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''of Fourier series with summable majorant of coefficients''
 
''of Fourier series with summable majorant of coefficients''
  
 
An algebra closely related to the Wiener algebra
 
An algebra closely related to the Wiener algebra
  
<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/b/b130/b130120/b1301201.png" /></td> </tr></table>
+
\begin{equation*} \mathcal{A} = \{ f : \| f \| _ { \mathcal{A} } = \sum _ { m = - \infty } ^ { \infty } | \hat { f } ( m ) | < \infty \}, \end{equation*}
  
 
where
 
where
  
<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/b/b130/b130120/b1301202.png" /></td> </tr></table>
+
\begin{equation*} \hat { f } ( m ) = ( 2 \pi ) ^ { - 1 } \int _ { - \pi } ^ { \pi } f ( u ) e ^ { - i m u } d u \end{equation*}
  
is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b1301203.png" />th Fourier coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b1301204.png" /> (cf. also [[Fourier coefficients|Fourier coefficients]]). The Beurling algebra is defined as
+
is the $m$th Fourier coefficient of $f$ (cf. also [[Fourier coefficients|Fourier coefficients]]). The Beurling algebra is defined as
  
<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/b/b130/b130120/b1301205.png" /></td> </tr></table>
+
\begin{equation*} \mathcal{A} ^ { * } = \left\{ f : \|\, f \| _ { \mathcal{A}^ { * } }  = \sum _ { k = 0 } ^ { \infty } \operatorname { sup } _ { k \leq |m | < \infty } |\, \hat { f } ( m ) | < \infty \right\}. \end{equation*}
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b1301206.png" /> was introduced by A. Beurling for establishing contraction properties of functions [[#References|[a2]]]: Let
+
The space $\mathcal{A} ^ { * }$ was introduced by A. Beurling for establishing contraction properties of functions [[#References|[a2]]]: 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/b/b130/b130120/b1301207.png" /></td> </tr></table>
+
\begin{equation*} f ( t ) = \sum _ { n = - \infty } ^ { \infty } a _ { n } e ^ { i n t } , a _ { 0 } = 0, \end{equation*}
  
be an absolutely convergent [[Fourier series|Fourier series]] such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b1301208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b1301209.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012010.png" /> is a non-increasing sequence of numbers with a finite sum. Then if
+
be an absolutely convergent [[Fourier series|Fourier series]] such that $| a _ { \pm n }  | \leq a _ { n } ^ { * }$, $n \geq 1$, where $\{ a _ { n } ^ { * } \}$ is a non-increasing sequence of numbers with a finite sum. Then 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/b/b130/b130120/b13012011.png" /></td> </tr></table>
+
\begin{equation*} g ( t ) \sim \sum _ { n = - \infty } ^ { \infty } b _ { n } e ^ { i n t } , b _ { 0 } = 0, \end{equation*}
  
is a contraction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012012.png" /> (that is, for any pair of arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012014.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012015.png" /> holds), then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012016.png" /> also converges absolutely, and
+
is a contraction of $f ( t )$ (that is, for any pair of arguments $t_{1}$, $t_2$ the inequality $| g ( t _ { 1 } ) - g ( t _ { 2 } ) | \leq | f ( t _ { 1 } ) - f ( t _ { 2 } ) |$ holds), then the Fourier series of $g ( t )$ also converges absolutely, and
  
<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/b/b130/b130120/b13012017.png" /></td> </tr></table>
+
\begin{equation*} \sum _ { n = - \infty } ^ { \infty } | b _ { n } | \leq 10 \sum _ { n = 1 } ^ { \infty } a _ { n } ^ { * }. \end{equation*}
  
 
A similar result was proved in [[#References|[a2]]] for the [[Fourier transform|Fourier transform]], whence integrability of the monotone majorant on the real line is considered. These two spaces coincide locally, hence have much in common.
 
