Let f: R R be a smooth function such that for all are f(0) we...

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Let f: R" → R be a smooth function such that for all are f¹(0) we have Vf(x) = grad f(x) = 0. Show that the the oscillatory integral = [efox d£ U= can be regularized, and describe and what sense it equals 276(f(x)), where & is the Dirac delta function viewed as a distribution. Hint: If € C(R") a compactly supported smooth test function (meaning it is smooth and its derivatives decay faster than any polynomial), then upon regularization (u, o) is given by an absolutely convergent double integral. Moreover, in a neighborhood of a point zo E (0), you can assume WLOG that , f(ro)0, then change variables to (f(x),x₂,...). Let f: R" → R be a smooth function such that for all are f¹(0) we have Vf(x) = grad f(x) = 0. Show that the the oscillatory integral = [efox d£ U= can be regularized, and describe and what sense it equals 276(f(x)), where & is the Dirac delta function viewed as a distribution. Hint: If € C(R") a compactly supported smooth test function (meaning it is smooth and its derivatives decay faster than any polynomial), then upon regularization (u, o) is given by an absolutely convergent double integral. Moreover, in a neighborhood of a point zo E (0), you can assume WLOG that , f(ro)0, then change variables to (f(x),x₂,...).

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