# 数学代写|傅里叶分析代写Fourier analysis代考|MATH3205

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## 数学代写|傅里叶分析代写Fourier analysis代考|Discrete Trigonometric Transforms

In this section we consider some real versions of the DFT. Discrete trigonometric transforms are widely used in applied mathematics, digital signal processing, and image compression. Examples of such real transforms are a discrete cosine transform (DCT), discrete sine transform (DST), and discrete Hartley transform (DHT). We will introduce these transforms and show that they are generated by orthogonal matrices.

Lemma 3.46 Let $N \geq 2$ be a given integer. Then the set of cosine vectors of type I
$$\mathbf{c}{k}^{\mathrm{I}}:=\sqrt{\frac{2}{N}} \varepsilon{N}(k)\left(\varepsilon_{N}(j) \cos \frac{j k \pi}{N}\right){j=0}^{N}, \quad k=0, \ldots, N,$$ forms an orthonormal basis of $\mathbb{R}^{N+1}$, where $\varepsilon{N}(0)=\varepsilon_{N}(N):=\frac{\sqrt{2}}{2}$ and $\varepsilon_{N}(j):=$ 1 for $j=1, \ldots, N-1$. The $(N+1)$-by-(N+1) cosine matrix of type I is defined by
$$\mathbf{C}{N+1}^{\mathrm{I}}:=\sqrt{\frac{2}{N}}\left(\varepsilon{N}(j) \varepsilon_{N}(k) \cos \frac{j k \pi}{N}\right){j, k=0}^{N},$$ i.e., it has the cosine vectors of type I as columns. Then $\mathbf{C}{N+1}^{\mathrm{I}}$ is symmetric and orthogonal, i.e., $\left(\mathbf{C}{N+1}^{\mathrm{I}}\right)^{-1}=\mathbf{C}{N+1}^{\mathrm{I}}$.
Proof By Example $1.14$ we know that for $x \in \mathbb{R} \backslash 2 \pi \mathbb{Z}$
$$\sum_{j=1}^{N-1} \cos (j x)=\frac{\sin \frac{(2 N-1) x}{2}}{2 \sin \frac{x}{2}}-\frac{1}{2} .$$
In particular, it follows for $x=\frac{2 \pi k}{N}$,
$$\sum_{j=1}^{N-1} \cos \frac{2 k j \pi}{N}=-1, \quad k \in \mathbb{Z} \backslash N \mathbb{Z},$$
and for $x=\frac{(2 k+1) \pi}{N}$,
$$\sum_{j=1}^{N-1} \cos \frac{(2 k+1) j \pi}{N}=0, \quad k \in \mathbb{Z}$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Multidimensional Fourier Methods

In this chapter, we considerr $d$-dimensional Fourier methods for fixed $d \in \mathbb{N}$. We start with Fourier series of $d$-variate, $2 \pi$-periodic functions $f: \mathbb{T}^{d} \rightarrow \mathbb{C}$ in Sect. 4.1, where we follow the lines of Chap. 1. In particular, we present hasic properties of the Fourier coefficients and learn about their decay for smonth functions.

Then, in Sect. 4.2, we deal with Fourier transforms of functions defined on $\mathbb{R}^{d}$. Here, we follow another path than in the case $d=1$ considered in Chap. 2. We show that the Fourier transform is a linear, bijective operator on the Schwartz space $\mathscr{S}\left(\mathbb{R}^{d}\right)$ of rapidly decaying functions. Using the density of $\mathscr{S}\left(\mathbb{R}^{d}\right)$ in $L_{1}\left(\mathbb{R}^{d}\right)$ and $L_{2}\left(\mathbb{R}^{d}\right.$ ), the Fourier transform on these spaces is discussed. The Poisson summation formula and the Fourier transforms of radial functions are also addressed.

In Sect. 4.3, we introduce tempered distributions as linear, continuous functionals on the Schwartz space $\mathscr{S}\left(\mathbb{R}^{d}\right)$. We consider Fourier transforms of tempered distributions and Fourier series of periodic tempered distributions. Further, we introduce the Hilbert transform and its multidimensional generalization, the Riesz transform.

As in the case $d=1$, any numerical application of $d$-dimensional Fourier series or Fourier transforms leads to $d$-dimensional discrete Fourier transforms handled in Sect. 4.4. We present the basic properties of the two-dimensional and higherdimensional $\mathrm{DFT}$, including the convolution property and the aliasing formula.

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Discrete Trigonometric Transforms

$$\mathbf{c} k^{\mathrm{I}}:=\sqrt{\frac{2}{N}} \varepsilon N(k)\left(\varepsilon_{N}(j) \cos \frac{j k \pi}{N}\right) j=0^{N}, \quad k=0, \ldots, N,$$

$$\mathbf{C} N+1^{\mathrm{I}}:=\sqrt{\frac{2}{N}}\left(\varepsilon N(j) \varepsilon_{N}(k) \cos \frac{j k \pi}{N}\right) j, k=0^{N},$$

$$\sum_{j=1}^{N-1} \cos (j x)=\frac{\sin \frac{(2 N-1) x}{2}}{2 \sin \frac{x}{2}}-\frac{1}{2} .$$

$$\sum_{j=1}^{N-1} \cos \frac{2 k j \pi}{N}=-1, \quad k \in \mathbb{Z} \backslash N \mathbb{Z},$$

$$\sum_{j=1}^{N-1} \cos \frac{(2 k+1) j \pi}{N}=0, \quad k \in \mathbb{Z}$$

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