DLMF:1.16.E36 (Q642)

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DLMF:1.16.E36
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    Statements

    Defining formula
    ⟨ ℱ ⁡ ( P ⁢ ( 𝐃 ) ⁢ u ) , ϕ ⟩ = ⟨ P - ⁢ ℱ ⁡ ( u ) , ϕ ⟩ = ⟨ ℱ ⁡ ( u ) , P - ⁢ ϕ ⟩ , distribution distributional-Fourier-transform 𝑃 𝐃 𝑢 italic-ϕ distribution subscript 𝑃 distributional-Fourier-transform 𝑢 italic-ϕ distribution distributional-Fourier-transform 𝑢 subscript 𝑃 italic-ϕ {\displaystyle{\displaystyle\left\langle\mathscr{F}\left(P(\mathbf{D})u\right)% ,\phi\right\rangle=\left\langle P_{-}\mathscr{F}\left(u\right),\phi\right% \rangle=\left\langle\mathscr{F}\left(u\right),P_{-}\phi\right\rangle,}}
    0 references
    DLMF id
    DLMF:1.16.E36
    0 references
    Symbols used
    Q10905
    Defining formula
    ⟨ Λ , ϕ ⟩ distribution Λ italic-ϕ {\displaystyle{\displaystyle\left\langle\NVar{\Lambda},\NVar{\phi}\right% \rangle}}
    xml-id
    C1.S16.SS1.p5.m5andec
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    Fourier transform of a tempered distribution
    Defining formula
    ℱ ⁡ ( u ) distributional-Fourier-transform 𝑢 {\displaystyle{\displaystyle\mathscr{F}\left(\NVar{u}\right)}}
    xml-id
    C1.S16.E35.m2aadec
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    Q10912
    Defining formula
    ϕ ⁢ ( x 1 , x 2 , … , x n ) italic-ϕ subscript 𝑥 1 subscript 𝑥 2 … subscript 𝑥 𝑛 {\displaystyle{\displaystyle\phi(x_{1},x_{2},\dots,x_{n})}}
    xml-id
    C1.S16.XMD13.m1gdec
    0 references
    vector differential operator (locally)
    Defining formula
    𝐃 𝐃 {\displaystyle{\displaystyle\mathbf{D}}}
    xml-id
    C1.S16.XMD16.m1ddec
    0 references
    Q10913
    Defining formula
    P 𝑃 {\displaystyle{\displaystyle P}}
    xml-id
    C1.S16.XMD17.m1ddec
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    Q10909
    Defining formula
    D ⁢ f 𝐷 𝑓 {\displaystyle{\displaystyle Df}}
    xml-id
    C1.S16.XMD6.m1kdec
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