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Difference between revisions of "Partial Fourier sum"

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 +
$$
 +
\newcommand{\vb}[1]{\mathbf{#1}}
 +
$$
 
A partial sum of the [[Fourier series|Fourier series]] of a given [[Function|function]].
 
A partial sum of the [[Fourier series|Fourier series]] of a given [[Function|function]].
  
Line 9: Line 12:
 
$$S_n(f ; x) = \frac{a_0}{2} + \sum_{k=1}^{n} (a_k \cos(k x) + b_k \sin(kx))$$
 
$$S_n(f ; x) = \frac{a_0}{2} + \sum_{k=1}^{n} (a_k \cos(k x) + b_k \sin(kx))$$
  
With the use of the sequence of partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p1200708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p1200709.png" />, the notion of convergence of the series $S[f]$ is introduced and its sum at a point $x$ is defined as follows:
+
With the use of the sequence of partial sums $S_n(f; x)$, $n=1, 2, \ldots$, the notion of convergence of the series $S[f]$ is introduced and its sum at a point $x$ is defined as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007012.png" /></td> </tr></table>
+
$$
 +
S(x) = \lim_{n\to\infty} S_n(f; x).
 +
$$
  
 
At every point $x$, the Dirichlet formula
 
At every point $x$, the Dirichlet formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007014.png" /></td> </tr></table>
+
$$
 +
S_n(f; x) = \frac1\pi \int_{-\pi}^\pi f(x+t) D_n() \, dt, \qquad n=0,1,\ldots,
 +
$$
  
 
is true; here,
 
is true; here,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007015.png" /></td> </tr></table>
+
$$
 +
D_n(t) = \frac12 + \sum_{k=1}^n \cos kt = \frac{\sin(n+\frac12)t}{2\sin \frac t2}
 +
$$
  
is the [[Dirichlet kernel|Dirichlet kernel]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007016.png" />. This formula plays a key role in many problems in the theory of [[Fourier series|Fourier series]].
+
is the [[Dirichlet kernel|Dirichlet kernel]] of order $n$. This formula plays a key role in many problems in the theory of [[Fourier series|Fourier series]].
  
If a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007017.png" /> is given in complex form, i.e., if
+
If a series $S[f]$ is given in complex form, i.e., if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007018.png" /></td> </tr></table>
+
$$
 +
\begin{gathered}
 +
S[f] = \sum_{k\in\Z} c_k e^{ikx}, \\
 +
c_k = \frac{1}{2\pi} \int_{-\pi}^\pi f(t) e^{-ikt} \, dt,
 +
\end{gathered}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007019.png" /></td> </tr></table>
+
where $\Z$ is the set of all integers, then
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007020.png" /> is the set of all integers, then
+
$$
 
+
S_n(f; x) = \sum_{|k| \le n} c_k e^{ikx}.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007021.png" /></td> </tr></table>
+
$$
  
 
In the multi-dimensional case, a notion of partial sum can be introduced in numerous different ways, none of which can be regarded as preferable.
 
In the multi-dimensional case, a notion of partial sum can be introduced in numerous different ways, none of which can be regarded as preferable.
  
One of the possible general approaches is as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007022.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007023.png" />-dimensional Euclidean space of points (vectors) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007024.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007025.png" /> be the integer lattice in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007026.png" />, i.e., the set of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007027.png" /> with integer coordinates. For vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007028.png" />, let
+
One of the possible general approaches is as follows: Let $\R^N$ be the $N$-dimensional Euclidean space of points (vectors) $\vb{x} = (x_1, \ldots, x_N)$, and let $\Z^N$ be the integer lattice in $\R^N$, i.e., the set of vectors $\vb{n} = (n_1, \ldots, n_N)$ with integer coordinates. For vectors $\vb{x}, \vb{y} \in \R^N$, let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007029.png" /></td> </tr></table>
+
$$
 +
(\vb{x}, \vb{y}) = x_1y_1 + \cdots + x_N y_N, \qquad |\vb{x}| = \sqrt{(\vb{x}, \vb{x})}.
 +
$$
  
