View source for Beurling algebra

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''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|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 [[#References|[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|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 [[#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, $\mathcal{A} ^ { * }$ 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 $f$ 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|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|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|Absolute continuity]]) with respect to the [[Lebesgue measure|Lebesgue measure]] and

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130120/b13012043.png"/></td> </tr></table>

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

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 $\mathcal{A} ^ { * }$ 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|Separable space]]) and thus the space $\mathcal{A} ^ { * }$, like $\mathcal{A}$, is not reflexive (cf. also [[Reflexive space|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|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|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|Nikol'skii space]] and [[Hankel operator|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|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|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 $\mathcal{A} ^ { * }$.

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====
<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>