varphi_{v, k}(x)=2^{\frac{v_n}{2}} \varphi\left(2^v x-k\right),$$
where $v$ ranges over $\mathbf{Z}$ and $k$ over $\mathbf{Z}^n$, is an orthonormal basis of $L^2\left(\mathbf{R}^n\right)$. Note that the Fourier transform of $\varphi_{v, k}$ is given by
$$\widehat{\varphi_{v, k}}(\xi)=2^{-\frac{v_n}{2}} \widehat{\varphi}\left(2^{-v} \xi\right) e^{-2 \pi i 2^{-v} \xi \cdot k} .$$
Rephrasing the question posed earlier, the main issue addressed in this section is whether smooth wavelets actually exist. Before we embark on this topic, we recall that we have already encountered examples of nonsmooth wavelets.

数学代写|傅里叶分析代写Fourier analysis代考|Some Preliminary Facts

Before we look at more examples, we make some observations. We begin with the following useful fact.
Proposition 5.6.3. Let $g \in L^1\left(\mathbf{R}^n\right)$. Then
$$\widehat{g}(m)=0 \quad \text { for all } m \in \mathbf{Z}^n \backslash{0}$$
if and only if
$$\sum_{k \in \mathbf{Z}^n} g(x+k)=\int_{\mathbf{R}^n} g(t) d t$$
for almost all $x \in \mathbf{T}^n$.
Proof. We define the periodized function
$$G(x)=\sum_{k \in \mathbf{Z}^n} g(x+k)$$
which is easily shown to be in $L^1\left(\mathbf{T}^n\right)$. Moreover, we have
$$\widehat{G}(m)=\widehat{g}(m)$$
for all $m \in \mathbf{Z}^n$, where $\widehat{G}(m)$ denotes the $m$ th Fourier coefficient of $G$ and $\widehat{g}(m)$ denotes the Fourier transform of $g$ at $\xi=m$. If $\widehat{g}(m)=0$ for all $m \in \mathbf{Z}^n \backslash{0}$, then all the Fourier coefficients of $G$ (except for $m=0$ ) vanish, which means that the sequence ${\widehat{G}}_{m \in \mathbf{Z}^n}$ lies in $\ell^1\left(\mathbf{Z}^n\right)$ and hence Fourier inversion applies. We conclude that for almost all $x \in \mathbf{T}^n$ we have
$$G(x)=\sum_{m \in \mathbf{Z}^n} \widehat{G}(m) e^{2 \pi i m \cdot x}=\widehat{G}(0)=\widehat{g}(0)=\int_{\mathbf{R}^n} g(t) d t$$

傅里叶分析代写

数学代写|傅里叶分析代写Fourier analysis代考|Wavelets

5.6.1.定义一个平方可积函数$\varphi$在$\mathbf{R}^n$上被称为一个小波，如果函数族
$$\varphi_{v, k}(x)=2^{\frac{v_n}{2}} \varphi\left(2^v x-k\right),$$

$$\widehat{\varphi_{v, k}}(\xi)=2^{-\frac{v_n}{2}} \widehat{\varphi}\left(2^{-v} \xi\right) e^{-2 \pi i 2^{-v} \xi \cdot k} .$$

数学代写|傅里叶分析代写Fourier analysis代考|Some Preliminary Facts

$$\widehat{g}(m)=0 \quad \text { for all } m \in \mathbf{Z}^n \backslash{0}$$

$$\sum_{k \in \mathbf{Z}^n} g(x+k)=\int_{\mathbf{R}^n} g(t) d t$$

$$G(x)=\sum_{k \in \mathbf{Z}^n} g(x+k)$$

$$\widehat{G}(m)=\widehat{g}(m)$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。