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Doubts about Fubini
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I am reading about Fourier Transform on $S(mathbbR)$ to have a first formal approach. The book I read is 'Fourier Analysis' by Rami Shakarchi and Elias.M.Stein. In the chapter about Fourier transform on $mathbbR$ I found that the great problem for all the main results is Fubini for improper integrals. I know that, at the end, there is a general Fubini theorem with really weak hypothesis called 'Fubini-Tonelli' so Fubini in general is not a problem. However, at this stage in the theory the book only uses Riemman Integrals and does not have a good Fubini Theorem. The book proves Fubini Theorem for continuos functions on a compact and after that extends it to improper integrals of functions with an adequate decay at infinity. I am really down with this because the proof is really hard and the hypothesis very strong, I can not imagine how difficult is the proof for Fubini Tonelli. Asking a friend of mine about it tell me that indeed Fubini TOnelli is not that hard because the proof is not analytic (as the proof for improper integrals of the book). So I do not understand why the book does not offer a non-analytic proof for improper Riemman Integrals because the actual Fubini is really terrible in the sense that the proof are horrible inequalities and what is more, given a function $F(x,y)$ it is difficult to check the hypothesis of work. For example I need to check the hypothesis to the function $F(x,y) = f(x)g(x-y)$ with $f,g in S(mathbbR)$. The hypothesis I need to check is that $|F(x,y)|leq fracA1+ (x^2 + y^2)^3/2$ I am not seeing how to get it, is the last inequality I need to check to end the use of Fubini in this chapter and I would be really grateful if someone could help me
analysis fourier-analysis fourier-transform
asked Jul 21 at 3:39
Ale.B
628
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up vote
0
down vote
favorite
I am reading about Fourier Transform on $S(mathbbR)$ to have a first formal approach. The book I read is 'Fourier Analysis' by Rami Shakarchi and Elias.M.Stein. In the chapter about Fourier transform on $mathbbR$ I found that the great problem for all the main results is Fubini for improper integrals. I know that, at the end, there is a general Fubini theorem with really weak hypothesis called 'Fubini-Tonelli' so Fubini in general is not a problem. However, at this stage in the theory the book only uses Riemman Integrals and does not have a good Fubini Theorem. The book proves Fubini Theorem for continuos functions on a compact and after that extends it to improper integrals of functions with an adequate decay at infinity. I am really down with this because the proof is really hard and the hypothesis very strong, I can not imagine how difficult is the proof for Fubini Tonelli. Asking a friend of mine about it tell me that indeed Fubini TOnelli is not that hard because the proof is not analytic (as the proof for improper integrals of the book). So I do not understand why the book does not offer a non-analytic proof for improper Riemman Integrals because the actual Fubini is really terrible in the sense that the proof are horrible inequalities and what is more, given a function $F(x,y)$ it is difficult to check the hypothesis of work. For example I need to check the hypothesis to the function $F(x,y) = f(x)g(x-y)$ with $f,g in S(mathbbR)$. The hypothesis I need to check is that $|F(x,y)|leq fracA1+ (x^2 + y^2)^3/2$ I am not seeing how to get it, is the last inequality I need to check to end the use of Fubini in this chapter and I would be really grateful if someone could help me
analysis fourier-analysis fourier-transform
asked Jul 21 at 3:39
Ale.B
628
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am reading about Fourier Transform on $S(mathbbR)$ to have a first formal approach. The book I read is 'Fourier Analysis' by Rami Shakarchi and Elias.M.Stein. In the chapter about Fourier transform on $mathbbR$ I found that the great problem for all the main results is Fubini for improper integrals. I know that, at the end, there is a general Fubini theorem with really weak hypothesis called 'Fubini-Tonelli' so Fubini in general is not a problem. However, at this stage in the theory the book only uses Riemman Integrals and does not have a good Fubini Theorem. The book proves Fubini Theorem for continuos functions on a compact and after that extends it to improper integrals of functions with an adequate decay at infinity. I am really down with this because the proof is really hard and the hypothesis very strong, I can not imagine how difficult is the proof for Fubini Tonelli. Asking a friend of mine about it tell me that indeed Fubini TOnelli is not that hard because the proof is not analytic (as the proof for improper integrals of the book). So I do not understand why the book does not offer a non-analytic proof for improper Riemman Integrals because the actual Fubini is really terrible in the sense that the proof are horrible inequalities and what is more, given a function $F(x,y)$ it is difficult to check the hypothesis of work. For example I need to check the hypothesis to the function $F(x,y) = f(x)g(x-y)$ with $f,g in S(mathbbR)$. The hypothesis I need to check is that $|F(x,y)|leq fracA1+ (x^2 + y^2)^3/2$ I am not seeing how to get it, is the last inequality I need to check to end the use of Fubini in this chapter and I would be really grateful if someone could help me
analysis fourier-analysis fourier-transform
asked Jul 21 at 3:39
Ale.B
628
I am reading about Fourier Transform on $S(mathbbR)$ to have a first formal approach. The book I read is 'Fourier Analysis' by Rami Shakarchi and Elias.M.Stein. In the chapter about Fourier transform on $mathbbR$ I found that the great problem for all the main results is Fubini for improper integrals. I know that, at the end, there is a general Fubini theorem with really weak hypothesis called 'Fubini-Tonelli' so Fubini in general is not a problem. However, at this stage in the theory the book only uses Riemman Integrals and does not have a good Fubini Theorem. The book proves Fubini Theorem for continuos functions on a compact and after that extends it to improper integrals of functions with an adequate decay at infinity. I am really down with this because the proof is really hard and the hypothesis very strong, I can not imagine how difficult is the proof for Fubini Tonelli. Asking a friend of mine about it tell me that indeed Fubini TOnelli is not that hard because the proof is not analytic (as the proof for improper integrals of the book). So I do not understand why the book does not offer a non-analytic proof for improper Riemman Integrals because the actual Fubini is really terrible in the sense that the proof are horrible inequalities and what is more, given a function $F(x,y)$ it is difficult to check the hypothesis of work. For example I need to check the hypothesis to the function $F(x,y) = f(x)g(x-y)$ with $f,g in S(mathbbR)$. The hypothesis I need to check is that $|F(x,y)|leq fracA1+ (x^2 + y^2)^3/2$ I am not seeing how to get it, is the last inequality I need to check to end the use of Fubini in this chapter and I would be really grateful if someone could help me
analysis fourier-analysis fourier-transform
asked Jul 21 at 3:39
Ale.B
628
asked Jul 21 at 3:39
Ale.B
628
asked Jul 21 at 3:39
Ale.B
628
asked Jul 21 at 3:39
Ale.B
628
628
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