\documentclass{article}
\usepackage{iftex}
\usepackage{ifthen}
\usepackage{epsfig}
\usepackage[utf8]{inputenc}
\usepackage{xcolor}
\color{white}
\pagecolor{black}
% Packages requested by user
\usepackage{amsmath}
\usepackage{amssymb}

\ifPDFTeX
\usepackage{newunicodechar}
  \makeatletter
    \def\doxynewunicodechar#1#2{%
    \@tempswafalse
    \edef\nuc@tempa{\detokenize{#1}}%
    \if\relax\nuc@tempa\relax
      \nuc@emptyargerr
    \else
      \edef\@tempb{\expandafter\@car\nuc@tempa\@nil}%
      \nuc@check
      \if@tempswa
        \@namedef{u8:\nuc@tempa}{#2}%
      \fi
    \fi
  }
  \makeatother
  \doxynewunicodechar{⁻}{${}^{-}$}% Superscript minus
  \doxynewunicodechar{²}{${}^{2}$}% Superscript two
  \doxynewunicodechar{³}{${}^{3}$}% Superscript three

\fi
\pagestyle{empty}
\begin{document}
\[  f_j= \sum_{k=-\frac{N}{2}}^{\frac{N}{2}-1} \hat{f}_{k}
{\rm e}^{-2\pi{\rm i}\frac{kj}{N}}
\]
\pagebreak

$j=-\frac{N}{2},\dots,\frac{N}{2}-1$
\pagebreak

$\hat{f}_{k}\in\mathbb{C}$
\pagebreak

${\cal O}(N^2)$
\pagebreak

${\cal O}(N\log N)$
\pagebreak

$\mathbf{k}\in\mathbb{Z}^d$
\pagebreak

\[I_{\mathbf{N}} := \left\{ \mathbf{k}=\left(k_t\right)_{t=0,\dots,d-1}
 \in \mathbb{Z}^d: -
  \frac{N_t}{2} \le k_t < \frac{N_t}{2} ,\;t=0,\dots,d-1\right\},
\]
\pagebreak

$\mathbf{N}=\left(N_t\right)_{t=0,\dots,d-1}$
\pagebreak

$N_t\in 2\mathbb{N}$
\pagebreak

$\hat f_{\mathbf{k}} \in \mathbb{C}$
\pagebreak

$\mathbf{k}\in I_{\mathbf{N}}$
\pagebreak

\[ f\left(\mathbf{x}\right)
 := \sum_{ \mathbf{k}\in I_{ N}} \hat{f}_{\mathbf{ k}}
 {\rm e}^{-2\pi{\rm i}\mathbf{k}\mathbf{ x}}
\]
\pagebreak

$\mathbf{x}_j \in \mathbb{T}^d$
\pagebreak

$j=0,\ldots, M-1$
\pagebreak

$ d$
\pagebreak

\[ \hat h_{\mathbf{k}} := \sum_{j=0}^{M-1} {f}_{j}
 {\rm e}^{2\pi{\rm i}\mathbf{k}\mathbf{ x}_j}.
\]
\pagebreak

$\alpha_n,\;\beta_n,\;\gamma_n,\;n=0,\dots,N$
\pagebreak

$P_n$
\pagebreak

$P_{-1}(x) = 0$
\pagebreak

$P_{0}(x) = 1$
\pagebreak

\[ P_n(x) = (\alpha_nx+\beta_n) P_{n-1}(x) + \gamma_n P_{n-2}(x)
 ,\qquad n=1,2,\dots
\]
\pagebreak