A similar result was proved in [[#References|[a2]]] for the [[Fourier transform|Fourier transform]], whence integrability of the monotone majorant on the real line is considered. These two spaces coincide locally, hence have much in common.
  
Subsequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012018.png" /> appeared in some other papers either in explicit or implicit form. See [[#References|[a1]]] for a detailed survey of the history and properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012019.png" />.
+
Subsequently, $\mathcal{A} ^ { * }$ appeared in some other papers either in explicit or implicit form. See [[#References|[a1]]] for a detailed survey of the history and properties of $\mathcal{A} ^ { * }$.
  
It turned out that the consideration of summability of Fourier series by linear methods at Lebesgue points leads to the same space of functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012020.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012021.png" />-periodic integrable function with Fourier series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012022.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012023.png" /> be a continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012024.png" />, representable as follows:
+
It turned out that the consideration of summability of Fourier series by linear methods at Lebesgue points leads to the same space of functions. Let $f$ be a $2 \pi$-periodic integrable function with Fourier series $\sum _ { k } \hat { f } ( k ) e ^ { i k x }$. Let $\lambda$ be a continuous function on $\mathbf{R} = ( - \infty , \infty )$, representable 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/b/b130/b130120/b13012025.png" /></td> </tr></table>
+
\begin{equation*} \lambda ( x ) = \int _ {  \mathbf{R} } e ^ { - i x t } d \mu ( t ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012026.png" /> is a finite [[Borel measure|Borel measure]] satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012027.png" />. Consider the means
+
where $\mu$ is a finite [[Borel measure|Borel measure]] satisfying $\int _ { \mathbf{R} } d \mu ( t ) = 1$. Consider the means
  
<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/b/b130/b130120/b13012028.png" /></td> </tr></table>
+
\begin{equation*} ( f  *  d \mu ) _ { N } : = \operatorname { lim } _ { h \rightarrow 0 } \int _ { \mathbf{R} } f _ { h } \left( \frac { x - u } { N } \right) d \mu ( u ), \end{equation*}
  
 
where
 
where
  
<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/b/b130/b130120/b13012029.png" /></td> </tr></table>
+
\begin{equation*} f _ { h } ( x ) = h ^ { - 1 } \int _ {\bf R } \varphi \left( \frac { t } { h } \right) f ( x - t ) d t. \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012030.png" /> is infinitely differentiable, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012032.png" />, vanishing for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012033.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012034.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012035.png" /> sufficiently smooth one has
+
Here, $\varphi ( t )$ is infinitely differentiable, equal to $1$ for $|t | < 1$, vanishing for $| t | > 2$ and such that $\int _ { \mathbf R } \varphi ( t ) d t = 1$. For $f$ sufficiently smooth one has
  
<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/b/b130/b130120/b13012036.png" /></td> </tr></table>
+
\begin{equation*} ( f ^ { * } d \mu ) _ { N } ( x ) = \sum _ { k } \lambda \left( \frac { k } { N } \right) \hat { f } ( k ) e ^ { i k x }, \end{equation*}
  
and these are the linear means of the [[Fourier series|Fourier series]] generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012037.png" />.
+
and these are the linear means of the [[Fourier series|Fourier series]] generated by $\lambda$.
  
The linear means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012038.png" /> converge to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012039.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012040.png" /> at all the Lebesgue points of each integrable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012041.png" /> if and only if the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012042.png" /> is absolutely continuous (cf. [[Absolute continuity|Absolute continuity]]) with respect to the [[Lebesgue measure|Lebesgue measure]] and
+
The linear means $( f ^ { * } d \mu ) _ { N } ( x )$ converge to $f ( x )$ as $N \rightarrow \infty$ at all the Lebesgue points of each integrable $f$ if and only if the measure $\mu$ is absolutely continuous (cf. [[Absolute continuity|Absolute continuity]]) with respect to the [[Lebesgue measure|Lebesgue measure]] and
  