 
Further, let
 
Further, let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007030.png" /></td> </tr></table>
+
$$
 +
Q_N = \{ \vb{x} \in \R^N : -\pi \le x_k \le \pi, k = 1, \ldots, N\},
 +
$$
  
let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007031.png" /> be a function that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007032.png" />-periodic in each variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007033.png" /> and integrable over a cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007034.png" />, and let
+
let $f(\vb{x}) = f(x_1, \ldots, x_N)$ be a function that is $2\pi$-periodic in each variable $x_k$ and integrable over a cube $Q_N$, and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007035.png" /></td> </tr></table>
+
$$
 
+
\begin{gathered}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007036.png" /></td> </tr></table>
+
S[f] = \sum_{\vb{k}\in\Z^N} c_{\vb{k}} e^{i (\vb{k},\vb{x})}, \\
 +
c_{\vb{k}} = \frac{1}{(2\pi)^N} \int_{Q_N} f(\vb{t}) e^{-i (\vb{k},\vb{t})} \, d\vb{t},
 +
\end{gathered}
 +
$$
  
 
be its Fourier series.
 
be its Fourier series.
  
Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007037.png" /> be a family of bounded domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007038.png" /> that depend on a real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007039.png" /> and are such that any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007040.png" /> belongs to all domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007041.png" /> for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007042.png" />. In this case, the expression
+
Further, let $\{G_\alpha\}$ be a family of bounded domains in $\R^N$ that depend on a real parameter $\alpha$ and are such that any vector $\vb{n} \in \R^N$ belongs to all domains $G_\alpha$ for sufficiently large $\alpha$. In this case, the expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007043.png" /></td> </tr></table>
+
$$
 +
S_\alpha(\vb{x}) = S_{G_\alpha} (f; \vb{x}) = \sum_{\vb{k} \in G_\alpha \cap \Z^N} e^{i(\vb{k},\vb{x})}
 +
$$
  
is called a partial Fourier sum of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007044.png" /> corresponding to the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007045.png" />, and the expression
+
is called a partial Fourier sum of the function $f$ corresponding to the domain $G_\alpha$, and the expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007046.png" /></td> </tr></table>
+
$$
 +
D_{G_\alpha}(\vb{t}) = \frac{1}{2^{N}} \sum_{\vb{k} \in G_\alpha \cap \Z^N} e^{-i(\vb{k}, \vb{t})}
 +
$$
  
is called the Dirichlet kernel corresponding to this domain. It is clear that, for any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007047.png" />, the following formula holds:
+
is called the Dirichlet kernel corresponding to this domain. It is clear that, for any vector $\vb{x} \in \R^N$, the following formula holds:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007048.png" /></td> </tr></table>
+
$$
 +
S_{G_{\alpha}}(f; \vb{x}) = \frac{1}{\pi^N} \int_{Q_N} f(\vb{x} + \vb{t}) D_{G_\alpha} (\vb{t}) \, d\vb{t}.
 +
$$
  
This definition allows one to consider the problem of the convergence (or summability) of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007049.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007050.png" />. By virtue of the boundedness of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007051.png" /> the expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007052.png" /> is always a [[Trigonometric polynomial|trigonometric polynomial]].
+
This definition allows one to consider the problem of the convergence (or summability) of the series $S[f]$ as $\alpha\to\infty$. By virtue of the boundedness of the domains $G_\alpha$ the expression for $S_\alpha(\vb{x})$ is always a [[Trigonometric polynomial|trigonometric polynomial]].
  
The cases where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007053.png" />-dimensional spheres or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007054.png" />-dimensional intervals centred at the origin are taken as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007055.png" /> are most often encountered and are well studied. The expressions
+
The cases where $N$-dimensional spheres or $N$-dimensional intervals centred at the origin are taken as $G_\alpha$ are most often encountered and are well studied. The expressions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007056.png" /></td> </tr></table>
+
$$
 +
\sum_{|\vb{k}| \le \alpha} c_{\vb{k}} e^{i (\vb{k}, \vb{x})}, \qquad \alpha > 1,
 +
$$
  
 
are called spherical partial sums, and the expressions
 
are called spherical partial sums, and the expressions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007057.png" /></td> </tr></table>
+
$$
 