$x\in[-1,1]$
\pagebreak

\[ T_n(x) = \cos(n\, \arccos x).
\]
\pagebreak

$f\colon [-1,1]\to\mathbb R$
\pagebreak

$N\in\mathbb N$
\pagebreak

$[x_n]_{n=0..N}$
\pagebreak

\[ f = \sum_{n=0}^N x_n P_n
\]
\pagebreak

$[y_n]_{n=0..N}$
\pagebreak

\[ f = \sum_{n=0}^N y_n T_n.
\]
\pagebreak

$M \in \mathbb{N}_0$
\pagebreak

$m \in
       \mathbb{N}_0$
\pagebreak

$m = 0,\ldots,M$
\pagebreak

$M+1$
\pagebreak

$t \in \mathbb{N}, t \ge 2$
\pagebreak

$N = 2^t \in \mathbb{N}, N \ge 4$
\pagebreak

$m \in \mathbb{N}, 0 \le m \le M$
\pagebreak

$\alpha_k \in
     \mathbb{R}$
\pagebreak

$k=0,\ldots,N$
\pagebreak

$=\alpha_k$
\pagebreak

$\beta_k \in \mathbb{R}$
\pagebreak

$=\beta_k$
\pagebreak

$\gamma_k \in
           \mathbb{R}$
\pagebreak

$=\gamma_k$
\pagebreak

$k_{\text{start}} \in \mathbb{N}_0,
             0 \le k_{\text{start}} \le N$
\pagebreak

$\kappa \in \mathbb{R}, \kappa > 0$
\pagebreak

$j=0,...,M_{total}-1$
\pagebreak

\[  f(x_j) = \sum_{k \in I_N^2} \hat{f}(k) {\rm e}^{\mbox{\rm\scriptsize i} t_j \omega(k)}
                     {\rm e}^{-2 \pi \mbox{\rm\scriptsize i} k x_j}
\]
\pagebreak

$N$
\pagebreak

$x$
\pagebreak

$k \in I_N^2$
\pagebreak

\[  \hat{f}(k) = \sum_{j=0}^{M_{total}-1} f(x_j) {\rm e}^{\mbox{\rm\scriptsize i} t_j \omega(k)}
                     {\rm e}^{-2 \pi \mbox{\rm\scriptsize i} k x_j}
\]
\pagebreak

$j=0,...,M\_total-1$
\pagebreak

$f_j^C(x_j) = \sum_{k \in I_0^{N,d}} \hat{f}_k^C * cos(2 \pi k x_j)$
\pagebreak

$k \in I_0^{N,d}$
\pagebreak

$h^C(k) = \sum_{j \in I_0^{(M\_total,1)}} f_j^C * cos(2 \pi k x_j)$
\pagebreak

\[  \mathbb{T}^d
   := \left\{ \mathbf{x}=\left(x_t\right)_{t=0,\dots,d-1}\in\mathbb{R}^{d}:
   \; - \frac{1}{2} \le x_t < \frac{1}{2},\; t=0,\dots,d-1 \right\}
\]
\pagebreak

$d$
\pagebreak

$\mathbf{x}$
\pagebreak

${\cal X}:=\{\mathbf{x}_j \in {\mathbb T}^d:
\,j=0,\dots,M-1\}$
\pagebreak

$\mathbf{k}$
\pagebreak

\[  I_{\mathbf{N}} := \left\{ \mathbf{k}=\left(k_t\right)_{t=0,\dots,d-1}\in
  \mathbb{Z}^d: - \frac{N_t}{2} \le k_t < \frac{N_t}{2} ,\;t=0,\dots,d-1
\right\}.
\]
\pagebreak

\[f_j = \sum_{\mathbf{k}\in I_{\mathbf{N}}}
\hat{f}_{\mathbf{k}} {\rm e}^{-2\pi{\rm i} \mathbf{k}\mathbf{x}_j}, \qquad
j=0,\dots,M-1.
\]
\pagebreak

\[  \hat f_{\mathbf{k}}=\sum_{j=0}^{M-1} f_j {\rm e}^{+2\pi{\rm i}
   \mathbf{k}\mathbf{x}_j}, \qquad \mathbf{k}\in I_{\mathbf{N}}.
\]
\pagebreak

${\cal O}(|I_{\mathbf{N}}|M)$
\pagebreak

${\cal O}(|I_{\mathbf{N}}|\log|I_{\mathbf{N}}|+M)$
\pagebreak

$\mathbf{D}$
\pagebreak

$\mathbf{B}$
\pagebreak

$(K+1)d$
\pagebreak

$K=2^{10}(m+1)$
\pagebreak

$2dM$
\pagebreak

$(2m+2)dM$
\pagebreak

$(2m+2)^dM$
\pagebreak

$\mathbb{R}^3$
\pagebreak

$(r,\vartheta,\varphi)^{\mathrm{T}}$
\pagebreak

$r \in \mathbb{R}^{+}$
\pagebreak

$\vartheta \in [0,\pi]$
\pagebreak

$\varphi \in [-\pi,\pi)$
\pagebreak

$\mathbb{S}^2$
\pagebreak

\[  \mathbb{S}^2 := \left\{\mathbf{x} \in \mathbb{R}^{3}:\;
  \|\mathbf{x}\|_2=1\right\}
\]
\pagebreak