<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/b/b130/b130120/b13012043.png" /></td> </tr></table>
+
$$
 +
\int_0^\infty \operatorname{esssup}_{u < |x| < \infty} |\mu'(x)| \, du < \infty.
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012044.png" /> possesses many properties which are similar to those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012045.png" />:
+
$\mathcal{A} ^ { * }$ possesses many properties which are similar to those of $\mathcal{A}$:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012046.png" /> is a [[Banach algebra|Banach algebra]] with the local property.
+
1) $\mathcal{A} ^ { * }$ is a [[Banach algebra|Banach algebra]] with the local property.
  
2) The space of maximal ideals (cf. also [[Maximal ideal|Maximal ideal]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012047.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012048.png" />.
+
2) The space of maximal ideals (cf. also [[Maximal ideal|Maximal ideal]]) of $\mathcal{A} ^ { * }$ coincides with $[- \pi , \pi ]$.
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012049.png" /> is a regular Banach algebra with trivial radical.
+
3) $\mathcal{A} ^ { * }$ is a regular Banach algebra with trivial radical.
  
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012051.png" /> is defined and analytic on a neighbourhood of the set of values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012052.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012053.png" /> (in particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012054.png" /> does not vanish anywhere, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012055.png" />).
+
4) If $f \in \mathcal{A} ^ { * }$ and $F ( z )$ is defined and analytic on a neighbourhood of the set of values of the function $f$, then $F \circ f \in \mathcal{A} ^ { * }$ (in particular, if $f$ does not vanish anywhere, then $1 / f \in \mathcal{A} ^ { * }$).
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012056.png" /> of all sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012057.png" /> with finite norm
+
The space ${\cal P M} ^ { * }$ of all sequences $d = \{ d_{ k } \} ^ { \infty } _ { k  = - \infty}$ with finite norm
  
<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/b/b130/b130120/b13012058.png" /></td> </tr></table>
+
\begin{equation*} \| d \| _ {\cal P M ^* } = \operatorname { sup } _ { n \geq 0 } \frac { 1 } { n + 1 } \sum _ { k = - n } ^ { n } | d _ { k } | \end{equation*}
  
is the dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012059.png" />.
+
is the dual space of $\mathcal{A} ^ { * }$.
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012060.png" /> is not separable (cf. also [[Separable space|Separable space]]) and thus the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012061.png" />, like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012062.png" />, is not reflexive (cf. also [[Reflexive space|Reflexive space]]).
+
The space ${\cal P M} ^ { * }$ is not separable (cf. also [[Separable space|Separable space]]) and thus the space $\mathcal{A} ^ { * }$, like $\mathcal{A}$, is not reflexive (cf. also [[Reflexive space|Reflexive space]]).
  
==Spectral properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012063.png" />.==
+
==Spectral properties of $\mathcal{A} ^ { * }$.==
The following two results are analogues for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012064.png" /> of the Herz theorem and of the Wiener–Ditkin theorem, respectively.
+
The following two results are analogues for $\mathcal{A} ^ { * }$ of the Herz theorem and of the Wiener–Ditkin theorem, respectively.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012065.png" /> be a function coinciding with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012066.png" /> at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012067.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012068.png" /> are integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012069.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012070.png" /> is linear on intervals. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012071.png" />. Then
+
Let $f _ { N }$ be a function coinciding with $f$ at each point $2 \pi k / N$, where $k , N > 0$ are integers, $k < N$, and $f _ { N }$ is linear on intervals. Suppose $f \in \mathcal{A} ^ { * }$. Then
  
