What is the difference working in $L^2$ and $C^2 $ for Fourier series expansion of functions?

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Once one of my lecturer said that if we worked in $L_2$, the complex inner product $$(u,v)=displaystyleint^L_0 u bar v dx $$ works flawless for Fourier series, instead of working in $C^2$.



Question: I do't see the general picture that what is the problem with working in $C^2$. I am taking differential equations under Mathematical Methods in Physics I (in my Physics department), so there is almost no rigor in our lectures, I want to learn all the underlying idea Fourier series and this mentioned idea between $C^2$ and $L^2$. Any idea, comment, source, guidance, will be appreciated.




differential-equations soft-question fourier-analysis online-resources




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  • One book I liked a lot for learning a bit more rigor behind all of the stuff I learned in Math methods was "Div, Grad, Curl, and all That"
    – Rushabh Mehta
    Aug 1 at 15:26










  • link
    – Rushabh Mehta
    Aug 1 at 15:26










  • math.stackexchange.com/questions/2621446/…
    – Dionel Jaime
    Aug 1 at 15:31






  • 2




    Also I'm slightly confused on your question. I could be wrong but $mathbbC^2$ is not the usual domain of fourier series for physicist. Are you sure you don't mean $C^2$, the set of twice continuously differentiable functions?
    – Dionel Jaime
    Aug 1 at 15:36










  • I meant C^2 and wrote but someone changed editing
    – user2312512851
    Aug 1 at 15:58














up vote
0
down vote

favorite












Once one of my lecturer said that if we worked in $L_2$, the complex inner product $$(u,v)=displaystyleint^L_0 u bar v dx $$ works flawless for Fourier series, instead of working in $C^2$.



Question: I do't see the general picture that what is the problem with working in $C^2$. I am taking differential equations under Mathematical Methods in Physics I (in my Physics department), so there is almost no rigor in our lectures, I want to learn all the underlying idea Fourier series and this mentioned idea between $C^2$ and $L^2$. Any idea, comment, source, guidance, will be appreciated.




differential-equations soft-question fourier-analysis online-resources




share|cite|improve this question





















  • One book I liked a lot for learning a bit more rigor behind all of the stuff I learned in Math methods was "Div, Grad, Curl, and all That"
    – Rushabh Mehta
    Aug 1 at 15:26










  • link
    – Rushabh Mehta
    Aug 1 at 15:26










  • math.stackexchange.com/questions/2621446/…
    – Dionel Jaime
    Aug 1 at 15:31






  • 2




    Also I'm slightly confused on your question. I could be wrong but $mathbbC^2$ is not the usual domain of fourier series for physicist. Are you sure you don't mean $C^2$, the set of twice continuously differentiable functions?
    – Dionel Jaime
    Aug 1 at 15:36










  • I meant C^2 and wrote but someone changed editing
    – user2312512851
    Aug 1 at 15:58












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Once one of my lecturer said that if we worked in $L_2$, the complex inner product $$(u,v)=displaystyleint^L_0 u bar v dx $$ works flawless for Fourier series, instead of working in $C^2$.



Question: I do't see the general picture that what is the problem with working in $C^2$. I am taking differential equations under Mathematical Methods in Physics I (in my Physics department), so there is almost no rigor in our lectures, I want to learn all the underlying idea Fourier series and this mentioned idea between $C^2$ and $L^2$. Any idea, comment, source, guidance, will be appreciated.




differential-equations soft-question fourier-analysis online-resources




share|cite|improve this question













Once one of my lecturer said that if we worked in $L_2$, the complex inner product $$(u,v)=displaystyleint^L_0 u bar v dx $$ works flawless for Fourier series, instead of working in $C^2$.



Question: I do't see the general picture that what is the problem with working in $C^2$. I am taking differential equations under Mathematical Methods in Physics I (in my Physics department), so there is almost no rigor in our lectures, I want to learn all the underlying idea Fourier series and this mentioned idea between $C^2$ and $L^2$. Any idea, comment, source, guidance, will be appreciated.





differential-equations soft-question fourier-analysis online-resources





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 3 at 11:05
























asked Aug 1 at 15:11









user2312512851

1,134520




1,134520











  • One book I liked a lot for learning a bit more rigor behind all of the stuff I learned in Math methods was "Div, Grad, Curl, and all That"
    – Rushabh Mehta
    Aug 1 at 15:26










  • link
    – Rushabh Mehta
    Aug 1 at 15:26










  • math.stackexchange.com/questions/2621446/…
    – Dionel Jaime
    Aug 1 at 15:31






  • 2




    Also I'm slightly confused on your question. I could be wrong but $mathbbC^2$ is not the usual domain of fourier series for physicist. Are you sure you don't mean $C^2$, the set of twice continuously differentiable functions?
    – Dionel Jaime
    Aug 1 at 15:36










  • I meant C^2 and wrote but someone changed editing
    – user2312512851
    Aug 1 at 15:58
