$(\vartheta,\varphi)^{\mathrm{T}}$
\pagebreak

$x_1 :=
\frac{\varphi}{2\pi}$
\pagebreak

$x_2 := \frac{\vartheta}{2\pi}$
\pagebreak

$\mathbf{x} := \left(x_1,x_2\right) \in
 [-\frac{1}{2}, \frac{1}{2}) \times [0,\frac{1}{2}]$
\pagebreak

$P_k : [-1,1]
\rightarrow \mathbb{R}$, $k \in \mathbb{N}_{0}$
\pagebreak

\[  P_k(t) := \frac{1}{2^k k!} \frac{\text{d}^k}{\text{d} t^k}
  \left(t^2-1\right)^k.
\]
\pagebreak

\[  (k+1)P_{k+1}(t) = (2k+1) x P_{k}(t) - k P_{k-1}(t) \quad (k \in
  \mathbb{N}_0).
\]
\pagebreak

\[  \left< f,g \right>_{\text{L}^2\left([-1,1]\right)} :=
  \int_{-1}^{1} f(t) g(t) \text{d} t
\]
\pagebreak

$\text{L}^2\left([-1,1]\right)$
\pagebreak

\[  \left< P_k,P_l \right>_{\text{L}^2\left([-1,1]\right)} = \frac{2}{2k+1}
  \delta_{k,l}.
\]
\pagebreak

$ c_k := \sqrt{\frac{2k+1}{2}}$
\pagebreak

$c_k P_k$
\pagebreak

\[  \|f\|_{\text{L}^2\left([-1,1]\right)} :=
  \left(<f,f>_{\text{L}^2\left([-1,1]\right)}\right)^{1/2} =
  \left(\int_{-1}^{1} |f(t)|^2 \; \text{d} t\right)^{1/2}.
\]
\pagebreak

$P_k^n : [-1,1] \rightarrow
\mathbb{R} $
\pagebreak

$n \in \mathbb{N}_0$
\pagebreak

$k \ge n$
\pagebreak

\[  P_k^n(t) := \left(\frac{(k-n)!}{(k+n)!}\right)^{1/2}
  \left(1-t^2\right)^{n/2} \frac{\text{d}^n}{\text{d} t^n} P_k(t).
\]
\pagebreak

$n = 0$
\pagebreak

$P_k^0 = P_k$
\pagebreak

\[  P_{k+1}^n(t) = v_{k}^n t P_k^n(t) + w_{k}^n P_{k-1}^n(t) \quad (k \ge n),
\]
\pagebreak

$P_{n-1}^n(t) := 0$
\pagebreak

$P_{n}^n(t) := \frac{\sqrt{(2n)!}}{2^n n!}
\left(1-t^2\right)^{n/2}$
\pagebreak

\[  v_{k}^n := \frac{2k+1}{((k-n+1)(k+n+1))^{1/2}}\; ,\qquad
  w_{k}^n := - \frac{((k-n)(k+n))^{1/2}}{((k-n+1)(k+n+1))^{1/2}}.
\]
\pagebreak

$n$
\pagebreak

$\left\{P_k^n:\: k
\ge n\right\}$
\pagebreak

\[  \left< P_k^n,P_l^n \right>_{\text{L}^2\left([-1,1]\right)} = \frac{2}{2k+1}
  \delta_{k,l} \quad (0 \le n \le k,l).
\]
\pagebreak

$ c_k = \sqrt{\frac{2k+1}{2}}$
\pagebreak

$c_k P_k^n$
\pagebreak

$\text{L}^2
\left(\mathbb{S}^2\right)$
\pagebreak

$\tilde{Y}_k^n : \mathbb{S}^2 \rightarrow \mathbb{C}$
\pagebreak

\[  \tilde{Y}_k^n(\vartheta,\varphi) := P_k^{|n|}(\cos\vartheta)
  \mathrm{e}^{\mathrm{i} n \varphi}
\]
\pagebreak