<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/b/b130/b130120/b13012072.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { lim } _ { N \rightarrow \infty } \| f - f _ { N } \| _ { \cal A ^ { * } }= 0. \end{equation*}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012073.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012076.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012077.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012079.png" />. Then
+
Let $\Delta _ { \varepsilon } ( t ) = ( 1 - | t | / \varepsilon ) _ { + }$ for $t | \leq \pi$ and $\varepsilon \in ( 0 , \pi / 2 )$, $\Delta _ { \varepsilon } ( t + 2 \pi ) = \Delta _ { \varepsilon } ( t )$, and $V _ { \varepsilon } = 2 \Delta _ { 2  \varepsilon} - \Delta _ { \varepsilon }$. Let $f \in \mathcal{A} ^ { * }$ and $f ( 0 ) = 0$. Then
  
<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/b/b130/b130120/b13012080.png" /></td> </tr></table>
+
\begin{equation*} \operatorname { lim } _ { \varepsilon \rightarrow 0 } \| f V _ { \varepsilon } \| _ { \mathcal{A} } * = 0. \end{equation*}
  
 
==Synthesis problems.==
 
==Synthesis problems.==
Writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012081.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012082.png" /> admits synthesis in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012083.png" /> (cf. also [[Synthesis problems|Synthesis problems]]), for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012084.png" /> the following statements hold.
+
Writing $f \in S$ if $f$ admits synthesis in the norm of $\mathcal{A} ^ { * }$ (cf. also [[Synthesis problems|Synthesis problems]]), for $f \in \mathcal{A} ^ { * }$ the following statements hold.
  
a) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012085.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012086.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012087.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012088.png" />).
+
a) If $f \in \operatorname { Lip } 1$, then $f \in S$ ($\operatorname { Lip } ( 1 / 2 )$ for $\mathcal{A}$).
  
b) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012089.png" /> is absolutely continuous and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012091.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012092.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012094.png" /> has to be of bounded variation).
+
b) If $f$ is absolutely continuous and in $\operatorname { Lip } \alpha$, $\alpha > 1 / 2$, then $f \in S$ (for $f \in \mathcal{A}$, $f$ has to be of bounded variation).
  
(Here, analogous conditions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012095.png" /> are given in parentheses; these assertions make up the Beurling–Pollard theorem.)
+
(Here, analogous conditions for $\mathcal{A}$ are given in parentheses; these assertions make up the Beurling–Pollard theorem.)
  
The following statements describe structural properties of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012096.png" />. In the sequel, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012097.png" /> stands for the modulus of continuity, in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012098.png" />, of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012099.png" /> (cf. also [[Continuity, modulus of|Continuity, modulus of]]).
+
The following statements describe structural properties of functions in $\mathcal{A} ^ { * }$. In the sequel, $\omega ( g , .)_p$ stands for the modulus of continuity, in the norm of $L ^ { p }$, of a function $g$ (cf. also [[Continuity, modulus of|Continuity, modulus of]]).
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120100.png" />, and the imbeddings into these Besov spaces (cf. also [[Nikol'skii space|Nikol'skii space]] and [[Hankel operator|Hankel operator]]) are both continuous.
+
i) $B _ { 1,1 } ^ { 1 } \subset \mathcal{A} ^ { * } \subset B _ { 2,1 } ^ { 1 / 2 }$, and the imbeddings into these Besov spaces (cf. also [[Nikol'skii space|Nikol'skii space]] and [[Hankel operator|Hankel operator]]) are both continuous.
  
ii) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120101.png" /> is absolutely continuous and
+
ii) If $f$ is absolutely continuous and
  
<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/b/b130/b130120/b130120102.png" /></td> </tr></table>
+
\begin{equation*} \int _ { 0 } ^ { 1 } \omega ( f ^ { \prime } ; t ) _ { p } \left( \operatorname { ln } \frac { 1 } { t } \right) ^ { - 1 / p ^ { \prime } } t ^ { - 1 } d t < \infty \end{equation*}
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120104.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120105.png" />.
+
for some $p \in [ 1,2 ]$, $p ^ { \prime } = p / ( p - 1 )$, then $f \in \mathcal{A} ^ { * }$.
  