  • One book I liked a lot for learning a bit more rigor behind all of the stuff I learned in Math methods was "Div, Grad, Curl, and all That"
    – Rushabh Mehta
    Aug 1 at 15:26










  • link
    – Rushabh Mehta
    Aug 1 at 15:26










  • math.stackexchange.com/questions/2621446/…
    – Dionel Jaime
    Aug 1 at 15:31






  • 2




    Also I'm slightly confused on your question. I could be wrong but $mathbbC^2$ is not the usual domain of fourier series for physicist. Are you sure you don't mean $C^2$, the set of twice continuously differentiable functions?
    – Dionel Jaime
    Aug 1 at 15:36










  • I meant C^2 and wrote but someone changed editing
    – user2312512851
    Aug 1 at 15:58















One book I liked a lot for learning a bit more rigor behind all of the stuff I learned in Math methods was "Div, Grad, Curl, and all That"
– Rushabh Mehta
Aug 1 at 15:26




One book I liked a lot for learning a bit more rigor behind all of the stuff I learned in Math methods was "Div, Grad, Curl, and all That"
– Rushabh Mehta
Aug 1 at 15:26












link
– Rushabh Mehta
Aug 1 at 15:26




link
– Rushabh Mehta
Aug 1 at 15:26












math.stackexchange.com/questions/2621446/…
– Dionel Jaime
Aug 1 at 15:31




math.stackexchange.com/questions/2621446/…
– Dionel Jaime
Aug 1 at 15:31




2




2




Also I'm slightly confused on your question. I could be wrong but $mathbbC^2$ is not the usual domain of fourier series for physicist. Are you sure you don't mean $C^2$, the set of twice continuously differentiable functions?
– Dionel Jaime
Aug 1 at 15:36




Also I'm slightly confused on your question. I could be wrong but $mathbbC^2$ is not the usual domain of fourier series for physicist. Are you sure you don't mean $C^2$, the set of twice continuously differentiable functions?
– Dionel Jaime
Aug 1 at 15:36












I meant C^2 and wrote but someone changed editing
– user2312512851
Aug 1 at 15:58




I meant C^2 and wrote but someone changed editing
– user2312512851
Aug 1 at 15:58










1 Answer
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3
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Your lecturer probably meant that the vector space $C^2([0,L])$, consisting of functions that are twice differentiable with continuous derivatives, is not complete if equipped with the scalar product
$$
langle f| grangle= int_0^L f(x)overlineg(x), dx.$$
Among the various unpleasant consequences of this, there's the Fourier series fact that if you take an arbitrary series
$$
S(x)=sum_k=-infty^infty c_k e^ifrac2piLkx $$
then, even if $sum_k=-infty^infty |c_k|^2<infty $ (corresponding to the physical fact that $S$ has finite energy), you cannot conclude that $Sin C^2$. (In fact, it is not trivial to find conditions on $c_k$ that ensure $Sin C^2$. I don't think that a necessary and sufficient condition is known.)



On the other hand, the vector space $L^2([0, L])$, consisting of all measurable functions $f$ such that $int_0^L |f(x)|^2, dx<infty$, is complete if equipped with the aforementioned scalar product $langlecdot|cdotrangle$. As a consequence of this, the finite energy Fourier series $S(x)$ always defines an element of $L^2([0, L])$. (Actually, all elements of $L^2([0, L])$ can be decomposed in a finite energy Fourier series.)



A good reference to be introduced to these things is the book by Stein and Shakarchi "Fourier analysis: an introduction", first volume.






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    You might mention explicitly that otoh it does follow that $Sin L^2$.
    – David C. Ullrich
    Aug 1 at 17:01










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Your lecturer probably meant that the vector space $C^2([0,L])$, consisting of functions that are twice differentiable with continuous derivatives, is not complete if equipped with the scalar product
$$
langle f| grangle= int_0^L f(x)overlineg(x), dx.$$
Among the various unpleasant consequences of this, there's the Fourier series fact that if you take an arbitrary series
$$
S(x)=sum_k=-infty^infty c_k e^ifrac2piLkx $$
then, even if $sum_k=-infty^infty |c_k|^2<infty $ (corresponding to the physical fact that $S$ has finite energy), you cannot conclude that $Sin C^2$. (In fact, it is not trivial to find conditions on $c_k$ that ensure $Sin C^2$. I don't think that a necessary and sufficient condition is known.)



On the other hand, the vector space $L^2([0, L])$, consisting of all measurable functions $f$ such that $int_0^L |f(x)|^2, dx<infty$, is complete if equipped with the aforementioned scalar product $langlecdot|cdotrangle$. As a consequence of this, the finite energy Fourier series $S(x)$ always defines an element of $L^2([0, L])$. (Actually, all elements of $L^2([0, L])$ can be decomposed in a finite energy Fourier series.)