$\text{L}^2\left(\mathbb{S}^2\right)$
\pagebreak

\[  \left< f,g \right>_{\mathrm{L}^2\left(\mathbb{S}^2\right)} :=
  \int_{\mathbb{S}^2} f(\vartheta,\varphi) \overline{g(\vartheta,\varphi)}
  \: \mathrm{d} \mathbf{\xi} := \int_{-\pi}^{\pi} \int_{0}^{\pi}
  f(\vartheta,\varphi) \overline{g(\vartheta,\varphi)} \sin \vartheta
  \; \mathrm{d} \vartheta \; \mathrm{d} \varphi.
\]
\pagebreak

$c_k^n := \sqrt{\frac{2k+1}{4\pi}}$
\pagebreak

\[  Y_k^n(\vartheta,\varphi) := c_k^n P_k^{|n|}(\cos\vartheta)
  \mathrm{e}^{\mathrm{i} n \varphi}
\]
\pagebreak

$\text{L}^2\left(\mathbb{S}^2
\right)$
\pagebreak

\[  \|f\|_{\text{L}^2\left(\mathbb{S}^2\right)} =
  \left(<f,f>_{\text{L}^2\left(\mathbb{S}^2\right)}\right)^{1/2} =
  \left(\int_{-\pi}^{\pi} \int_{0}^{\pi} |f(\vartheta,\varphi)|^2 \sin
  \vartheta \; \mathrm{d} \vartheta \; \mathrm{d} \varphi\right)^{1/2}.
\]
\pagebreak

$f \in \mathrm{L}^2\left(\mathbb{S}^2\right)$
\pagebreak

\[  f = \sum_{k=0}^{\infty} \sum_{n=-k}^{k} \hat{f}(k,n) Y_k^n,
\]
\pagebreak

$\hat{f}(k,n) := \left< f, Y_k^{n}
\right>_{\mathrm{L}^2\left(\mathbb{S}^2\right)}$
\pagebreak

$\mathrm{L}^2$
\pagebreak

\[    \begin{array}{rcl}
      \text{\textbf{Input}} & : & \text{coefficients }
        \hat{f}(k,n) \in \mathbb{C} \text{ for } k=0,\ldots,N,\;n=-k,
        \ldots,k,\; N \in \mathbb{N}_0,\\[1ex]
                            &   & \text{arbitrary nodes } \mathbf{x}(m) \in
        [-\frac{1}{2},\frac{1}{2}] \times [0,\frac{1}{2}]
        \text{ for } m=0,\ldots,M-1, M \in \mathbb{N}. \\[1ex]
      \text{\textbf{Task}}  & : & \text{evaluate } f(m) := f\left(
      \mathbf{x}(m)\right) = \sum_{k=0}^N \sum_{n=-k}^k \hat{f}_k^n
        Y_k^n\left(\mathbf{x}(m)\right) \text{ for } m=0,\ldots,M-1.
        \\[1ex]
      \text{\textbf{Output}} & : & \text{coefficients } f(m) \in
        \mathbb{C} \text{ for } m=0,\ldots,M-1.\\
    \end{array}
\]
\pagebreak

\[    \begin{array}{rcl}
      \text{\textbf{Input}} & : & \text{coefficients } f(m) \in
        \mathbb{C} \text{ for } m=0,\ldots,M-1, M \in \mathbb{N},\\
                            &   & \text{arbitrary nodes } \mathbf{x}(m) \in
        [-\frac{1}{2},\frac{1}{2}] \times [0,\frac{1}{2}] \text{ for }
        m=0,\ldots,M-1, N \in \mathbb{N}_0.\\[1ex]
      \text{\textbf{Task}}  & : & \text{evaluate } \hat{f}(k,n)
        := \sum_{m=0}^{M-1} f(m) \overline{Y_k^n\left(\mathbf{x}(m)\right)}cd Do
        \text{ for } k=0,\ldots,N,\;n=-k,\ldots,k.\\[1ex]
      \text{\textbf{Output}} & : & \text{coefficients }
        \hat{f}(k,n) \in \mathbb{C} \text{ for }
        k=0,\ldots,N,\;n=-k,\ldots,k.\\[1ex]
    \end{array}
\]
\pagebreak