iii) There exists a continuously differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120106.png" /> for which
+
iii) There exists a continuously differentiable function $f \notin \mathcal{A} ^ { * }$ for which
  
<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/b/b130/b130120/b130120107.png" /></td> </tr></table>
+
\begin{equation*} \omega ( f ^ { \prime } ; t ) _ { \infty } = O \left( \left( \operatorname { ln } \frac { 1 } { t } \right) ^ { - 1 / 2 } \right). \end{equation*}
  
The second inclusion in i) is sharp; indeed, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120108.png" /> there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120109.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120110.png" />. As for ii), it is sharp for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120111.png" />. The question is open for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120112.png" /> (as of 2000).
+
The second inclusion in i) is sharp; indeed, for each $\varepsilon > 0$ there exists a function $f \notin B _ { 2 , \infty } ^ { \varepsilon + 1 / 2 }$ such that $f \in \mathcal{A} ^ { * }$. As for ii), it is sharp for $p = 1$. The question is open for $p > 1$ (as of 2000).
  
There exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120113.png" /> that is not of bounded variation (cf. also [[Function of bounded variation|Function of bounded variation]]).
+
There exists a function $f \in \mathcal{A} ^ { * }$ that is not of bounded variation (cf. also [[Function of bounded variation|Function of bounded variation]]).
  
The following condition, although being very simple, is surprisingly sharp. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120114.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120115.png" />. If the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120116.png" /> is lacunary in the sense of Hadamard (cf. also [[Lacunary sequence|Lacunary sequence]]), then the converse statement holds.
+
The following condition, although being very simple, is surprisingly sharp. If $f ^ { \prime } \in \mathcal{A}$, then $f \in \mathcal{A} ^ { * }$. If the Fourier series of $f$ is lacunary in the sense of Hadamard (cf. also [[Lacunary sequence|Lacunary sequence]]), then the converse statement holds.
  
Note that the central problem of [[Spectral synthesis|spectral synthesis]], that is, existence of sets that are not of synthesis as well as existence of functions not admitting synthesis, is still open (as of 2000) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120117.png" />.
+
Note that the central problem of [[Spectral synthesis|spectral synthesis]], that is, existence of sets that are not of synthesis as well as existence of functions not admitting synthesis, is still open (as of 2000) for $\mathcal{A} ^ { * }$.
  
Another open question is connected with Beurling's initial result. It would be interesting to know whether the following statement, converse to that given above, is true or not: If for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120118.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120119.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120120.png" />.
+
Another open question is connected with Beurling's initial result. It would be interesting to know whether the following statement, converse to that given above, is true or not: If for every $f \in \mathcal{A} ^ { * }$ one has $F \circ f \in \mathcal{A}$, then $F \in \operatorname { Lip } 1$.
  
Some of the results known for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120121.png" /> have been generalized to the multi-dimensional case.
+
Some of the results known for $\mathcal{A} ^ { * }$ have been generalized to the multi-dimensional case.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Belinsky,  E. Liflyand,  R. Trigub,  "The Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b130120122.png" /> and its properties"  ''J. Fourier Anal. Appl.'' , '''3'''  (1997)  pp. 103–129</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Beurling,  "On the spectral synthesis of bounded functions"  ''Acta Math.'' , '''81'''  (1949)  pp. 225–238</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.-P. Kahane,  "Séries de Fourier absolument convergentes" , Springer  (1970)</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  E. Belinsky,  E. Liflyand,  R. Trigub,  "The Banach algebra $A ^ { * }$ and its properties"  ''J. Fourier Anal. Appl.'' , '''3'''  (1997)  pp. 103–129</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A. Beurling,  "On the spectral synthesis of bounded functions"  ''Acta Math.'' , '''81'''  (1949)  pp. 225–238</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J.-P. Kahane,  "Séries de Fourier absolument convergentes" , Springer  (1970)</td></tr>
 +
</table>

