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Multiplier (Fourier analysis)
Encyclopedia
In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply the Fourier transform
of a function by a specified function known as the multiplier or symbol. Occasionally, the term "multiplier operator" itself is shortened simply to "multiplier" . In simple terms, the multiplier reshapes the frequencies involved in any function. This class of operators turns out to be broad: general theory shows that a translation-invariant operator on a group which obeys some (very mild) regularity conditions can be expressed as a multiplier operator, and conversely. Many familiar operators, such as translations and differentiation, are multiplier operators, although there are many more complicated examples such as the Hilbert transform
.
In signal processing
, a multiplier operator is called a "filter
", and the multiplier is the filter's frequency response
(or transfer function
).
In the wider context, multiplier operators are special cases of spectral multiplier operators, which arise from the functional calculus
of an operator (or family of commuting operators). They are also special cases of pseudo-differential operator
s, and more generally Fourier integral operator
s. There are natural questions in this field that are still open, such as characterizing the
bounded multiplier operators (see below). In this context, multipliers are unrelated to Lagrange multipliers, except for the fact that they both involve the multiplication operation.
For the necessary background on the Fourier transform
, see that page. Additional important background may be found on the pages operator norm
and Lp space
.
.
So, formally, it follows that the Fourier series for the derivative is simply in multiplied by the Fourier series for ƒ. This is the same as saying that differentiation is a multiplier operator with multiplier in.
An example of a multiplier operator acting on functions on the real line is the Hilbert transform
. It can be shown that the Hilbert transform is a multiplier operator whose multiplier is given by the m(ξ) = −i sgn(ξ), where sgn is the signum function
.
Finally another important example of a multiplier is the characteristic function of the unit ball in ℝn which arises in the study of "partial sums" for the Fourier transform (see Convergence of Fourier series
).
). The general definition is as follows. If
is a sufficiently regular function, let 
denote its Fourier transform (where 
is the Pontryagin dual of G). Let 
denote another function, which we shall call the multiplier. Then the multiplier operator 
associated to this symbol m is defined via the formula
In other words, the Fourier transform of
at a frequency 
is given by the Fourier transform of 
at that frequency, multiplied by the value of the multiplier at that frequency. This explains the terminology "multiplier".
Note that the above definition only defines
implicitly; in order to recover 
explicitly one needs to invert the Fourier transform. This can be easily done if both f and m are sufficiently smooth and integrable. One of the major problems in the subject is to determine, for any specified multiplier m, whether the corresponding Fourier
multiplier operator continues to be well-defined when f has very low regularity, for instance if it is only assumed to lie in an
space. See the discussion on the "boundedness problem" below. As a bare minimum, one usually requires the multiplier m to be bounded and measurable; this is sufficient to establish boundedness on 
but is in general not strong enough to give boundedness on other spaces.
One can view the multiplier operator T as the composition of three operators, namely the Fourier transform, the operation of pointwise multiplication by m, and then the inverse Fourier transform. Equivalently, T is the conjugation of the
pointwise multiplication operator by the Fourier transform. Thus one can think of multiplier operators as operators which are diagonalized by the Fourier transform.
; functions on G can thus be thought of as 
-periodic functions on the real line. In this group, the Pontryagin dual is the group of integers, 
. The Fourier transform (for sufficiently regular functions f) is given by
and the inverse Fourier transform is given by
A multiplier in this setting is simply a sequence
of numbers, and the operator
associated to this multiplier is then given by the formula
at least for sufficiently well-behaved choices of the multiplier
and the function f.
Now let G be a Euclidean space
. Here the dual group is also Euclidean, 
, and the Fourier and inverse Fourier transforms are given by the formulae
A multiplier in this setting is a function
, and the associated multiplier operator 
is defined by
again assuming sufficiently strong regularity and boundedness assumptions on the multiplier and function.
In the sense of distribution
s, there is no difference between multiplier operators and convolution operators; every multiplier T can also be expressed in the form
for some distribution K, known as the convolution kernel of T. In this view, translation by an amount x0 is convolution with a Dirac delta function
δ(· − x0), differentiation is convolution with δ'. Further examples are given in the table below.
.
The following table shows some common examples of multiplier operators on Euclidean space
.