$N \in \mathbb{N}_0$
\pagebreak

$ = (2N+2)^2$
\pagebreak

$M$
\pagebreak

$(2N+2)^2$
\pagebreak

$i \in
       \mathbb{N}_0$
\pagebreak

$i$
\pagebreak

$i=0,1,\ldots,(2N+2)^2-1$
\pagebreak

$\hat{f}(k,n)$
\pagebreak

$0 \le k \le M$
\pagebreak

$-k \le n \le k$
\pagebreak

$i = (N+2)(N-n+1)+N+k+1$
\pagebreak

$k,n$
\pagebreak

$f(m)$
\pagebreak

$m=0,\ldots,M-1$
\pagebreak

$m$
\pagebreak

$\mathbf{x}(m) \in
       [-\frac{1}{2},\frac{1}{2}] \times [0,\frac{1}{2}]$
\pagebreak

$m = 0,
       \ldots,M-1$
\pagebreak

$2m$
\pagebreak

$x_1$
\pagebreak

$2m+1$
\pagebreak

$x_2$
\pagebreak

$N_{\text{max}}$
\pagebreak

$M \in \mathbb{N}$
\pagebreak

$N \in \mathbb{N}_2$
\pagebreak

$\kappa \in
\mathbb{R}^{+}$
\pagebreak

$\kappa \in \mathbb{R}^{+}$
\pagebreak

$m = 0,\ldots,M-1$
\pagebreak

\[  f(m) = \sum_{k=0}^N \sum_{n=-k}^k \hat{f}(k,n) Y_k^n\left(2\pi x_1(m),
  2\pi x_2(m)\right).
\]
\pagebreak

$k=0,\ldots,N;
n=-k,\ldots,k$
\pagebreak

\[  \hat{f}(k,n) = \sum_{m = 0}^{M-1} f(m) Y_k^n\left(2\pi x_1(m),
  2\pi x_2(m)\right).
\]
\pagebreak

\[  \tilde{Y}_k^n(\vartheta,\varphi) = P_k^{|n|}(\cos\vartheta)
  \mathrm{e}^{\mathrm{i} n \varphi}.
\]
\pagebreak

$L_2$
\pagebreak

\[  Y_k^n(\vartheta,\varphi) = \sqrt{\frac{2k+1}{4\pi}} P_k^{|n|}(\cos\vartheta)
  \mathrm{e}^{\mathrm{i} n \varphi}.
\]
\pagebreak

\[ \varphi_i = 2\pi \frac{i}{2N+2}, \qquad i=-N-1,\dots,N, \]
\pagebreak

\[ \vartheta_j = 2\pi \frac{j}{2N+2}, \qquad j=0,\dots,N+1. \]
\pagebreak

$\mathtt{f\_hat}(k,n)$
\pagebreak

$0 \le k \le N$
\pagebreak

\[  (N+2)(N-n+1)+N+k+1
\]
\pagebreak

$= (2N+2)\,\sigma$
\pagebreak

$\sigma\ge1$
\pagebreak

\[  f(g_m) = \sum_{l=0}^B \sum_{m=-l}^l \sum_{n=-l}^l \hat{f}^{mn}_l
           D_l^{mn}\left( \alpha_m,\beta_m,\gamma_m\right).
\]
\pagebreak

$l=0,\ldots,B;
m,n=-l,\ldots,l$
\pagebreak

\[  \hat{f}^{mn}_l = \sum_{m = 0}^{M-1} f(g_m)
                   D_l^{mn}\left( \alpha_m,\beta_m,\gamma_m\right)
\]
\pagebreak

\[  D_{mn}^l(\alpha,\beta,\gamma) = d^{mn}_{l}(\cos\beta)
  \mathrm{e}^{-\mathrm{i} m \alpha}\mathrm{e}^{-\mathrm{i} n \gamma}.
\]
\pagebreak

\[ \tilde D_{mn}^l(\alpha,\beta,\gamma) = \sqrt{\frac{2l+1}{8\pi^2}}d^{mn}_{l}(\cos\beta)
  \mathrm{e}^{-\mathrm{i} m \alpha}\mathrm{e}^{-\mathrm{i} n \gamma}
\]
\pagebreak

$\alpha, \beta, \gamma$
\pagebreak

\[ \sum_{m=-l}^l D_{mn}^l(\alpha,\beta,\gamma) Y_m^l(x) = Y_n^l(A^{-1} x)
\]
\pagebreak