Latest revision as of 08:15, 15 February 2024

of Fourier series with summable majorant of coefficients

An algebra closely related to the Wiener algebra

\begin{equation*} \mathcal{A} = \{ f : \| f \| _ { \mathcal{A} } = \sum _ { m = - \infty } ^ { \infty } | \hat { f } ( m ) | < \infty \}, \end{equation*}

where

\begin{equation*} \hat { f } ( m ) = ( 2 \pi ) ^ { - 1 } \int _ { - \pi } ^ { \pi } f ( u ) e ^ { - i m u } d u \end{equation*}

is the $m$th Fourier coefficient of $f$ (cf. also Fourier coefficients). The Beurling algebra is defined as

\begin{equation*} \mathcal{A} ^ { * } = \left\{ f : \|\, f \| _ { \mathcal{A}^ { * } } = \sum _ { k = 0 } ^ { \infty } \operatorname { sup } _ { k \leq |m | < \infty } |\, \hat { f } ( m ) | < \infty \right\}. \end{equation*}

The space $\mathcal{A} ^ { * }$ was introduced by A. Beurling for establishing contraction properties of functions [a2]: Let

\begin{equation*} f ( t ) = \sum _ { n = - \infty } ^ { \infty } a _ { n } e ^ { i n t } , a _ { 0 } = 0, \end{equation*}

be an absolutely convergent Fourier series such that $| a _ { \pm n } | \leq a _ { n } ^ { * }$, $n \geq 1$, where $\{ a _ { n } ^ { * } \}$ is a non-increasing sequence of numbers with a finite sum. Then if

\begin{equation*} g ( t ) \sim \sum _ { n = - \infty } ^ { \infty } b _ { n } e ^ { i n t } , b _ { 0 } = 0, \end{equation*}

is a contraction of $f ( t )$ (that is, for any pair of arguments $t_{1}$, $t_2$ the inequality $| g ( t _ { 1 } ) - g ( t _ { 2 } ) | \leq | f ( t _ { 1 } ) - f ( t _ { 2 } ) |$ holds), then the Fourier series of $g ( t )$ also converges absolutely, and

\begin{equation*} \sum _ { n = - \infty } ^ { \infty } | b _ { n } | \leq 10 \sum _ { n = 1 } ^ { \infty } a _ { n } ^ { * }. \end{equation*}

A similar result was proved in [a2] for the Fourier transform, whence integrability of the monotone majorant on the real line is considered. These two spaces coincide locally, hence have much in common.

Subsequently, $\mathcal{A} ^ { * }$ appeared in some other papers either in explicit or implicit form. See [a1] for a detailed survey of the history and properties of $\mathcal{A} ^ { * }$.

It turned out that the consideration of summability of Fourier series by linear methods at Lebesgue points leads to the same space of functions. Let $f$ be a $2 \pi$-periodic integrable function with Fourier series $\sum _ { k } \hat { f } ( k ) e ^ { i k x }$. Let $\lambda$ be a continuous function on $\mathbf{R} = ( - \infty , \infty )$, representable as follows:

\begin{equation*} \lambda ( x ) = \int _ { \mathbf{R} } e ^ { - i x t } d \mu ( t ), \end{equation*}

where $\mu$ is a finite Borel measure satisfying $\int _ { \mathbf{R} } d \mu ( t ) = 1$. Consider the means

\begin{equation*} ( f * d \mu ) _ { N } : = \operatorname { lim } _ { h \rightarrow 0 } \int _ { \mathbf{R} } f _ { h } \left( \frac { x - u } { N } \right) d \mu ( u ), \end{equation*}

where

\begin{equation*} f _ { h } ( x ) = h ^ { - 1 } \int _ {\bf R } \varphi \left( \frac { t } { h } \right) f ( x - t ) d t. \end{equation*}

Here, $\varphi ( t )$ is infinitely differentiable, equal to $1$ for $|t | < 1$, vanishing for $| t | > 2$ and such that $\int _ { \mathbf R } \varphi ( t ) d t = 1$. For $f$ sufficiently smooth one has

\begin{equation*} ( f ^ { * } d \mu ) _ { N } ( x ) = \sum _ { k } \lambda \left( \frac { k } { N } \right) \hat { f } ( k ) e ^ { i k x }, \end{equation*}

and these are the linear means of the Fourier series generated by $\lambda$.

