Greetings.

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I think the link to the download is broken. I get an error message from Aalto Uni.

I liked your previous notes, but would really enjoy having a revised PDF version.

Cheers

]]>How to prove that “any function {\phi} which is positive and radially decreasing can be approximated monotonically from below by a sequence of simple functions of the form {\sum a_j \chi_{B_j}}.

Thank you very much

]]>You’re right, the Fourier transform of a finite Borel measure can be defined directly, without appealing to the theory of distributions. As you commented, one thinks of an function as the density of the Borel measure . Replace this by and you have a perfectly meaningful definition of the Fourier transform on the class of finite Borel measures. One can quite easily check that this definition coincides with the definition given by distribution theory. Indeed we have (by definition)

I can’t see the connection with its transform viewed as a tempered distribution. Should it be the same or the definition is made by analogy to functions thinking as the measure’s density?

Great post btw. ]]>

My question was general, I did not assume that the function is compactly supported, only that it is square integrable on R.

Yet, if it is simpler, let us assume this hypothesis at first. ]]>

to be honest I don’t know of this specific result from the top of my head. I would try to argue via the Fourier transform of H(f) though. I will try to come back with a more precise answer. To clear out one thing, do you mean that your function is also compactly supported on [-a,a]?

yannis

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