Difference between revisions of "Multiplier theory"
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Given a [[Fourier series|Fourier series]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655302.png" /> say, and a (doubly infinite) sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655303.png" />, one may form a new Fourier series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655304.png" />. The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655305.png" /> is called a Fourier multiplier. The principal problem about Fourier multipliers is to determine conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655306.png" /> which guarantee that, when the old Fourier series corresponds to an element of some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655307.png" /> of functions or generalized functions (cf. [[Generalized function|Generalized function]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655308.png" />, then the new series corresponds to an element of some other given space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655309.png" /> of functions or generalized functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553010.png" />. Typically, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553012.png" /> are Lebesgue spaces, Sobolev spaces or similar function spaces (cf. [[Lebesgue space|Lebesgue space]]; [[Sobolev space|Sobolev space]]). Particular cases of the problem were first solved by W.H. Young (1913), H. Steinhaus (1915) and S. Sidon (1921), the most significant of these solutions being that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553013.png" /> is a multiplier from the space of integrable functions to itself or from the space of continuous functions to itself if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553014.png" /> is a [[Fourier–Stieltjes series|Fourier–Stieltjes series]]. Equivalently, one can seek to characterize generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553016.png" /> with the property that, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553017.png" />, then the convolution product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553018.png" />; the corresponding Fourier multiplier is the sequence of Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553019.png" />. | Given a [[Fourier series|Fourier series]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655302.png" /> say, and a (doubly infinite) sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655303.png" />, one may form a new Fourier series, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655304.png" />. The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655305.png" /> is called a Fourier multiplier. The principal problem about Fourier multipliers is to determine conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655306.png" /> which guarantee that, when the old Fourier series corresponds to an element of some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655307.png" /> of functions or generalized functions (cf. [[Generalized function|Generalized function]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655308.png" />, then the new series corresponds to an element of some other given space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m0655309.png" /> of functions or generalized functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553010.png" />. Typically, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553012.png" /> are Lebesgue spaces, Sobolev spaces or similar function spaces (cf. [[Lebesgue space|Lebesgue space]]; [[Sobolev space|Sobolev space]]). Particular cases of the problem were first solved by W.H. Young (1913), H. Steinhaus (1915) and S. Sidon (1921), the most significant of these solutions being that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553013.png" /> is a multiplier from the space of integrable functions to itself or from the space of continuous functions to itself if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553014.png" /> is a [[Fourier–Stieltjes series|Fourier–Stieltjes series]]. Equivalently, one can seek to characterize generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553016.png" /> with the property that, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553017.png" />, then the convolution product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553018.png" />; the corresponding Fourier multiplier is the sequence of Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553019.png" />. | ||
− | The analogous problem, of characterizing operators which map one space to another and which correspond to a pointwise multiplication of the Fourier transform by a fixed object, can be posed in the context of Fourier integrals rather than series, and in one or several variables. (Indeed, the theory can even be developed in the general context of locally compact groups.) | + | The analogous problem, of characterizing operators which map one space to another and which correspond to a [[pointwise multiplication]] of the Fourier transform by a fixed object, can be posed in the context of Fourier integrals rather than series, and in one or several variables. (Indeed, the theory can even be developed in the general context of locally compact groups.) |
− | The most important results on Fourier multipliers are connected with the theories of singular integral operators and pseudo-differential operators (cf. [[Pseudo-differential operator|Pseudo-differential operator]]). The general style of these is exemplified by the theorem that a bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553020.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553021.png" /> whose total variation (cf. [[ | + | The most important results on Fourier multipliers are connected with the theories of singular integral operators and pseudo-differential operators (cf. [[Pseudo-differential operator|Pseudo-differential operator]]). The general style of these is exemplified by the theorem that a bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553020.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553021.png" /> whose total variation (cf. [[Variation of a function]]) on each dyadic interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553022.png" /> is bounded is a Fourier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553023.png" />-multiplier (i.e. the associated operator maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553024.png" /> into itself) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553025.png" />. Perhaps the other most important result in the field is C. Fefferman's theorem that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553027.png" />, the characteristic function of the unit ball is a Fourier <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553028.png" />-multiplier only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553029.png" />. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hörmander, "Estimates for translation-invariant operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553030.png" /> spaces" ''Acta Math.'' , '''104''' (1960) pp. 93–139</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C. Fefferman, "The multiplier problem for the ball" ''Ann. of Math.'' , '''94''' (1971) pp. 330–336</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''2''' , Cambridge Univ. Press (1988)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Hörmander, "Estimates for translation-invariant operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065530/m06553030.png" /> spaces" ''Acta Math.'' , '''104''' (1960) pp. 93–139</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C. Fefferman, "The multiplier problem for the ball" ''Ann. of Math.'' , '''94''' (1971) pp. 330–336</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Zygmund, "Trigonometric series" , '''2''' , Cambridge Univ. Press (1988)</TD></TR></table> |
Revision as of 18:20, 1 December 2014
Given a Fourier series on ,
say, and a (doubly infinite) sequence
, one may form a new Fourier series,
. The sequence
is called a Fourier multiplier. The principal problem about Fourier multipliers is to determine conditions on
which guarantee that, when the old Fourier series corresponds to an element of some space
of functions or generalized functions (cf. Generalized function) on
, then the new series corresponds to an element of some other given space
of functions or generalized functions on
. Typically,
and
are Lebesgue spaces, Sobolev spaces or similar function spaces (cf. Lebesgue space; Sobolev space). Particular cases of the problem were first solved by W.H. Young (1913), H. Steinhaus (1915) and S. Sidon (1921), the most significant of these solutions being that
is a multiplier from the space of integrable functions to itself or from the space of continuous functions to itself if and only if
is a Fourier–Stieltjes series. Equivalently, one can seek to characterize generalized functions
on
with the property that, if
, then the convolution product
; the corresponding Fourier multiplier is the sequence of Fourier coefficients of
.
The analogous problem, of characterizing operators which map one space to another and which correspond to a pointwise multiplication of the Fourier transform by a fixed object, can be posed in the context of Fourier integrals rather than series, and in one or several variables. (Indeed, the theory can even be developed in the general context of locally compact groups.)
The most important results on Fourier multipliers are connected with the theories of singular integral operators and pseudo-differential operators (cf. Pseudo-differential operator). The general style of these is exemplified by the theorem that a bounded function on
whose total variation (cf. Variation of a function) on each dyadic interval
is bounded is a Fourier
-multiplier (i.e. the associated operator maps
into itself) if
. Perhaps the other most important result in the field is C. Fefferman's theorem that in
, where
, the characteristic function of the unit ball is a Fourier
-multiplier only if
.
References
[a1] | L. Hörmander, "Estimates for translation-invariant operators in ![]() |
[a2] | C. Fefferman, "The multiplier problem for the ball" Ann. of Math. , 94 (1971) pp. 330–336 |
[a3] | A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988) |
Multiplier theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplier_theory&oldid=14879