电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|ELEN30012

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|The Stieltjes and multiplicative Stieltjes classes

An important subclass of the Pick class is the Stieltjes class denoted here by $\mathcal{S}(\widehat{\mathcal{G}})$, consisting of functions $S$ in the Pick class $\mathcal{P}(\widehat{\mathcal{G}})$ with analytic continuation across the negative half-axis $\mathbb{R}^{-}$and taking positive semidefinite values on $\mathbb{R}^{-}$:
$$\frac{S(z)-S(z)^{}}{z-\bar{z}} \succeq 0(z \notin \mathbb{R}), \quad S(x) \succeq 0 \quad(x<0) .$$ Stieltjes functions made their first explicit appearance in [44] as continued fractions of certain type and as Cauchy transforms of positive measures on $\mathbb{R}^{+}=[0, \infty)$. Being special instances of absolutely monotone functions, operator monotone functions and Pick functions, they have been extensively studied in various contexts $[12,29,30,34,33,37,43,45]$. Such functions have the alternative characterization as being those functions $S \in \mathcal{P}(\widehat{\mathcal{G}})$ such that the function $z \mapsto z S(z)$ is also in $\mathcal{P}(\widehat{\mathcal{G}})$ (see [33] for the scalar case – the operator-valued case is similar). This leads to the kernel characterization of the Stieltjes class: an $\mathcal{L}(\mathcal{G})$-valued function $S$ is in $\mathcal{S}(\mathcal{G})$ if and only if both kernels $$\mathfrak{K}(z, \omega)=\frac{S(z)-S(\omega)^{}}{z-\bar{\omega}} \text { and } \widetilde{\mathfrak{K}}(z, \omega)=\frac{z S(z)-\bar{\omega} S(\omega)^{*}}{z-\bar{\omega}}$$
are positive on the upper half-plane.

电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Connections with interpolation theory

The importance of multiplicative Pick functions for interpolation theory arises from the fact that the linear fractional map based on a function $\Theta \in \mathcal{M} \mathcal{P}(\widehat{\mathcal{G}})$ maps the class $\mathcal{P}(\mathcal{G})$ into itself. Choosing $\Theta$ with a suitable pole/zero structure then implies that the linear-fractional map based on $\Theta$ gives rise to a parametrization (with free parameter from the Pick class $\mathcal{P}(\mathcal{G})$ ) of the solution set of a given interpolation problem in the class $\mathcal{P}(\mathcal{G})$; we refer to $[11,42]$ for specific examples. It turns out the multiplicative Stieltjes class $\mathcal{M S}(\mathcal{G}, \mathbb{C})$ has similar applications in interpolation theory for the additive Stieltjes class $\mathcal{S}(\mathcal{G})$ as the linear fractional map based on a function $\Theta \in \mathcal{M} \mathcal{S}(\mathcal{G})$ not only maps the class $\mathcal{P}(\mathcal{G})$ into itself, but also the class $\mathcal{S}(\mathcal{G})$ into itself. In the context of the Nevanlinna-Pick interpolation problem, multiplicative Stieltjes functions appeared explicitly in the series of papers $[23,25,26]$; see also $[2,13,14,15,24,25,26]$ for other examples and far-reaching generalizations. From the integral representation (1.27) for the Stieltjes class, we see that the Stieltjes moment problem going back to the nineteenth century [44] can be seen as a boundary version of a Stieltjes interpolation problem. The Stieltjes class also arises in the recent work of Agler-Tully-Doyle-Young [1] on characterizing boundary directional derivatives of Schur-class functions on the bidisk.

电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|The Stieltjes and multiplicative Stieltjes classes

Pick 类的一个重要子类是 Stieltjes 类，在此表示为 $S(\widehat{\mathcal{G}})$ ，由函数组成 $S$ 在 Pick 类中 $\mathcal{P}(\widehat{\mathcal{G}})$ 在负半轴上具有解析延拓 $\mathbb{R}^{-}$并取半正定值 $\mathbb{R}^{-}$:
$$\frac{S(z)-S(z)}{z-\bar{z}} \succeq 0(z \notin \mathbb{R}), \quad S(x) \succeq 0 \quad(x<0) .$$
Stieltjes 函数在 [44] 中作为某种类型的连分数和正测量的柯西变换在 [44] 中首次明确出现 $\mathbb{R}^{+}=[0, \infty)$. 作为绝 对单调函数、算子单调函数和 Pick 函数的特例，它们在各种情况下得到了广泛的研究
$[12,29,30,34,33,37,43,45]$. 此类功能具有作为这些功能的替代特征 $S \in \mathcal{P}(\widehat{\mathcal{G}})$ 使得函数 $z \mapsto z S(z)$ 也在 $\mathcal{P}(\widehat{\mathcal{G}})$ (有关标量情况，请参见 [33] – 运算符值情况类似) 。这导致了 Stieltjes 类的内核特征: $\mathcal{L}(\mathcal{G})$ 值函数 $S$ 在 $\mathcal{S}(\mathcal{G})$ 当且仅当两个内核
$$\mathfrak{K}(z, \omega)=\frac{S(z)-S(\omega)}{z-\bar{\omega}} \text { and } \widetilde{\Re}(z, \omega)=\frac{z S(z)-\bar{\omega} S(\omega)^{*}}{z-\bar{\omega}}$$

广义线性模型代考

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MATLAB代写

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