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# 数学代写|数值分析代写Numerical analysis代考|MATH/CS514 The Algorithms of Goertzel and Reinsch

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## 数学代写|数值分析代写Numerical analysis代考|The Algorithms of Goertzel and Reinsch

The problem of evaluating phase polynomials $p(x)$ from (2.3.1.2) or trigonometric expressions $\Psi(x)$ from (2.3.1.1) for some arbitrary argument $x=\xi$ is called Fourier synthesis. For phase polynomials, there are Horner-type evaluation schemes, as there are for expressions (2.3.1.1a) when written in the form $\Psi(x)=\sum_{j=-M}^M \beta_j e^{j i x}$. The numerical behavior of such evaluation schemes, however, should be examined carefully.

For example, Goertzel (1958) proposed an algorithm for a problem closely related to Fourier synthesis, namely, for simultaneously evaluating the two sums
$$\sum_{k=0}^{N-1} y_k \cos k \xi, \quad \sum_{k=1}^{N-1} y_k \sin k \xi$$
for a given argument $\xi$ and given values $y_k, k=0, \ldots, N-1$. This algorithm is not numerically stable unless it is suitably modified. The algoithm is based on the following:
(2.3.3.1) Theorem. For $\xi \neq r \pi, r=0, \pm 1, \pm 2, \ldots$, define the quantities
\begin{aligned} U_j &:=\frac{1}{\sin \xi} \sum_{k=j}^{N-1} y_k \sin (k-j+1) \xi, \quad j=0,1, \ldots, N-1, \ U_N &:=U_{N+1}:=0 . \end{aligned}

## 数学代写|数值分析代写Numerical analysis代考|The Calculation of Fourier Coefficients. Attenuation Factors

Let $\mathcal{K}$ be the set of all absolutely continuous ${ }^3$ real functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which are periodic with period $2 \pi$. It is well known [see for instance Achieser (1956)] that every function $f \in \mathcal{K}$ can be expanded into a Fourier series
$$f(x)=\sum_{j=-\infty}^{\infty} c_j e^{j i x}$$
which converges towards $f(x)$ for every $x \in \mathbb{R}$. The coefficients $c_j=c_j(f)$ of this series are given by
(2.3.4.2) $\quad c_j=c_j(f):=\frac{1}{2 \pi} \int_0^{2 \pi} f(x) e^{-j i x} d x, \quad j=0, \pm 1, \pm 2, \ldots$.
In practice, frequently all one knows of a function $f$ are its values $f_k:=$ $f\left(x_k\right)$ at equidistant arguments $x_k:=2 \pi k / N$, where $N$ is a given fixed positive integer. The problem then is to find, under these circumstances, reasonable approximate values for the Fourier coefficients $c_j(f)$. We will show how the methods of trigonometric interpolation can be applied to this problem.

By Theorem (2.3.1.9), the coefficients $\beta_j$ of the interpolating phase polynomial
$$p(x)=\beta_0+\beta_1 e^{i x}+\cdots+\beta_{N-1} e^{(N-1) i x},$$
with
$$p\left(x_k\right)=f_k$$
for $k=0, \pm 1, \pm 2, \ldots$, are given by
$$\beta_j=\frac{1}{N} \sum_{k=0}^{N-1} f_k e^{-j i x_k}, \quad j=0,1, \ldots, N-1$$

## 数学代写数值分析代写Numerical analysis代考|The Algorithms of Goertzel and Reinsch

$$\sum_{k=0}^{N-1} y_k \cos k \xi, \quad \sum_{k=1}^{N-1} y_k \sin k \xi$$

(2.3.3.1) 定理。为了 $\xi \neq r \pi, r=0, \pm 1, \pm 2, \ldots$, 定义数量
$$U_j:=\frac{1}{\sin \xi} \sum_{k=j}^{N-1} y_k \sin (k-j+1) \xi, \quad j=0,1, \ldots, N-1, U_N \quad:=U_{N+1}:=0 .$$

## 数学代写|数值分析代写Numerical analysis代考|The Calculation of Fourier Coefficients. Attenuation Factors

$$f(x)=\sum_{j=-\infty}^{\infty} c_j e^{j i x}$$

(2.3.4.2)给出 $c_j=c_j(f):=\frac{1}{2 \pi} \int_0^{2 \pi} f(x) e^{-j i x} d x, \quad j=0, \pm 1, \pm 2, \ldots$.

$$p(x)=\beta_0+\beta_1 e^{i x}+\cdots+\beta_{N-1} e^{(N-1) i x}$$

$$p\left(x_k\right)=f_k$$

$$\beta_j=\frac{1}{N} \sum_{k=0}^{N-1} f_k e^{-j i x_k}, \quad j=0,1, \ldots, N-1$$

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