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# 数学代写|傅里叶分析代写Fourier Analysis代考|The Schwartz Class and the Fourier Transform

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## 数学代写|傅里叶分析代写Fourier Analysis代考|The Schwartz Class and the Fourier Transform

In this section we introduce the single most important tool in harmonic analysis, the Fourier transform. It is often the case that the Fourier transform is introduced as an operation on $L^1$ functions. In this exposition we first define the Fourier transform on a smaller class, the space of Schwartz functions, which turns out to be a very natural environment. Once the basic properties of the Fourier transform are derived, we extend its definition to other spaces of functions.

We begin with some preliminaries. Given $x=\left(x_1, \ldots, x_n\right) \in \mathbf{R}^n$, we set $|x|=$ $\left(x_1^2+\cdots+x_n^2\right)^{1 / 2}$. The first partial derivative of a function $f$ on $\mathbf{R}^n$ with respect to the $j$ th variable $x_j$ is denoted by $\partial_j f$ while the $m$ th partial derivative with respect to the $j$ th variable is denoted by $\partial_j^m f$. A multi-index $\alpha$ is an ordered $n$-tuple of nonnegative integers. For a multi-index $\alpha=\left(\alpha_1, \ldots, \alpha_n\right), \partial^\alpha f$ denotes the derivative $\partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n} f$. If $\alpha=\left(\alpha_1, \ldots, \alpha_n\right)$ is a multi-index, $|\alpha|=\alpha_1+\cdots+\alpha_n$ denotes its size

and $\alpha !=\alpha_{1} ! \cdots \alpha_n$ ! denotes the product of the factorials of its entries. The number $|\alpha|$ indicates the total order of differentiation of $\partial^\alpha f$. The space of functions in $\mathbf{R}^n$ all of whose derivatives of order at most $N \in \mathbf{Z}^{+}$are continuous is denoted by $\mathscr{C}^N\left(\mathbf{R}^n\right)$ and the space of all infinitely differentiable functions on $\mathbf{R}^n$ by $\mathscr{C}^{\infty}\left(\mathbf{R}^n\right)$. The space of $\mathscr{C}^{\infty}$ functions with compact support on $\mathbf{R}^n$ is denoted by $\mathscr{C}_0^{\infty}\left(\mathbf{R}^n\right)$. This space is nonempty; see Exercise 2.2.1(a).

For $x \in \mathbf{R}^n$ and $\alpha=\left(\alpha_1, \ldots, \alpha_n\right)$ a multi-index, we set $x^\alpha=x_1^{\alpha_1} \cdots x_n^{\alpha_n}$. It is a simple fact to verify that
$$\left|x^\alpha\right| \leq c_{n, \alpha}|x|^{|\alpha|}$$
for some constant that depends on the dimension $n$ and on $\alpha$. In fact, $c_{n, \alpha}$ is the maximum of the continuous function $\left(x_1, \ldots, x_n\right) \mapsto\left|x_1^{\alpha_1} \cdots x_n^{\alpha_n}\right|$ on the sphere $\mathbf{S}^{n-1}=\left{x \in \mathbf{R}^n:|x|=1\right}$. The converse inequality in (2.2.1) fails. However, the following substitute of the converse of (2.2.1) is of great use: for $k \in \mathbf{Z}^{+}$we have
$$|x|^k \leq C_{n, k} \sum_{|\beta|=k}\left|x^\beta\right|$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Interpolation of Analytic Families of Operators

Theorem 1.3.4 can now be extended to the case in which the interpolated operators are allowed to vary. In particular, if a family of operators depends analytically on a parameter $z$, then the proof of this theorem can be adapted to work in this setting.
We now describe the setup for this theorem. Let $(X, \mu)$ and $(Y, v)$ be measure spaces. Suppose that for every $z$ in the closed strip $\bar{S}={z \in \mathbf{C}: 0 \leq \operatorname{Re} z \leq 1}$ there is an associated linear operator $T_z$ defined on the space of simple functions on $X$ and taking values in the space of measurable functions on $Y$ such that
$$\int_Y\left|T_z(f) g\right| d v<\infty$$
whenever $f$ and $g$ are simple functions on $X$ and $Y$, respectively. The family $\left{T_z\right}_z$ is said to be analytic if the function
$$z \mapsto \int_Y T_z(f) g d v$$
is analytic in the open strip $S={z \in \mathbf{C}: 0<\operatorname{Re} z<1}$ and continuous on its closure. Finally, the analytic family is of admissible growth if there is a constant $a<\pi$ and a constant $C_{f, g}$ such that
$$e^{-a|\operatorname{Im} z|} \log \left|\int_Y T_z(f) g d v\right| \leq C_{f, g}<\infty$$
for all $z$ satisfying $0 \leq \operatorname{Re} z \leq 1$. The extension of the Riesz-Thorin interpolation theorem is now stated.

## MATLAB代写

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