The linear means $( f ^ { * } d \mu ) _ { N } ( x )$ converge to $f ( x )$ as $N \rightarrow \infty$ at all the Lebesgue points of each integrable $f$ if and only if the measure $\mu$ is absolutely continuous (cf. Absolute continuity) with respect to the Lebesgue measure and

$$ \int_0^\infty \operatorname{esssup}_{u < |x| < \infty} |\mu'(x)| \, du < \infty. $$

$\mathcal{A} ^ { * }$ possesses many properties which are similar to those of $\mathcal{A}$:

1) $\mathcal{A} ^ { * }$ is a Banach algebra with the local property.

2) The space of maximal ideals (cf. also Maximal ideal) of $\mathcal{A} ^ { * }$ coincides with $[- \pi , \pi ]$.

3) $\mathcal{A} ^ { * }$ is a regular Banach algebra with trivial radical.

4) If $f \in \mathcal{A} ^ { * }$ and $F ( z )$ is defined and analytic on a neighbourhood of the set of values of the function $f$, then $F \circ f \in \mathcal{A} ^ { * }$ (in particular, if $f$ does not vanish anywhere, then $1 / f \in \mathcal{A} ^ { * }$).

The space ${\cal P M} ^ { * }$ of all sequences $d = \{ d_{ k } \} ^ { \infty } _ { k = - \infty}$ with finite norm

\begin{equation*} \| d \| _ {\cal P M ^* } = \operatorname { sup } _ { n \geq 0 } \frac { 1 } { n + 1 } \sum _ { k = - n } ^ { n } | d _ { k } | \end{equation*}

is the dual space of $\mathcal{A} ^ { * }$.

The space ${\cal P M} ^ { * }$ is not separable (cf. also Separable space) and thus the space $\mathcal{A} ^ { * }$, like $\mathcal{A}$, is not reflexive (cf. also Reflexive space).

Spectral properties of $\mathcal{A} ^ { * }$.

The following two results are analogues for $\mathcal{A} ^ { * }$ of the Herz theorem and of the Wiener–Ditkin theorem, respectively.

Let $f _ { N }$ be a function coinciding with $f$ at each point $2 \pi k / N$, where $k , N > 0$ are integers, $k < N$, and $f _ { N }$ is linear on intervals. Suppose $f \in \mathcal{A} ^ { * }$. Then

\begin{equation*} \operatorname { lim } _ { N \rightarrow \infty } \| f - f _ { N } \| _ { \cal A ^ { * } }= 0. \end{equation*}

Let $\Delta _ { \varepsilon } ( t ) = ( 1 - | t | / \varepsilon ) _ { + }$ for $t | \leq \pi$ and $\varepsilon \in ( 0 , \pi / 2 )$, $\Delta _ { \varepsilon } ( t + 2 \pi ) = \Delta _ { \varepsilon } ( t )$, and $V _ { \varepsilon } = 2 \Delta _ { 2 \varepsilon} - \Delta _ { \varepsilon }$. Let $f \in \mathcal{A} ^ { * }$ and $f ( 0 ) = 0$. Then

\begin{equation*} \operatorname { lim } _ { \varepsilon \rightarrow 0 } \| f V _ { \varepsilon } \| _ { \mathcal{A} } * = 0. \end{equation*}

Synthesis problems.