{| border="1" cellpadding="2"
!Name
!Multiplier
!Operator
!Kernel
|-
|Identity operator
|1
|f(x)
|
|-
|Multiplication by a constant c
|c
|cf(x)
|
|-
|Translation by y
|
|f(x − y)
|
|-
|Derivative
(one dimension only)
|
|
|
|-
|Partial derivative
|
|
|
|-
|Laplacian
|
|
|
|-
|Constant coefficient differential operator
|
|
|
|-
|Fractional derivative of order
|
|
|
|-
|Riesz potential
of order
|
|
|
|-
|Bessel potential
of order
|
|
|
|-
|Heat flow operator
|
|
|Heat kernel
|-
|Schrödinger equation
evolution operator
|
|
|Schrödinger kernel
|-
|Hilbert transform
H (one dimension only)
|
|
|
|-
|Riesz transforms Rj
|
|
|
|-
|Partial Fourier integral
(one dimension only)
|
|
|
|-
|Disk multiplier
|
|
|
(J is a Bessel function
)
|-
|Bochner–Riesz operators
|
|
|
|-
|General multiplier
|
|
|
|-
|General convolution operator
|
|
|
|}
is a homomorphism
of C*-algebras. This follows because the sum of two multiplier operators
and 
is a multiplier operators with multiplier 
, the composition of these two multiplier operators is a multiplier operator with multiplier 
, and the adjoint
of a multiplier operator
is another multiplier operator with multiplier 
.
In particular, we see that any two multiplier operators commute with each other. It is known that multiplier operators are translation-invariant. Conversely, one can show that any translation-invariant linear operator which is bounded on
is a multiplier operator.
boundedness problem (for any particular p) for a given group G is, stated simply, to identify the multipliers 
such that the corresponding multiplier operator is bounded from 
to 
. Such multipliers are usually simply referred to as "
multipliers". Note that as multiplier operators are always linear, such operators are bounded if and only if they are continuous
. This problem is considered to be extremely difficult in general, but many special cases can be treated. The problem depends greatly on p, although there is a duality relationship
: if
and 
, then a multiplier operator is bounded on 
if and only if it is bounded on 
.
The Riesz-Thorin theorem
shows that if a multiplier operator is bounded on two different
spaces, then it is also bounded on all intermediate spaces. Hence we get that the space of multipliers is smallest for 
and L∞ and grows as one approaches 
, which has the largest multiplier space.
allows to solve this problem completely and obtain that a function m is an
multiplier if and only if it is bounded and measurable.
(
) case, but is fully resolved. The following is true:
Theorem: In the euclidean space
ℝn, a function
is an 
multiplier (equivalently an 
multiplier) if and only if there exists a finite Borel measure μ such that 
is the Fourier transform of μ.
(The "if" part is a simple calculation. The "only if" part here is more complicated.)
space, the multiplier must be bounded and measurable (this follows from the characterisation of 
multipliers above and the inclusion property). However, this is not sufficient except when 
.
Results that give sufficient conditions for boundedness are known as multiplier theorems. Two such results are given below.
be a bounded function that is continuously differentiable in every set of the form 
for 
and has derivative such that
.
Then m is an
multiplier for all 
.
be a bounded function on ℝn which is smooth except possibly at the origin, and such that the function 
is bounded for all integers 
: then m is an 
multiplier for all 
.
This is a special case of the Hörmander-Mikhlin multiplier theorem.
The proof of these two theorems are fairly tricky, involving techniques from Calderón–Zygmund theory
and the Marcinkiewicz interpolation theorem
: for the original proof, see or .
. Differentiation is not bounded on any 
. The Hilbert transform
is bounded only for p strictly between 1 and ∞. The fact that it is unbounded on L∞ is easy, since it is well known that the Hilbert transform of a step function is unbounded. Duality gives the same for p = 1. However, both the Marcinkiewicz and Mikhlin multiplier theorems show that the Hilbert transform is bounded in
for all 
.
Another interesting case on the unit circle is when the sequence
is constant on the intervals 
and 
. From the Marcinkiewicz multiplier theorem (adapted to the context of the unit circle) we see that any such sequence (bounded, of course) is a multiplier for every 1 < p < ∞.
