Return to Question

I can't understand a lemma in "An introduction to harmonic analysis" (http://www.mat.uniroma2.it/~picard/SMC/didattica/materiali_did/Anal.Armon./Katznelson/Katznelson.pdf) by Yitzhak Katznelson which is stated as follows:

Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\mathbb T}|\mu(\{\tau\})|^2=\lim\limits_{N\rightarrow\infty }\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|^2.$$

It looks bizzare to me since the left hand sum up all variable $\tau$, while the other side not. What I only know is that $\mu(\{\tau\})=\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|,$ but it seems impossible to derive the corollary from this relation.

(note added by YC: the result is stated without proof at the end of Chapter I Section 7.11 in the 1976 Dover edition)

I can't understand a lemma in "An introduction to harmonic analysis" by Yitzhak Katznelson which is stated as follows:

Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\mathbb T}|\mu(\{\tau\})|^2=\lim\limits_{N\rightarrow\infty }\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|^2.$$

It looks bizzare to me since the left hand sum up all variable $\tau$, while the other side not. What I only know is that $\mu(\{\tau\})=\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|,$ but it seems impossible to derive the corollary from this relation.

(note added by YC: the result is stated without proof at the end of Chapter I Section 7.11 in the 1976 Dover edition)

I can't understand a lemma in "An introduction to harmonic analysis" (http://www.mat.uniroma2.it/~picard/SMC/didattica/materiali_did/Anal.Armon./Katznelson/Katznelson.pdf) by Yitzhak Katznelson which is stated as follows:

Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\mathbb T}|\mu(\{\tau\})|^2=\lim\limits_{N\rightarrow\infty }\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|^2.$$

It looks bizzare to me since the left hand sum up all variable $\tau$, while the other side not. What I only know is that $\mu(\{\tau\})=\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|,$ but it seems impossible to derive the corollary from this relation.

I can't understand a lemma in "An introduction to harmonic analysis" (http://www.mat.uniroma2.it/~picard/SMC/didattica/materiali_did/Anal.Armon./Katznelson/Katznelson.pdf) by Yitzhak Katznelson which is stated as follows:

Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\mathbb T}|\mu(\{\tau\})|^2=\lim\limits_{N\rightarrow\infty }\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|^2.$$

It looks bizzare to me since the left hand sum up all variable $\tau$, while the other side not. What I only know is that $\mu(\{\tau\})=\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|,$ but it seems impossible to derive the corollary from this relation.

(note added by YC: the result is stated without proof at the end of Chapter I Section 7.11 in the 1976 Dover edition)

I can't understand a lemma in "An introduction to harmonic analysis" which is stated as follows:

Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\mathbb T}|\mu(\{\tau\})|^2=\lim\limits_{N\rightarrow\infty }\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|^2.$$

It looks bizzare to me since the left hand sum up all variable $\tau$, while the other side not. What I only know is that $\mu(\{\tau\})=\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|,$ but it seems impossible to derive the corollary from this relation.

I can't understand a lemma in "An introduction to harmonic analysis" (http://www.mat.uniroma2.it/~picard/SMC/didattica/materiali_did/Anal.Armon./Katznelson/Katznelson.pdf) by Yitzhak Katznelson which is stated as follows:

Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\mathbb T}|\mu(\{\tau\})|^2=\lim\limits_{N\rightarrow\infty }\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|^2.$$

It looks bizzare to me since the left hand sum up all variable $\tau$, while the other side not. What I only know is that $\mu(\{\tau\})=\frac{1}{2N+1}\sum\limits_{-N}^N|\hat\mu(n)|,$ but it seems impossible to derive the corollary from this relation.