+
\sum_{|k_j| \le n_j} c_{\vb{k}} e^{i (\vb{k}, \vb{x})},  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007058.png" /> is an arbitrary vector from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007059.png" /> with positive coordinates, are called rectangular partial sums. In recent years, in connection with problems in the approximation of functions from Sobolev spaces, partial Fourier sums constructed by  "hyperbolic crosses" , namely, expressions of the form
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007060.png" /></td> </tr></table>
+
where $\vb{n} = (n_1, \ldots, n_N)$ is an arbitrary vector from $\Z^N$ with positive coordinates, are called rectangular partial sums. In recent years, in connection with problems in the approximation of functions from Sobolev spaces, partial Fourier sums constructed by  "hyperbolic crosses" , namely, expressions of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120070/p12007061.png" /></td> </tr></table>
+
$$
 +
\begin{gathered}
 +
\sum_{\vb{k} \in \Gamma_{r,\alpha}} c_{\vb{k}} e^{(\vb{k}, \vb{x})},  \\
 +
\Gamma_{r,\alpha} = \left\{ \vb{k} \in \Z^N : \prod_{i=1}^N |k_i|^{r_i} < \alpha, r_i > 0\right\},
 +
\end{gathered}
 +
$$
  
 
have been extensively used. For Fourier series in general orthonormal systems of functions, partial Fourier series are constructed analogously. (Cf. also [[Orthonormal system|Orthonormal system]].)
 
have been extensively used. For Fourier series in general orthonormal systems of functions, partial Fourier series are constructed analogously. (Cf. also [[Orthonormal system|Orthonormal system]].)

Latest revision as of 11:37, 13 February 2024

$$ \newcommand{\vb}[1]{\mathbf{#1}} $$ A partial sum of the Fourier series of a given function.

In the classical one-dimensional case where a function $f$ is integrable on the segment $[-\pi,\pi]$ and

$$S[f] = \frac{a_0}{2} + \sum_{k=1}^{\infty} (a_k \cos(k x) + b_k \sin(kx))$$

is its trigonometric Fourier series, the partial Fourier sum $S_n(f ; x)$ of order $n$ of $f$ is the trigonometric polynomial

$$S_n(f ; x) = \frac{a_0}{2} + \sum_{k=1}^{n} (a_k \cos(k x) + b_k \sin(kx))$$

With the use of the sequence of partial sums $S_n(f; x)$, $n=1, 2, \ldots$, the notion of convergence of the series $S[f]$ is introduced and its sum at a point $x$ is defined as follows:

$$ S(x) = \lim_{n\to\infty} S_n(f; x). $$

At every point $x$, the Dirichlet formula

$$ S_n(f; x) = \frac1\pi \int_{-\pi}^\pi f(x+t) D_n() \, dt, \qquad n=0,1,\ldots, $$

is true; here,

$$ D_n(t) = \frac12 + \sum_{k=1}^n \cos kt = \frac{\sin(n+\frac12)t}{2\sin \frac t2} $$

is the Dirichlet kernel of order $n$. This formula plays a key role in many problems in the theory of Fourier series.

If a series $S[f]$ is given in complex form, i.e., if

$$ \begin{gathered} S[f] = \sum_{k\in\Z} c_k e^{ikx}, \\ c_k = \frac{1}{2\pi} \int_{-\pi}^\pi f(t) e^{-ikt} \, dt, \end{gathered} $$

where $\Z$ is the set of all integers, then

$$ S_n(f; x) = \sum_{|k| \le n} c_k e^{ikx}. $$

In the multi-dimensional case, a notion of partial sum can be introduced in numerous different ways, none of which can be regarded as preferable.

One of the possible general approaches is as follows: Let $\R^N$ be the $N$-dimensional Euclidean space of points (vectors) $\vb{x} = (x_1, \ldots, x_N)$, and let $\Z^N$ be the integer lattice in $\R^N$, i.e., the set of vectors $\vb{n} = (n_1, \ldots, n_N)$ with integer coordinates. For vectors $\vb{x}, \vb{y} \in \R^N$, let

$$ (\vb{x}, \vb{y}) = x_1y_1 + \cdots + x_N y_N, \qquad |\vb{x}| = \sqrt{(\vb{x}, \vb{x})}. $$

Further, let

$$ Q_N = \{ \vb{x} \in \R^N : -\pi \le x_k \le \pi, k = 1, \ldots, N\}, $$

let $f(\vb{x}) = f(x_1, \ldots, x_N)$ be a function that is $2\pi$-periodic in each variable $x_k$ and integrable over a cube $Q_N$, and let

$$ \begin{gathered} S[f] = \sum_{\vb{k}\in\Z^N} c_{\vb{k}} e^{i (\vb{k},\vb{x})}, \\ c_{\vb{k}} = \frac{1}{(2\pi)^N} \int_{Q_N} f(\vb{t}) e^{-i (\vb{k},\vb{t})} \, d\vb{t}, \end{gathered} $$

be its Fourier series.

