Difference between revisions of "Wiener-Hopf operator"
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− | An operator which arises from another operator by compression to a subspace. Given a [[Linear operator|linear operator]] | + | <!-- |
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+ | An operator which arises from another operator by compression to a subspace. Given a [[Linear operator|linear operator]] $ A : X \rightarrow X $ | ||
+ | and a projection $ P = P ^ {2} $ | ||
+ | on $ X $ | ||
+ | with range $ { \mathop{\rm Im} } P $, | ||
+ | the corresponding Wiener–Hopf operator $ W _ {P} ( A ) $ | ||
+ | is defined as the operator on $ { \mathop{\rm Im} } P $ | ||
+ | that sends $ x \in { \mathop{\rm Im} } P $ | ||
+ | to $ P ( Ax ) \in { \mathop{\rm Im} } P $. | ||
+ | Thus, $ W _ {P} ( A ) = PA \mid _ { { \mathop{\rm Im} } P } $. | ||
A Wiener–Hopf integral operator formally acts by the rule | A Wiener–Hopf integral operator formally acts by the rule | ||
− | + | $$ | |
+ | ( Wf ) ( x ) = cf ( x ) + \int\limits _ { 0 } ^ \infty {k ( x - t ) f ( t ) } {dt } | ||
+ | $$ | ||
− | ( | + | ( $ x > 0 $) |
+ | on some space of functions over $ \mathbf R _ {+} = ( 0, \infty ) $, | ||
+ | say on $ L _ {p} ( \mathbf R _ {+} ) $( | ||
+ | $ 1 \leq p \leq \infty $). | ||
+ | It may be regarded as the restriction to $ L _ {p} ( \mathbf R _ {+} ) $ | ||
+ | of a convolution integral operator on $ L _ {p} ( \mathbf R ) $. | ||
+ | The operator $ W $ | ||
+ | is bounded if, for example, $ c \in \mathbf C $ | ||
+ | and $ k \in L _ {1} ( \mathbf R ) $. | ||
+ | Many properties of $ W $ | ||
+ | can be read off from its symbol. This is the function $ a $ | ||
+ | given by | ||
− | + | $$ | |
+ | a ( \xi ) = c + \int\limits _ { \mathbf R } {k ( t ) e ^ {i \xi t } } {dt } \quad ( \xi \in \mathbf R ) . | ||
+ | $$ | ||
− | The operator | + | The operator $ W $ |
+ | is Fredholm (cf. [[Fredholm-operator(2)|Fredholm operator]]), i.e. invertible modulo compact operators, if and only if its symbol $ a $ | ||
+ | has no zeros on the one-point compactification $ \mathbf R \cup \{ \infty \} $( | ||
+ | cf. also [[Aleksandrov compactification|Aleksandrov compactification]]) of $ \mathbf R $. | ||
+ | In that case the kernel and cokernel dimensions of $ W $ | ||
+ | are: | ||
− | + | $$ | |
+ | { \mathop{\rm dim} } { \mathop{\rm Ker} } W = \max \{ - \kappa, 0 \} | ||
+ | $$ | ||
− | + | $$ | |
+ | { \mathop{\rm dim} } { \mathop{\rm Coker} } W = \max \{ \kappa, 0 \} , | ||
+ | $$ | ||
− | where | + | where $ \kappa $ |
+ | is the [[Winding number|winding number]] of the symbol $ a $ | ||
+ | about the origin. The equation $ Wf = g $ | ||
+ | can be solved by Wiener–Hopf factorization, which means that one represents $ a $ | ||
+ | in the form | ||
− | + | $$ | |
+ | a ( \xi ) = a _ {-} ( \xi ) \left ( { | ||
+ | \frac{\xi - i }{\xi + i } | ||
+ | } \right ) ^ \kappa a _ {+} ( \xi ) | ||
+ | $$ | ||
− | such that | + | such that $ a _ {-} $ |
+ | and $ a _ {+} $ | ||
+ | extend to analytic functions without zeros in the lower and upper complex half-planes, respectively. | ||