In one dimension, the disk multiplier operator
is bounded on 
for every 
. However, in 1972, Charles Fefferman
showed the surprising result that in two and higher dimensions the disk multiplier operator
is unbounded on 
for every 
. The corresponding problem for Bochner–Riesz multipliers is only partially solved; see also Bochner–Riesz operator and Bochner–Riesz conjecture.
A final result concerns a random
:
Theorem: Let
be a symbol consisting of independent
variables uniform
on [0,1]. Then almost surely
the multiplier operator corresponding to this symbol is bounded only
.
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of a function by a specified function known as the multiplier or symbol. Occasionally, the term "multiplier operator" itself is shortened simply to "multiplier" . In simple terms, the multiplier reshapes the frequencies involved in any function. This class of operators turns out to be broad: general theory shows that a translation-invariant operator on a group which obeys some (very mild) regularity conditions can be expressed as a multiplier operator, and conversely. Many familiar operators, such as translations and differentiation, are multiplier operators, although there are many more complicated examples such as the Hilbert transform
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
.
In signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, a multiplier operator is called a "filter
Filter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
", and the multiplier is the filter's frequency response
Frequency response
Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...
(or transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
).
In the wider context, multiplier operators are special cases of spectral multiplier operators, which arise from the functional calculus
Functional calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch of the field of functional analysis, connected with spectral theory. In mathematics, a functional calculus is a theory allowing one to apply mathematical...
of an operator (or family of commuting operators). They are also special cases of pseudo-differential operator
Pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory....
s, and more generally Fourier integral operator
Fourier integral operator
In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operators as well as classical integral operators as special cases....
s. There are natural questions in this field that are still open, such as characterizing the
bounded multiplier operators (see below). In this context, multipliers are unrelated to Lagrange multipliers, except for the fact that they both involve the multiplication operation.For the necessary background on the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
, see that page. Additional important background may be found on the pages operator norm
Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
and Lp space
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
.
Examples
In the setting of periodic functions defined on the unit circle, the Fourier transform of a function is simply the sequence of its Fourier coefficients. To see that differentiation can be realized as multiplier, consider the Fourier series for the derivative of a periodic function ƒ(t). After using integration by parts in the definition of the Fourier coefficient we have that.So, formally, it follows that the Fourier series for the derivative is simply in multiplied by the Fourier series for ƒ. This is the same as saying that differentiation is a multiplier operator with multiplier in.
An example of a multiplier operator acting on functions on the real line is the Hilbert transform
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
. It can be shown that the Hilbert transform is a multiplier operator whose multiplier is given by the m(ξ) = −i sgn(ξ), where sgn is the signum function
Sign function
In mathematics, the sign function is an odd mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function ....
.
Finally another important example of a multiplier is the characteristic function of the unit ball in ℝn which arises in the study of "partial sums" for the Fourier transform (see Convergence of Fourier series
Convergence of Fourier series
In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics...
).
Definition
Multiplier operators can be defined on any group G for which the Fourier transform is also defined (in particular, on any locally compact amenable abelian groupAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
). The general definition is as follows. If
is a sufficiently regular function, let denote its Fourier transform (where is the Pontryagin dual of G). Let denote another function, which we shall call the multiplier. Then the multiplier operator associated to this symbol m is defined via the formulaIn other words, the Fourier transform of
at a frequency is given by the Fourier transform of at that frequency, multiplied by the value of the multiplier at that frequency. This explains the terminology "multiplier".Note that the above definition only defines
implicitly; in order to recover explicitly one needs to invert the Fourier transform. This can be easily done if both f and m are sufficiently smooth and integrable. One of the major problems in the subject is to determine, for any specified multiplier m, whether the corresponding Fouriermultiplier operator continues to be well-defined when f has very low regularity, for instance if it is only assumed to lie in an
space. See the discussion on the "boundedness problem" below. As a bare minimum, one usually requires the multiplier m to be bounded and measurable; this is sufficient to establish boundedness on but is in general not strong enough to give boundedness on other spaces.One can view the multiplier operator T as the composition of three operators, namely the Fourier transform, the operation of pointwise multiplication by m, and then the inverse Fourier transform. Equivalently, T is the conjugation of the
pointwise multiplication operator by the Fourier transform. Thus one can think of multiplier operators as operators which are diagonalized by the Fourier transform.