Further, let $\{G_\alpha\}$ be a family of bounded domains in $\R^N$ that depend on a real parameter $\alpha$ and are such that any vector $\vb{n} \in \R^N$ belongs to all domains $G_\alpha$ for sufficiently large $\alpha$. In this case, the expression

$$ S_\alpha(\vb{x}) = S_{G_\alpha} (f; \vb{x}) = \sum_{\vb{k} \in G_\alpha \cap \Z^N} e^{i(\vb{k},\vb{x})} $$

is called a partial Fourier sum of the function $f$ corresponding to the domain $G_\alpha$, and the expression

$$ D_{G_\alpha}(\vb{t}) = \frac{1}{2^{N}} \sum_{\vb{k} \in G_\alpha \cap \Z^N} e^{-i(\vb{k}, \vb{t})} $$

is called the Dirichlet kernel corresponding to this domain. It is clear that, for any vector $\vb{x} \in \R^N$, the following formula holds:

$$ S_{G_{\alpha}}(f; \vb{x}) = \frac{1}{\pi^N} \int_{Q_N} f(\vb{x} + \vb{t}) D_{G_\alpha} (\vb{t}) \, d\vb{t}. $$

This definition allows one to consider the problem of the convergence (or summability) of the series $S[f]$ as $\alpha\to\infty$. By virtue of the boundedness of the domains $G_\alpha$ the expression for $S_\alpha(\vb{x})$ is always a trigonometric polynomial.

The cases where $N$-dimensional spheres or $N$-dimensional intervals centred at the origin are taken as $G_\alpha$ are most often encountered and are well studied. The expressions

$$ \sum_{|\vb{k}| \le \alpha} c_{\vb{k}} e^{i (\vb{k}, \vb{x})}, \qquad \alpha > 1, $$

are called spherical partial sums, and the expressions

$$ \sum_{|k_j| \le n_j} c_{\vb{k}} e^{i (\vb{k}, \vb{x})}, $$

where $\vb{n} = (n_1, \ldots, n_N)$ is an arbitrary vector from $\Z^N$ with positive coordinates, are called rectangular partial sums. In recent years, in connection with problems in the approximation of functions from Sobolev spaces, partial Fourier sums constructed by "hyperbolic crosses" , namely, expressions of the form

$$ \begin{gathered} \sum_{\vb{k} \in \Gamma_{r,\alpha}} c_{\vb{k}} e^{(\vb{k}, \vb{x})}, \\ \Gamma_{r,\alpha} = \left\{ \vb{k} \in \Z^N : \prod_{i=1}^N |k_i|^{r_i} < \alpha, r_i > 0\right\}, \end{gathered} $$

have been extensively used. For Fourier series in general orthonormal systems of functions, partial Fourier series are constructed analogously. (Cf. also Orthonormal system.)

Various properties of partial Fourier sums and their applications to the theory of approximation and other fields of science can be found in, e.g., [a1], [a3], [a4], [a5], [a7]. [a6], [a2],

References

[a1] N. Bary, "Treatise on trigonometric series" , 1; 2 , Pergamon (1964) Zbl 0129.28002
[a2] A. Zygmund, "Trigonometrical series" , 1; 2 , Cambridge Univ. Press (1959)
[a3] R. Edwards, "Fourier series: A modern introduction" , 1; 2 , Springer (1979)
[a4] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1; 2 , Springer (1963/70)
[a5] W. Rudin, "Fourier analysis on groups" , Interscience (1962)
[a6] G. Szegő, "Orthogonal polynomials" , Amer. Math. Soc. (1959)
[a7] A. Stepanets, "Classification and approximation of periodic functions" , Kluwer Acad. Publ. (1995)
How to Cite This Entry:
Partial Fourier sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_Fourier_sum&oldid=53046
This article was adapted from an original article by Alexander Stepanets (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article