Difference between revisions of "Fourier-Stieltjes transform"
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| − | + | One of the integral transforms (cf. [[Integral transform|Integral transform]]) related to the [[Fourier transform|Fourier transform]]. Let the function $ F $ | |
| + | have bounded variation on $ (- \infty , + \infty ) $. | ||
| + | The function | ||
| − | + | $$ \tag{* } | |
| + | \phi ( x) = \ | ||
| + | { | ||
| + | \frac{1}{\sqrt {2 \pi } } | ||
| + | } | ||
| + | \int\limits _ {- \infty } ^ {+\infty } | ||
| + | e ^ {-} ixy dF ( y) | ||
| + | $$ | ||
| − | + | is called the Fourier–Stieltjes transform of $ F $. | |
| + | The function $ \phi $ | ||
| + | determined by the integral (*) is bounded and continuous. Every periodic function $ \phi $ | ||
| + | that can be expanded in an absolutely-convergent Fourier series $ \sum _ {- \infty } ^ {+ \infty } a _ {n} e ^ {inx} $ | ||
| + | can be written as an integral (*) with $ F ( x) = \sum _ {n \leq x } a _ {n} $. | ||
| + | |||
| + | Formula (*) can be inverted: If $ F $ | ||
| + | has bounded variation and if | ||
| + | |||
| + | $$ | ||
| + | F ^ \bullet ( x) = \ | ||
| + | { | ||
| + | \frac{F ( x + 0) + F ( x - 0) }{2} | ||
| + | } , | ||
| + | $$ | ||
then | then | ||
| − | + | $$ | |
| + | F ^ \bullet ( x) - F ^ \bullet ( 0) = \ | ||
| + | { | ||
| + | \frac{1}{\sqrt {2 \pi } } | ||
| + | } | ||
| + | \int\limits _ {- \infty } ^ {+\infty } | ||
| + | \phi ( \xi ) | ||
| + | |||
| + | \frac{e ^ {i \xi x } - 1 }{i \xi } | ||
| + | \ | ||
| + | d \xi ,\ \ | ||
| + | x \in (- \infty , + \infty ), | ||
| + | $$ | ||
| − | where the integral is taken to mean the principal value at | + | where the integral is taken to mean the principal value at $ \infty $. |
| − | If one only allows non-decreasing functions of bounded variation as the function | + | If one only allows non-decreasing functions of bounded variation as the function $ F $ |
| + | in formula (*), then the set of continuous functions $ \phi $ | ||
| + | thus obtained is completely characterized by the property: For any system of real numbers $ t _ {1} \dots t _ {n} $, | ||
| − | + | $$ | |
| + | \sum _ {i, j = 1 } ^ { n } | ||
| + | \phi ( t _ {i} - t _ {j} ) | ||
| + | \xi _ {i} \overline \xi \; _ {j} \geq 0, | ||
| + | $$ | ||
| − | whatever the complex numbers | + | whatever the complex numbers $ \xi _ {1} \dots \xi _ {n} $( |
| + | the Bochner–Khinchin theorem). Such functions are called positive definite. The Fourier–Stieltjes transform is extensively applied in probability theory, where the non-decreasing function | ||
| − | + | $$ | |
| + | P ( x) = \ | ||
| + | { | ||
| + | \frac{1}{\sqrt {2 \pi } } | ||
| + | } | ||
| + | F ( x) | ||
| + | $$ | ||
| − | is subjected to the additional restrictions | + | is subjected to the additional restrictions $ \lim\limits _ {x \rightarrow - \infty } P ( x) = 0 $, |
| + | $ \lim\limits _ {x \rightarrow + \infty } P ( x) = 1 $ | ||
| + | and $ P $ | ||
| + | is continuous on the left; it is called a distribution, and | ||
| − | + | $$ | |
| + | \Phi(x) = \int\limits_{-\infty}^{+\infty} e^{ixy} dP(y) | ||
| + | $$ | ||
| − | is called the characteristic function (of the distribution | + | is called the characteristic function (of the distribution $ P $). |
| + | The Bochner–Khinchin theorem then expresses a necessary and sufficient condition for a continuous function $ \Phi $( | ||
| + | for which $ \Phi ( 0) = 1 $) | ||
| + | to be the characteristic function of a certain distribution. | ||
| − | The Fourier–Stieltjes transform has also been developed in the | + | The Fourier–Stieltjes transform has also been developed in the $ n $- |
| + | dimensional case. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''2''' , Cambridge Univ. Press (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.V. Gnedenko, "The theory of probability" , Chelsea, reprint (1962) (Translated from Russian)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''2''' , Cambridge Univ. Press (1988)</TD></TR> | ||
| + | <TR><TD valign="top">[3]</TD> <TD valign="top"> B.V. Gnedenko, [[Gnedenko, "A course in the theory of probability"|"The theory of probability"]], Chelsea, reprint (1962) (Translated from Russian)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 20:16, 16 January 2024
One of the integral transforms (cf. Integral transform) related to the Fourier transform. Let the function $ F $
have bounded variation on $ (- \infty , + \infty ) $.