A good reference to be introduced to these things is the book by Stein and Shakarchi "Fourier analysis: an introduction", first volume.






share|cite|improve this answer



















  • 1




    You might mention explicitly that otoh it does follow that $Sin L^2$.
    – David C. Ullrich
    Aug 1 at 17:01














up vote
3
down vote













Your lecturer probably meant that the vector space $C^2([0,L])$, consisting of functions that are twice differentiable with continuous derivatives, is not complete if equipped with the scalar product
$$
langle f| grangle= int_0^L f(x)overlineg(x), dx.$$
Among the various unpleasant consequences of this, there's the Fourier series fact that if you take an arbitrary series
$$
S(x)=sum_k=-infty^infty c_k e^ifrac2piLkx $$
then, even if $sum_k=-infty^infty |c_k|^2<infty $ (corresponding to the physical fact that $S$ has finite energy), you cannot conclude that $Sin C^2$. (In fact, it is not trivial to find conditions on $c_k$ that ensure $Sin C^2$. I don't think that a necessary and sufficient condition is known.)



On the other hand, the vector space $L^2([0, L])$, consisting of all measurable functions $f$ such that $int_0^L |f(x)|^2, dx<infty$, is complete if equipped with the aforementioned scalar product $langlecdot|cdotrangle$. As a consequence of this, the finite energy Fourier series $S(x)$ always defines an element of $L^2([0, L])$. (Actually, all elements of $L^2([0, L])$ can be decomposed in a finite energy Fourier series.)



A good reference to be introduced to these things is the book by Stein and Shakarchi "Fourier analysis: an introduction", first volume.






share|cite|improve this answer



















  • 1




    You might mention explicitly that otoh it does follow that $Sin L^2$.
    – David C. Ullrich
    Aug 1 at 17:01












up vote
3
down vote










up vote
3
down vote









Your lecturer probably meant that the vector space $C^2([0,L])$, consisting of functions that are twice differentiable with continuous derivatives, is not complete if equipped with the scalar product
$$
langle f| grangle= int_0^L f(x)overlineg(x), dx.$$
Among the various unpleasant consequences of this, there's the Fourier series fact that if you take an arbitrary series
$$
S(x)=sum_k=-infty^infty c_k e^ifrac2piLkx $$
then, even if $sum_k=-infty^infty |c_k|^2<infty $ (corresponding to the physical fact that $S$ has finite energy), you cannot conclude that $Sin C^2$. (In fact, it is not trivial to find conditions on $c_k$ that ensure $Sin C^2$. I don't think that a necessary and sufficient condition is known.)



On the other hand, the vector space $L^2([0, L])$, consisting of all measurable functions $f$ such that $int_0^L |f(x)|^2, dx<infty$, is complete if equipped with the aforementioned scalar product $langlecdot|cdotrangle$. As a consequence of this, the finite energy Fourier series $S(x)$ always defines an element of $L^2([0, L])$. (Actually, all elements of $L^2([0, L])$ can be decomposed in a finite energy Fourier series.)



A good reference to be introduced to these things is the book by Stein and Shakarchi "Fourier analysis: an introduction", first volume.






share|cite|improve this answer















Your lecturer probably meant that the vector space $C^2([0,L])$, consisting of functions that are twice differentiable with continuous derivatives, is not complete if equipped with the scalar product
$$
langle f| grangle= int_0^L f(x)overlineg(x), dx.$$
Among the various unpleasant consequences of this, there's the Fourier series fact that if you take an arbitrary series
$$
S(x)=sum_k=-infty^infty c_k e^ifrac2piLkx $$
then, even if $sum_k=-infty^infty |c_k|^2<infty $ (corresponding to the physical fact that $S$ has finite energy), you cannot conclude that $Sin C^2$. (In fact, it is not trivial to find conditions on $c_k$ that ensure $Sin C^2$. I don't think that a necessary and sufficient condition is known.)



On the other hand, the vector space $L^2([0, L])$, consisting of all measurable functions $f$ such that $int_0^L |f(x)|^2, dx<infty$, is complete if equipped with the aforementioned scalar product $langlecdot|cdotrangle$. As a consequence of this, the finite energy Fourier series $S(x)$ always defines an element of $L^2([0, L])$. (Actually, all elements of $L^2([0, L])$ can be decomposed in a finite energy Fourier series.)



A good reference to be introduced to these things is the book by Stein and Shakarchi "Fourier analysis: an introduction", first volume.







share|cite|improve this answer















share|cite|improve this answer



share|cite|improve this answer








edited Aug 1 at 17:40


























answered Aug 1 at 16:48









Giuseppe Negro

15.6k328117




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  • 1




    You might mention explicitly that otoh it does follow that $Sin L^2$.
    – David C. Ullrich
    Aug 1 at 17:01












  • 1




    You might mention explicitly that otoh it does follow that $Sin L^2$.
    – David C. Ullrich
    Aug 1 at 17:01







1




1




You might mention explicitly that otoh it does follow that $Sin L^2$.
– David C. Ullrich
Aug 1 at 17:01




You might mention explicitly that otoh it does follow that $Sin L^2$.
– David C. Ullrich
Aug 1 at 17:01












 

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