$\hat f^{mn}_l$
\pagebreak

$l=0,...,B$
\pagebreak

$m,n =-l,...,l$
\pagebreak

$f_j^S(x_j) = \sum_{k \in I_1^{N,d}} \hat{f}_k^S * sin(2 \pi k x_j)$
\pagebreak

$k \in I_1^{N,d}$
\pagebreak

$h^S(k) = \sum_{j \in I_0^{M\_total,1}} f_j^S * cos(2 \pi k x_j)$
\pagebreak

$v$
\pagebreak

\[  f(x_j) = \sum_{k = 0}^{N_{total}-1} \hat{f}(v_k) {\rm e}^{-2 \pi
           \mbox{\rm\scriptsize i} v_k x_j \odot N}
\]
\pagebreak

$k=0,...,N_{total}-1$
\pagebreak

\[  \hat{f}(v_k) = \sum_{j = 0}^{M_{total}-1} f(x_j) {\rm e}^{2 \pi
                 \mbox{\rm\scriptsize i} v_k x_j \odot N}
\]
\pagebreak

\[  \hat{f}(v_k) = \sum_{j = 0}^{M_{tota}l-1} f(x_j) {\rm e}^{2 \pi
                 \mbox{\rm\scriptsize i} v_k x_j \odot N}
\]
\pagebreak

$j=0,\dots,M-1$
\pagebreak

\[  f_j = \sum_{k\in H_N^d}\hat f_k {\rm e}^{-2\pi{\rm\scriptsize i}k x_j}
\]
\pagebreak

$k\in H_N^d$
\pagebreak

\[  \hat f_k = \sum_{j=0,\dots,M-1} f_j {\rm e}^{+2\pi{\rm\scriptsize i}k x_j}
\]
\pagebreak

$x^H x$
\pagebreak

$x^H (w \odot x)$
\pagebreak

$x^H (w\odot w2\odot w2 \odot x)$
\pagebreak

$x^H (w2\odot w2 \odot x)$
\pagebreak

$x \leftarrow y$
\pagebreak

$x \leftarrow a y$
\pagebreak

$x \leftarrow w\odot y$
\pagebreak

$x \leftarrow a x + y$
\pagebreak

$x \leftarrow x + a y$
\pagebreak

$x \leftarrow a x + b y$
\pagebreak

$x \leftarrow x + a w\odot y$
\pagebreak

$x \leftarrow a x +  w\odot y$
\pagebreak

$[0,1]\times[0,1]{\rm i}$
\pagebreak

$[-1/2,1/2]$
\pagebreak

$\mathbf{x}(m) = \left(x_1,x_2\right) \in     [-\frac{1}{2},\frac{1}{2}) \times [0,\frac{1}{2}]$
\pagebreak

$m=0,\ldots,     M-1$
\pagebreak

$M \in \mathbb{N},$
\pagebreak

$P_k : \mathbb{R} \rightarrow
\mathbb{R}$
\pagebreak

\[  f(x) = \sum_{k=0}^N a_k P_k(x) \quad (N \in \mathbb{N}_0)
\]
\pagebreak

\[  P_{k+1}(x) = (alpha_k * x + beta_k)*P_{k}(x) + gamma_k P_{k-1}(x) \quad
  (alpha_k, beta_k, gamma_k \in \mathbb{R},\; k \ge 0)
\]
\pagebreak

$P_{-1}(x) := 0$
\pagebreak

$P_0(x) := \lambda$
\pagebreak

$\left(a_k\right)_{k=0}^N \in
\mathbb{C}^{N+1}$
\pagebreak

$\left(x_j\right)_{j=0}^M \in
\mathbb{R}^{M+1}$
\pagebreak

$\alpha_{k-1}^n$
\pagebreak

$k,n = 0,1,\ldots,N$
\pagebreak

$(N+1)^2$
\pagebreak

$\beta_{k-1}^n$
\pagebreak

$\gamma_{k-1}^n$
\pagebreak

$P_k^n(x,c)$
\pagebreak

$k$
\pagebreak

$\alpha_c^n,\ldots,\alpha_{c+k}^n$
\pagebreak

$\beta_c^n,\ldots,\beta_{c+k}^n$
\pagebreak

$\gamma_c^n,\ldots,\gamma_{c+k}^n$
\pagebreak

$\left(b_k^n\right)_{k=0}^M$
\pagebreak

$-M \le n \le M$
\pagebreak

\[  f(\cos\vartheta) = \sum_{k=0}^{2\lfloor\frac{M}{2}\rfloor}
  a_k (\sin\vartheta)^{n\;\mathrm{mod}\;2} T_k(\cos\vartheta)
\]
\pagebreak