− | Many interesting operators are Wiener–Hopf integral operators with discontinuous symbols. For example, the Cauchy singular integral operator | + | Many interesting operators are Wiener–Hopf integral operators with discontinuous symbols. For example, the Cauchy singular integral operator $ S _ {+} $( |
+ | cf. also [[Singular integral|Singular integral]]) on $ L _ {p} ( \mathbf R _ {+} ) $( | ||
+ | $ 1 < p < \infty $), | ||
− | + | $$ | |
+ | ( S _ {+} f ) ( x ) = { | ||
+ | \frac{1}{\pi i } | ||
+ | } \int\limits _ {\mathbf R _ {+} } { { | ||
+ | \frac{f ( t ) }{t - x } | ||
+ | } } {dt } \quad ( x \in \mathbf R _ {+} ) , | ||
+ | $$ | ||
− | can be interpreted as the Wiener–Hopf integral operator with symbol | + | can be interpreted as the Wiener–Hopf integral operator with symbol $ - { \mathop{\rm sign} } \xi $. |
+ | The spectrum of $ S _ {+} $ | ||
+ | on $ L _ {p} ( \mathbf R _ {+} ) $ | ||
+ | is the set of all $ \lambda \in \mathbf C $ | ||
+ | at which the line segment $ [ - 1,1 ] $ | ||
+ | is seen at an angle of at least $ \max \{ { {2 \pi } / p } , { {2 \pi } / q } \} $, | ||
+ | where $ {1 / p } + {1 / q } = 1 $. | ||
Wiener–Hopf integral operators with matrix-valued symbols, on finite intervals, or on higher-dimensional domains (including the quarter-plane) have also been extensively studied. | Wiener–Hopf integral operators with matrix-valued symbols, on finite intervals, or on higher-dimensional domains (including the quarter-plane) have also been extensively studied. | ||
− | A discrete Wiener–Hopf operator, or a Toeplitz operator (cf. also [[Toeplitz matrix|Toeplitz matrix]]), is given by a matrix of the form | + | A discrete Wiener–Hopf operator, or a Toeplitz operator (cf. also [[Toeplitz matrix|Toeplitz matrix]]), is given by a matrix of the form $ ( a _ {j - k } ) _ {j,k = 0 } ^ \infty $ |
+ | on some sequence space, e.g. on $ l ^ {p} ( \mathbf Z _ {+} ) $. | ||
+ | In this case the symbol is the function on the complex unit circle whose Fourier coefficients constitute the sequence $ ( a _ {n} ) _ {n \in \mathbf Z } $. | ||
There is a rich literature on Wiener–Hopf operators. A good introduction is the classical monograph [[#References|[a1]]]; [[#References|[a2]]] and [[#References|[a3]]] provide an overview of some recent developments. | There is a rich literature on Wiener–Hopf operators. A good introduction is the classical monograph [[#References|[a1]]]; [[#References|[a2]]] and [[#References|[a3]]] provide an overview of some recent developments. |
Latest revision as of 08:29, 6 June 2020
An operator which arises from another operator by compression to a subspace. Given a linear operator $ A : X \rightarrow X $
and a projection $ P = P ^ {2} $
on $ X $
with range $ { \mathop{\rm Im} } P $,
the corresponding Wiener–Hopf operator $ W _ {P} ( A ) $
is defined as the operator on $ { \mathop{\rm Im} } P $
that sends $ x \in { \mathop{\rm Im} } P $
to $ P ( Ax ) \in { \mathop{\rm Im} } P $.
Thus, $ W _ {P} ( A ) = PA \mid _ { { \mathop{\rm Im} } P } $.