Multiplier operators on common groups
We now specialize the above general definition to specific groups G. First consider the unit circle; functions on G can thus be thought of as -periodic functions on the real line. In this group, the Pontryagin dual is the group of integers, . The Fourier transform (for sufficiently regular functions f) is given byand the inverse Fourier transform is given by
A multiplier in this setting is simply a sequence
of numbers, and the operator associated to this multiplier is then given by the formulaat least for sufficiently well-behaved choices of the multiplier
and the function f.Now let G be a Euclidean space
. Here the dual group is also Euclidean, , and the Fourier and inverse Fourier transforms are given by the formulaeA multiplier in this setting is a function
, and the associated multiplier operator is defined byagain assuming sufficiently strong regularity and boundedness assumptions on the multiplier and function.
In the sense of distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
s, there is no difference between multiplier operators and convolution operators; every multiplier T can also be expressed in the form
for some distribution K, known as the convolution kernel of T. In this view, translation by an amount x0 is convolution with a Dirac delta functionDirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
δ(· − x0), differentiation is convolution with δ'. Further examples are given in the table below.
Further Examples
The following table shows some common examples of multiplier operators on the unit circle.| Name | Multiplier  |
Operator  |
Kernel  |
|---|---|---|---|
| Identity operator | 1 | f(t) | Dirac delta function Dirac delta function The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...  |
| Multiplication by a constant c | c | cf(t) |  |
| Translation by s |  |
f(t − s) |  |
| Differentiation Derivative In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a... |
in | f |
 |
| k-fold differentiation |  |
 |
 |
| Constant coefficient differential operator Differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,... |
 |
 |
 |
Fractional derivative of order  |
|||
| Mean value |  |
 |
1 |
| Mean-free component |  |
 |
 |
| Integration (of mean-free component) |  |
 |
Sawtooth function  |
| Periodic Hilbert transform Hilbert transform In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the... H |
 |
 |
 |
Dirichlet summation  |
 |
 |
Dirichlet kernel  |
Fejér summation  |
Fejér kernel  |
||
| General multiplier |  |
 |
 |
| General convolution Convolution In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation... operator |
 |
 |
 |
The following table shows some common examples of multiplier operators on Euclidean space
.{| border="1" cellpadding="2"
!Name
!Multiplier
!Operator
!Kernel
|-
|Identity operator
|1
|f(x)
|
|-
|Multiplication by a constant c
|c
|cf(x)
|
|-
|Translation by y
|
|f(x − y)
|
|-
|Derivative
(one dimension only)|
|
|
|-
|Partial derivative
|
|
|
|-
|Laplacian
|
|
|
|-
|Constant coefficient differential operator
|
|
|
|-
|Fractional derivative of order
|
|
|
|-
|Riesz potential
Riesz potential
In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space...
of order
|
|
|
|-
|Bessel potential
Bessel potential
In mathematics, the Bessel potential is a potential similar to the Riesz potential but with better decay properties at infinity....
of order
|
|
|
|-
|Heat flow operator
|
|
|Heat kernel
Heat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a particular domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some...
|-
|Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
evolution operator
|
|
|Schrödinger kernel
|-
|Hilbert transform
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
H (one dimension only)
|
|
|
|-
|Riesz transforms Rj
|
|
|
|-
|Partial Fourier integral
(one dimension only)|
|
|
|-
|Disk multiplier
|
|
|
(J is a Bessel functionBessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
)
|-
|Bochner–Riesz operators
|
|
|
|-
|General multiplier
|
|
|
|-
|General convolution operator
|
|
|
|}
General considerations
The map is a homomorphismHomomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
of C*-algebras. This follows because the sum of two multiplier operators
and is a multiplier operators with multiplier , the composition of these two multiplier operators is a multiplier operator with multiplier , and the adjointAdjoint
In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type = .Specifically, adjoint may mean:...
of a multiplier operator
is another multiplier operator with multiplier .In particular, we see that any two multiplier operators commute with each other. It is known that multiplier operators are translation-invariant. Conversely, one can show that any translation-invariant linear operator which is bounded on
is a multiplier operator.The Lp boundedness problem
The boundedness problem (for any particular p) for a given group G is, stated simply, to identify the multipliers such that the corresponding multiplier operator is bounded from to . Such multipliers are usually simply referred to as " multipliers". Note that as multiplier operators are always linear, such operators are bounded if and only if they are continuousContinuous linear operator
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces....