$\left(c_k^n\right)_{k=0}^M$
\pagebreak

\[  f(\cos\vartheta) = \sum_{k=-M}^{M} c_k \mathrm{e}^{\mathrm{i}k\vartheta}
\]
\pagebreak

$n=-M,\ldots,M$
\pagebreak

$\left(b_k^n\right)_{k=0,\ldots,M;n=-M,\ldots,M}$
\pagebreak

$\left(c_k^n\right)_{k=-M,\ldots,M;n=-M,\ldots,M}$
\pagebreak

$\alpha_l^{km}$
\pagebreak

$l$
\pagebreak

$\beta_l^{km}$
\pagebreak

$\gamma_l^{km}$
\pagebreak

$ \alpha_{l}^{km}$
\pagebreak

$ l= 0,\ldots,N $
\pagebreak

$(2N+1)^2(N+1)$
\pagebreak

$ \beta_{l}^{km}$
\pagebreak

$ \gamma_{l}^{km}$
\pagebreak

$ m = -N,\ldots,N $
\pagebreak

$\alpha_{l}^{km}$
\pagebreak

$ k,m = -N,\ldots,N $
\pagebreak

$\beta_{l}^{km}$
\pagebreak

$\gamma_{l}^{km}$
\pagebreak

$d_l^{km}(x,c)$
\pagebreak

$\alpha_c^{km},\ldots,\alpha_{c+l}^{km}$
\pagebreak

$\beta_c^{km},\ldots,\beta_{c+l}^{km}$
\pagebreak

$\gamma_c^{km},\ldots,\gamma_{c+l}^{km}$
\pagebreak

$ d^{km}_l(cos(theta)) $
\pagebreak

$B, B^{\rm T}$
\pagebreak

$\frac{\|x-y\|_{\infty}}{\|x\|_{\infty}} $
\pagebreak

$\frac{\|x-y\|_{\infty}}{\|z\|_1} $
\pagebreak

$\frac{\|x-y\|_2}{\|x\|_2} $
\pagebreak

\[ f(g_q)=\sum^{N}_{l=0}\sum_{m,n=-l}^l \hat f^{mn}_l D_{mn}^l(\alpha_q,\beta_q,\gamma_q)
\]
\pagebreak

\[  f(y_j) = \sum_{k=1}^N \alpha_k K(x_k-y_j),\quad   j=1\dots M.
\]
\pagebreak

$I_{n+\alpha}(x)$
\pagebreak

$\mathrm{e}^{-x}$
\pagebreak

$x,alpha$
\pagebreak

$0 \le \alpha < 1$
\pagebreak

$n=0,1,\ldots,nb-1$
\pagebreak

$|x| \ge exparg$
\pagebreak

$\min(nb,0)-1$
\pagebreak

$n=0,1,\ldots,ncalc$
\pagebreak

$J_{\nu}(x)$
\pagebreak

$I_{\nu}(x)$
\pagebreak

$\nu$
\pagebreak

$\varphi_1 \in [-\pi,\pi)$
\pagebreak

$\vartheta_1 \in [0,\pi]$
\pagebreak

$\varphi_2 \in [-\pi,\pi)$
\pagebreak

$\vartheta_2 \in [0,\pi]$
\pagebreak

$\cos \vartheta_1 \cos \vartheta_2 +
        \sin \vartheta_1 \sin(\vartheta_2 \cos(\varphi_1 - \varphi_2)$
\pagebreak

$Q_h: [-1,1] \rightarrow \mathbb{R}$
\pagebreak

$x \in [-1,1]$
\pagebreak

$h \in (0,1)$
\pagebreak

$Q_h(x)$
\pagebreak

$S_h: [-1,1] \rightarrow \mathbb{R}$
\pagebreak

$S_h(x)$
\pagebreak

$L_{h,\lambda}: [-1,1] \rightarrow
\mathbb{R}$
\pagebreak

$\lambda \in \mathbb{N}_0$
\pagebreak

$L_{h,\lambda}(x)$
\pagebreak

$G_\sigma: [-1,1] \rightarrow
\mathbb{R}$
\pagebreak

$\sigma \in \mathbb{R}_+$
\pagebreak

$G_\sigma(x)$
\pagebreak

\end{document}
