# Multiplier theory

Given a Fourier series on $[ - \pi , \pi )$, $\sum _ {n = - \infty } ^ \infty c _ {n} e ^ {inx}$ say, and a (doubly infinite) sequence $\{ \lambda _ {n} \}$, one may form a new Fourier series, $\sum _ {n = - \infty } ^ \infty \lambda _ {n} c _ {n} e ^ {inx}$. The sequence $\{ \lambda _ {n} \}$ is called a Fourier multiplier. The principal problem about Fourier multipliers is to determine conditions on $\{ \lambda _ {n} \}$ which guarantee that, when the old Fourier series corresponds to an element of some space ${\mathcal E}$ of functions or generalized functions (cf. Generalized function) on $[ - \pi , \pi )$, then the new series corresponds to an element of some other given space ${\mathcal F}$ of functions or generalized functions on $[ \pi , \pi )$. Typically, ${\mathcal E}$ and ${\mathcal F}$ 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 $\{ \lambda _ {n} \}$ is a multiplier from the space of integrable functions to itself or from the space of continuous functions to itself if and only if $\sum _ {n = - \infty } ^ \infty \lambda _ {n} e ^ {inx}$ is a Fourier–Stieltjes series. Equivalently, one can seek to characterize generalized functions $\phi$ on $[ - \pi , \pi )$ with the property that, if $f \in {\mathcal E}$, then the convolution product $\phi \star f \in {\mathcal F}$; the corresponding Fourier multiplier is the sequence of Fourier coefficients of $\phi$.
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 $m$ on $\mathbf R$ whose total variation (cf. Variation of a function) on each dyadic interval $\pm [ 2 ^ {k} , 2 ^ {k+} 1 ]$ is bounded is a Fourier $L _ {p}$- multiplier (i.e. the associated operator maps $L _ {p} ( \mathbf R )$ into itself) if $1 < p < \infty$. Perhaps the other most important result in the field is C. Fefferman's theorem that in $\mathbf R ^ {n}$, where $n \geq 2$, the characteristic function of the unit ball is a Fourier $L _ {p}$- multiplier only if $p = 2$.