. This problem is considered to be extremely difficult in general, but many special cases can be treated. The problem depends greatly on p, although there is a duality relationship
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
: if
and , then a multiplier operator is bounded on if and only if it is bounded on .The Riesz-Thorin theorem
Riesz-Thorin theorem
In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin.This theorem bounds the norms of linear maps...
shows that if a multiplier operator is bounded on two different
spaces, then it is also bounded on all intermediate spaces. Hence we get that the space of multipliers is smallest for and L∞ and grows as one approaches , which has the largest multiplier space.Boundedness on L2
This is the easiest case. Parseval's theoremParseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...
allows to solve this problem completely and obtain that a function m is an
multiplier if and only if it is bounded and measurable.Boundedness on L1 or L∞
This case is more complicated than the HilbertianHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
(
) case, but is fully resolved. The following is true:Theorem: In the euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
ℝn, a function
is an multiplier (equivalently an multiplier) if and only if there exists a finite Borel measure μ such that is the Fourier transform of μ.(The "if" part is a simple calculation. The "only if" part here is more complicated.)
Boundedness on Lp for 1 < p < ∞
In this general case, necessary and sufficient conditions for boundedness have not been established, even for Euclidean space or the unit circle. However, several necessary conditions and several sufficient conditions are known. For instance it is known that in order for a multiplier operator to be bounded on even a single space, the multiplier must be bounded and measurable (this follows from the characterisation of multipliers above and the inclusion property). However, this is not sufficient except when .Results that give sufficient conditions for boundedness are known as multiplier theorems. Two such results are given below.
Marcinkiewicz multiplier theorem
Let be a bounded function that is continuously differentiable in every set of the form for and has derivative such that.Then m is an
multiplier for all .Mikhlin multiplier theorem
Let be a bounded function on ℝn which is smooth except possibly at the origin, and such that the function is bounded for all integers : then m is an multiplier for all .This is a special case of the Hörmander-Mikhlin multiplier theorem.
The proof of these two theorems are fairly tricky, involving techniques from Calderón–Zygmund theory
and the Marcinkiewicz interpolation theorem
Marcinkiewicz theorem
In mathematics, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on Lp spaces....
: for the original proof, see or .
Examples
Translations are bounded operators on any. Differentiation is not bounded on any . The Hilbert transformHilbert transform
In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
is bounded only for p strictly between 1 and ∞. The fact that it is unbounded on L∞ is easy, since it is well known that the Hilbert transform of a step function is unbounded. Duality gives the same for p = 1. However, both the Marcinkiewicz and Mikhlin multiplier theorems show that the Hilbert transform is bounded in
for all .Another interesting case on the unit circle is when the sequence
is constant on the intervals and . From the Marcinkiewicz multiplier theorem (adapted to the context of the unit circle) we see that any such sequence (bounded, of course) is a multiplier for every 1 < p < ∞.In one dimension, the disk multiplier operator
is bounded on for every . However, in 1972, Charles FeffermanCharles Fefferman
Charles Louis Fefferman is an American mathematician at Princeton University. His primary field of research is mathematical analysis....
showed the surprising result that in two and higher dimensions the disk multiplier operator
is unbounded on for every . The corresponding problem for Bochner–Riesz multipliers is only partially solved; see also Bochner–Riesz operator and Bochner–Riesz conjecture.A final result concerns a random
:Theorem: Let
be a symbol consisting of independentStatistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
variables uniform
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
on [0,1]. Then almost surely
Almost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
the multiplier operator corresponding to this symbol is bounded only
.See also
- Calderón–Zygmund lemma
- Lars HörmanderLars HörmanderLars Valter Hörmander is a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". He was awarded the Fields Medal in 1962, and the Wolf Prize in 1988...
- Marcinkiewicz theoremMarcinkiewicz theoremIn mathematics, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on Lp spaces....
- Solomon MikhlinSolomon MikhlinSolomon Grigor'evich Mikhlin was a Soviet mathematician of Jewish origin, working in the fields of linear elasticity, singular integrals and numerical analysis: he is best known for the introduction of the concept of "symbol of a singular integral operator", which eventually led to...
- Singular integrals