The function
$$ \tag{* } \phi ( x) = \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ {- \infty } ^ {+\infty } e ^ {-} ixy dF ( y) $$
is called the Fourier–Stieltjes transform of $ F $. The function $ \phi $ determined by the integral (*) is bounded and continuous. Every periodic function $ \phi $ that can be expanded in an absolutely-convergent Fourier series $ \sum _ {- \infty } ^ {+ \infty } a _ {n} e ^ {inx} $ can be written as an integral (*) with $ F ( x) = \sum _ {n \leq x } a _ {n} $.
Formula (*) can be inverted: If $ F $ has bounded variation and if
$$ F ^ \bullet ( x) = \ { \frac{F ( x + 0) + F ( x - 0) }{2} } , $$
then
$$ F ^ \bullet ( x) - F ^ \bullet ( 0) = \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ {- \infty } ^ {+\infty } \phi ( \xi ) \frac{e ^ {i \xi x } - 1 }{i \xi } \ d \xi ,\ \ x \in (- \infty , + \infty ), $$
where the integral is taken to mean the principal value at $ \infty $.
If one only allows non-decreasing functions of bounded variation as the function $ F $ in formula (*), then the set of continuous functions $ \phi $ thus obtained is completely characterized by the property: For any system of real numbers $ t _ {1} \dots t _ {n} $,
$$ \sum _ {i, j = 1 } ^ { n } \phi ( t _ {i} - t _ {j} ) \xi _ {i} \overline \xi \; _ {j} \geq 0, $$
whatever the complex numbers $ \xi _ {1} \dots \xi _ {n} $( the Bochner–Khinchin theorem). Such functions are called positive definite. The Fourier–Stieltjes transform is extensively applied in probability theory, where the non-decreasing function
$$ P ( x) = \ { \frac{1}{\sqrt {2 \pi } } } F ( x) $$
is subjected to the additional restrictions $ \lim\limits _ {x \rightarrow - \infty } P ( x) = 0 $, $ \lim\limits _ {x \rightarrow + \infty } P ( x) = 1 $ and $ P $ is continuous on the left; it is called a distribution, and
$$ \Phi(x) = \int\limits_{-\infty}^{+\infty} e^{ixy} dP(y) $$
is called the characteristic function (of the distribution $ P $). The Bochner–Khinchin theorem then expresses a necessary and sufficient condition for a continuous function $ \Phi $( for which $ \Phi ( 0) = 1 $) to be the characteristic function of a certain distribution.
The Fourier–Stieltjes transform has also been developed in the $ n $- dimensional case.
References
| [1] | S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German) |
| [2] | A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988) |
| [3] | B.V. Gnedenko, "The theory of probability", Chelsea, reprint (1962) (Translated from Russian) |
Fourier-Stieltjes transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier-Stieltjes_transform&oldid=16936