A Wiener–Hopf integral operator formally acts by the rule
$$ ( Wf ) ( x ) = cf ( x ) + \int\limits _ { 0 } ^ \infty {k ( x - t ) f ( t ) } {dt } $$
( $ x > 0 $) on some space of functions over $ \mathbf R _ {+} = ( 0, \infty ) $, say on $ L _ {p} ( \mathbf R _ {+} ) $( $ 1 \leq p \leq \infty $). It may be regarded as the restriction to $ L _ {p} ( \mathbf R _ {+} ) $ of a convolution integral operator on $ L _ {p} ( \mathbf R ) $. The operator $ W $ is bounded if, for example, $ c \in \mathbf C $ and $ k \in L _ {1} ( \mathbf R ) $. Many properties of $ W $ can be read off from its symbol. This is the function $ a $ given by
$$ a ( \xi ) = c + \int\limits _ { \mathbf R } {k ( t ) e ^ {i \xi t } } {dt } \quad ( \xi \in \mathbf R ) . $$
The operator $ W $ is Fredholm (cf. Fredholm operator), i.e. invertible modulo compact operators, if and only if its symbol $ a $ has no zeros on the one-point compactification $ \mathbf R \cup \{ \infty \} $( cf. also Aleksandrov compactification) of $ \mathbf R $. In that case the kernel and cokernel dimensions of $ W $ are:
$$ { \mathop{\rm dim} } { \mathop{\rm Ker} } W = \max \{ - \kappa, 0 \} $$
$$ { \mathop{\rm dim} } { \mathop{\rm Coker} } W = \max \{ \kappa, 0 \} , $$
where $ \kappa $ is the winding number of the symbol $ a $ about the origin. The equation $ Wf = g $ can be solved by Wiener–Hopf factorization, which means that one represents $ a $ in the form
$$ a ( \xi ) = a _ {-} ( \xi ) \left ( { \frac{\xi - i }{\xi + i } } \right ) ^ \kappa a _ {+} ( \xi ) $$
such that $ a _ {-} $ and $ a _ {+} $ extend to analytic functions without zeros in the lower and upper complex half-planes, respectively.
Many interesting operators are Wiener–Hopf integral operators with discontinuous symbols. For example, the Cauchy singular integral operator $ S _ {+} $( cf. also Singular integral) on $ L _ {p} ( \mathbf R _ {+} ) $( $ 1 < p < \infty $),
$$ ( S _ {+} f ) ( x ) = { \frac{1}{\pi i } } \int\limits _ {\mathbf R _ {+} } { { \frac{f ( t ) }{t - x } } } {dt } \quad ( x \in \mathbf R _ {+} ) , $$
can be interpreted as the Wiener–Hopf integral operator with symbol $ - { \mathop{\rm sign} } \xi $. The spectrum of $ S _ {+} $ on $ L _ {p} ( \mathbf R _ {+} ) $ is the set of all $ \lambda \in \mathbf C $ at which the line segment $ [ - 1,1 ] $ is seen at an angle of at least $ \max \{ { {2 \pi } / p } , { {2 \pi } / q } \} $, where $ {1 / p } + {1 / q } = 1 $.
Wiener–Hopf integral operators with matrix-valued symbols, on finite intervals, or on higher-dimensional domains (including the quarter-plane) have also been extensively studied.
A discrete Wiener–Hopf operator, or a Toeplitz operator (cf. also Toeplitz matrix), is given by a matrix of the form $ ( a _ {j - k } ) _ {j,k = 0 } ^ \infty $ on some sequence space, e.g. on $ l ^ {p} ( \mathbf Z _ {+} ) $. In this case the symbol is the function on the complex unit circle whose Fourier coefficients constitute the sequence $ ( a _ {n} ) _ {n \in \mathbf Z } $.
There is a rich literature on Wiener–Hopf operators. A good introduction is the classical monograph [a1]; [a2] and [a3] provide an overview of some recent developments.
References
[a1] | I. Gohberg, I.A. Feldman, "Convolution equations and projection methods for their solution" , Amer. Math. Soc. (1974) |
[a2] | A. Böttcher, B. Silbermann, "Analysis of Toeplitz operators" , Springer (1990) |
[a3] | I. Gohberg, S. Goldberg, M.A. Kaashoek, "Classes of linear operators" , I–II , Birkhäuser (1990–1993) |
Wiener-Hopf operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener-Hopf_operator&oldid=23145