About: Multiplier (Fourier analysis)

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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 a

Attributes	Values
rdf:type	company
rdfs:label	Multiplier (Fourier analysis) (en)
rdfs:comment	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 a (en)
foaf:depiction
dcterms:subject	Fourier analysis
Wikipage page ID	995746 (xsd:integer)
Wikipage revision ID	1044467147 (xsd:integer)
Link from a Wikipage to another Wikipage	Calderón–Zygmund lemma Pseudo-differential operator Schrödinger equation Convolution operator Bessel function Derivative Fourier analysis Andreas Seeger Homomorphism Bessel potential Riesz potential Riesz transform Unit circle Continuous linear operator Convolution Measurable Russian language Operator norm Radial function Convergence of Fourier series Linear operator Lp space Bochner–Riesz conjecture Sign function Signal processing Singular integral operators of convolution type Fedor Nazarov Fejér kernel Functional calculus Pergamon Press Adjoint Locally compact abelian group Dual space Euclidean space Fourier analysis Fourier coefficient Fourier transform Parseval's theorem Differential operator Dirichlet kernel Fourier integral operator Differentiation (mathematics) Regular function Group (mathematics) Heat kernel Hilbert space Hilbert transform Riesz-Thorin theorem Charles Fefferman Lagrange multiplier Laplacian Transfer function Translation Dirac delta function Distribution (mathematics) Borel measure C*-algebra Filter (signal processing) Frequency response Continuously differentiable Doklady Akademii Nauk SSSR Indicator function Integration by parts Marcinkiewicz theorem Convolution kernel Periodic function Sawtooth function Bochner–Riesz operator Fractional derivative Mathematical function Singular integrals Pontryagin dual Commutative operation
sameAs	Multiplier (Fourier analysis) Multiplier (Fourier analysis) Multiplier (Fourier analysis)
dbp:wikiPageUsesTemplate	dbt:= dbt:Citation dbt:Clarify dbt:Harvtxt dbt:More_footnotes dbt:Reflist
thumbnail	wiki-commons:Special:FilePath/Fourier_multiplier_diagram.png?width=300
date	February 2017 (en)
reason	And are jump discontinuities at 2^ allowed? (en) Hope this is right. (en)
has abstract	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 operators, and more generally Fourier integral operators. There are natural questions in this field that are still open, such as characterizing the Lp bounded multiplier operators (see below). Multiplier operators are unrelated to Lagrange multipliers, except 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. (en)
gold:hypernym	Operator
prov:wasDerivedFrom	wikipedia-en:Multiplier_(Fourier_analysis)?oldid=1044467147&ns=0
page length (characters) of wiki page	25044 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Multiplier_(Fourier_analysis)
is rdfs:seeAlso of	Singular integral operators of convolution type

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