Writing $f \in S$ if $f$ admits synthesis in the norm of $\mathcal{A} ^ { * }$ (cf. also Synthesis problems), for $f \in \mathcal{A} ^ { * }$ the following statements hold.

a) If $f \in \operatorname { Lip } 1$, then $f \in S$ ($\operatorname { Lip } ( 1 / 2 )$ for $\mathcal{A}$).

b) If $f$ is absolutely continuous and in $\operatorname { Lip } \alpha$, $\alpha > 1 / 2$, then $f \in S$ (for $f \in \mathcal{A}$, $f$ has to be of bounded variation).

(Here, analogous conditions for $\mathcal{A}$ are given in parentheses; these assertions make up the Beurling–Pollard theorem.)

The following statements describe structural properties of functions in $\mathcal{A} ^ { * }$. In the sequel, $\omega ( g , .)_p$ stands for the modulus of continuity, in the norm of $L ^ { p }$, of a function $g$ (cf. also Continuity, modulus of).

i) $B _ { 1,1 } ^ { 1 } \subset \mathcal{A} ^ { * } \subset B _ { 2,1 } ^ { 1 / 2 }$, and the imbeddings into these Besov spaces (cf. also Nikol'skii space and Hankel operator) are both continuous.

ii) If $f$ is absolutely continuous and

\begin{equation*} \int _ { 0 } ^ { 1 } \omega ( f ^ { \prime } ; t ) _ { p } \left( \operatorname { ln } \frac { 1 } { t } \right) ^ { - 1 / p ^ { \prime } } t ^ { - 1 } d t < \infty \end{equation*}

for some $p \in [ 1,2 ]$, $p ^ { \prime } = p / ( p - 1 )$, then $f \in \mathcal{A} ^ { * }$.

iii) There exists a continuously differentiable function $f \notin \mathcal{A} ^ { * }$ for which

\begin{equation*} \omega ( f ^ { \prime } ; t ) _ { \infty } = O \left( \left( \operatorname { ln } \frac { 1 } { t } \right) ^ { - 1 / 2 } \right). \end{equation*}

The second inclusion in i) is sharp; indeed, for each $\varepsilon > 0$ there exists a function $f \notin B _ { 2 , \infty } ^ { \varepsilon + 1 / 2 }$ such that $f \in \mathcal{A} ^ { * }$. As for ii), it is sharp for $p = 1$. The question is open for $p > 1$ (as of 2000).

There exists a function $f \in \mathcal{A} ^ { * }$ that is not of bounded variation (cf. also Function of bounded variation).

The following condition, although being very simple, is surprisingly sharp. If $f ^ { \prime } \in \mathcal{A}$, then $f \in \mathcal{A} ^ { * }$. If the Fourier series of $f$ is lacunary in the sense of Hadamard (cf. also Lacunary sequence), then the converse statement holds.

Note that the central problem of spectral synthesis, that is, existence of sets that are not of synthesis as well as existence of functions not admitting synthesis, is still open (as of 2000) for $\mathcal{A} ^ { * }$.

Another open question is connected with Beurling's initial result. It would be interesting to know whether the following statement, converse to that given above, is true or not: If for every $f \in \mathcal{A} ^ { * }$ one has $F \circ f \in \mathcal{A}$, then $F \in \operatorname { Lip } 1$.

Some of the results known for $\mathcal{A} ^ { * }$ have been generalized to the multi-dimensional case.

References

[a1] E. Belinsky, E. Liflyand, R. Trigub, "The Banach algebra $A ^ { * }$ and its properties" J. Fourier Anal. Appl. , 3 (1997) pp. 103–129
[a2] A. Beurling, "On the spectral synthesis of bounded functions" Acta Math. , 81 (1949) pp. 225–238
[a3] J.-P. Kahane, "Séries de Fourier absolument convergentes" , Springer (1970)
How to Cite This Entry:
Beurling algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beurling_algebra&oldid=16258
This article was adapted from an original article by E.S. BelinskyE.R. Liflyand (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article