Let f: R R be a smooth function such that for all are f(0) we...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
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,...).
Expert Answer:
Related Book For
Posted Date:
Students also viewed these mathematics questions
-
help pls brothers asap 30 min left onlyyyy
-
4.1 For Youroil field the PVT data is given in Table 2 below. Calculate the recovery factor at the bubble point. Comment on the error you would make if you were to neglect water and formation...
-
1.2 The following data are read from a contour map of a gas reservoir with a gas specific gravity of 0.667. It shows area as a function of the depth of a production zone. Calculate the reservoir...
-
Why is it important to have a defined project scope? Why is it important to make sure there is agreement about the scope? Is there anything in the "Why Should You Use the WBS?
-
What controls can you identify in the management of your school or at a company where you now work (or recently worked)? If you can, interview a manager or employee of the organization to learn more...
-
What is public relations? Who can an organization reach through public relations?
-
Some corporate executives have reportedly attempted to manage their firms operating expenses through such actions as front-end loading or rear-end loading of their operating expenses. Explain the...
-
As noted in this chapter, because of global competition, companies have become increasingly focused on reducing costs. To reduce costs and remain competitive, many companies are turning to...
-
b) Formulate the element equations for axisymetric element shown in fig. Z r
-
Cecil C. Seymour is a 64-year-old widower. He had income for 2022 as follows: The cost of the annuity was $46,800, and Cecil was expected to receive a total of 260 monthly payments of $450. Cecil has...
-
Data on the weights (lb) of the contents of cans of diet soda versus the contents of cans of the regular version of the soda is summarized to the right. Assume that the two samples are independent...
-
April and Andy were married on May 16, 1998. They had two children. Noah Andrew was born on July 19, 1999. Emma Katherine was born on July 30, 2000 . . . In the spring of 2003, April and Andy...
-
The record supports the following facts. The petitioner filed for divorce from the respondent on grounds of irreconcilable differences. He subsequently moved to amend the petition to assert the fault...
-
What is the difference between physical and legal custody? Between sole and joint custody?
-
List some BOs for this term.
-
Explain the ALI approximation rule. How does it respond to the concerns about the best interest standard?
-
What would be good ideas for an app that could aid in teaching and/or learning?
-
Explain briefly what is meant by electronic data interchange (EDI). How does EDI affect a companys audit trail?
-
In contrast to the situation of Exercise 40, experiments show that the reaction H2 + Br2 2HBr satisfies the rate law D[HBr/dt] = k[H2][Br2]1/2 And so for this reaction the differential equation...
-
Let Find the values of m and that make f differentiable everywhere. f(x) = mx + b if x> 2
-
Find the value of c if (1 + c) - n = 2
-
Describe and explain how a firm can increase its profits through price discrimination.
-
What is a monopoly?
-
How does the demand curve for a monopolist differ from that for a perfectly competitive firm?
Study smarter with the SolutionInn App