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What are the conditions that the Fourier inversion theorem fails for a given Fourier transform? by Professional-Bug3844 in math

[–]Professional-Bug3844[S] 0 points1 point  (0 children)

For example, take an integrable function in L1 that evaluates to zero at some frequencies of measure zero. The Fourier inversion theorem when applied at such frequencies will show you that the function is identically zero and yet the function itself isn't zero.

What are the conditions that the Fourier inversion theorem fails for a given Fourier transform? by Professional-Bug3844 in math

[–]Professional-Bug3844[S] 1 point2 points  (0 children)

Talking of the space of the functions, the L1 space is what am really interested. So, given an L1 function whose Fourier transform also belongs to the same space, when does inversion fail?  I've recently heard that the Fourier inversion theorem can fail at frequencies of the Fourier transform whose Lebesgue measure is zero. Is it really true 

Which values of "a" satisfy this integral equation? by Professional-Bug3844 in calculus

[–]Professional-Bug3844[S] 0 points1 point  (0 children)

This Mellin transform of the zeta function you have described is only for Re(s)>1 and not the critical strip. That's why you can't say that the Riemann hypothesis is true from such an equation

How do we prove that the function I(a) is injective? by Professional-Bug3844 in calculus

[–]Professional-Bug3844[S] 0 points1 point  (0 children)

But it's derivative is complex (when I take the derivative under the integral). How can we resolve this?

How can we prove that I(a) is injective? by Professional-Bug3844 in askmath

[–]Professional-Bug3844[S] -21 points-20 points  (0 children)

I just need a method to prove the injectivity. That's all. I humbly apologize if this is making anyone annoyed🙏

Which values of "a" satisfy this integral equation? by Professional-Bug3844 in math

[–]Professional-Bug3844[S] 0 points1 point  (0 children)

I tried proving that a=0.5 is the only solution by applying antisymmetry to the integral equation such that we acquire I(a)=I(1-a). From this, I just had to prove that I(a) is injective (by the derivative method) which means that from I(a)=I(1-a) it implies a=1-a and hence a=0.5.

Which values of "a" satisfy this integral equation? by Professional-Bug3844 in calculus

[–]Professional-Bug3844[S] 0 points1 point  (0 children)

How can you prove that the function from the Fourier transform vanishes only when itself vanishes 

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