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nLab > Latest Changes: product of distributions

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 2nd 2017

    I gave product of distributions its own entry. For the moment it just points to the definition in Hörmander’s book.

    This should eventually supercede the section “Multiplication of distributions” at distributions, which I find suboptimal: that section starts very vaguely referring to physics as if the issue only appears there, and it keeps being very vague, with its three sub-subsections being little more than a pointer to one reference by Colombeau.

    I suggest to

    1. remove that whole subsection at distribution and leave just a pointer to product of distributions

    2. move the mentioning of Colombeau’s reference to product of distributions and say how it relates to Hörmander’s definition

    3. remove all vague mentioning of application in physics and instead add a pointer to Wick algebra and microcausal functional, which I will create shortly.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2017
    • (edited Aug 7th 2017)

    Since nobody objected to the suggestion #1, I went ahead and implemented it. Details on what I did are here.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2017
    • (edited Aug 7th 2017)

    I have spelled out the definition of the product of distributions with compatible wave front sets (from Hörmander 90) and the proof that this is well defined: here.

    That proof however relies on a little hierarchy of lemmas. I have added statement (not the proof yet, but pointers to Hörmander’s proofs) of the next lower lemmas at tensor product of distributions and at pullback of distributions.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 1st 2017

    I have given product of distributions an Idea-section (here) which means to explain, informally but accurately, how the product is defined via convolution of Fourier transforms “around any point” and how that immediately implies Hörmander’s condition on the wave front sets of the two factors.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2017

    for ease of linking, I gave Hörmander’s criterion a little entry of its own.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 27th 2017
    • (edited Dec 28th 2017)

    added statement and proof of the estimate of WF(u⋅v)WF(u\cdot v) (here)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2018

    I have added the statement for partial product of distributions of several variables, and composition of integral kernels: here

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2018
    • (edited Jan 16th 2018)

    Did some fixing. I hope now I have the signs right (still here):

    WF(K 1∘K 2)⊂{(x,z,k x,k z)|((x,y,k x,−k y)∈WF(K 1)and(y,z,k y,k z)∈WF(K 2)) or (k x=0and(y,z,0,−k z)∈WF(K 2)) or (k z=0and(x,y,k x,0)∈WF(K 1))} WF(K_1 \circ K_2) \;\subset\; \left\{ (x,z, k_x, k_z) \;\vert\; \array{ \left( (x,y,k_x,-k_y) \in WF(K_1) \,\, \text{and} \,\, (y,z,k_y, k_z) \in WF(K_2) \right) \\ \text{or} \\ \left( k_x = 0 \,\text{and}\, (y,z,0,-k_z) \in WF(K_2) \right) \\ \text{or} \\ \left( k_z = 0 \,\text{and}\, (x,y,k_x,0) \in WF(K_1) \right) } \right\}

1 to 8 of 8

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