Is there any two variable function which has no representation of the form $sumlimits_{n=1}^{infty}...
$begingroup$
Is there any function$f:Bbb R^2toBbb R$ which has no representation of the form below?
$$f(x,y)=sum_{n=1}^{infty} g_n(x)h_n(y)quad(x,yin Bbb R).$$Editor's note: A possible source of motivation for this question lies in a trick used to find solutions to linear partial diferential equations, in particular Laplace's equation in two dimensions. As a first step, a solution of the form $f(x,y)=g(x)h(y)$ is sought; and then more-general solutions of the form displayed above can be generated. While it is far from obvious that all solutions to the 2-D Laplace equation must be of this form, it is also hard to find a counterexample, namely a solution that cannot be thus structured. (Indeed, is such a solution known at all?)
If we relax the conditions so as not to require even continuity of $f$ (let alone its being a solution of Laplace's equation), then there is in principle more room to find a counterexample. Thus it seems that the question posed above may be answerable. In fact, while this editor is unable to come up with a counterexample, his (admittedly unreliable) intuition says that there probably is one and that the answer is yes.
real-analysis
edited Dec 15 '18 at 19:02
John Bentin
11.3k22554
asked Dec 15 '18 at 7:33
ZeinabZeinab
563
$endgroup$
|
show 12 more comments
$begingroup$
Is there any function$f:Bbb R^2toBbb R$ which has no representation of the form below?
$$f(x,y)=sum_{n=1}^{infty} g_n(x)h_n(y)quad(x,yin Bbb R).$$Editor's note: A possible source of motivation for this question lies in a trick used to find solutions to linear partial diferential equations, in particular Laplace's equation in two dimensions. As a first step, a solution of the form $f(x,y)=g(x)h(y)$ is sought; and then more-general solutions of the form displayed above can be generated. While it is far from obvious that all solutions to the 2-D Laplace equation must be of this form, it is also hard to find a counterexample, namely a solution that cannot be thus structured. (Indeed, is such a solution known at all?)
If we relax the conditions so as not to require even continuity of $f$ (let alone its being a solution of Laplace's equation), then there is in principle more room to find a counterexample. Thus it seems that the question posed above may be answerable. In fact, while this editor is unable to come up with a counterexample, his (admittedly unreliable) intuition says that there probably is one and that the answer is yes.
real-analysis
edited Dec 15 '18 at 19:02
John Bentin
11.3k22554
asked Dec 15 '18 at 7:33
ZeinabZeinab
563
$endgroup$
$begingroup$
Try $e^{xy}$.${}{}$
$endgroup$
– Seewoo Lee
Dec 15 '18 at 8:22
$begingroup$
@SeewooLee $=displaystylesum_{n=1}^{infty}frac{x^{n-1}}{(n-1)!}y^{n-1}$. But still OP goes far too broad.
$endgroup$
– metamorphy
Dec 15 '18 at 8:36
1
$begingroup$
The problem is related to (but not solved by) the Kolmogorov-Arnold representation theorem. I agree if the OP is interested in the problem, then they should add context beyond mere problem statement to their Question. I would also accept context added by @JohnBentin if so motivated.
$endgroup$
– hardmath
Dec 15 '18 at 16:37
1
$begingroup$
@jgon: For me, the question is incomparably more interesting than any details of how or why the OP came up with the question. Also, I can't see how such details would help in answering the question.
$endgroup$
– John Bentin
Dec 15 '18 at 16:46
1
$begingroup$
@jgon: In principle, this question could be stolen, dressed up with some plausible-sounding "context", and re-posted, probably to gain a good response and some upvotes. But I think that the OP should get the credit. I simply hope that enough people who dig what a lovely question it is come along and vote to restore it.
$endgroup$
– John Bentin
Dec 15 '18 at 17:54
|
show 12 more comments
$begingroup$
Is there any function$f:Bbb R^2toBbb R$ which has no representation of the form below?
$$f(x,y)=sum_{n=1}^{infty} g_n(x)h_n(y)quad(x,yin Bbb R).$$Editor's note: A possible source of motivation for this question lies in a trick used to find solutions to linear partial diferential equations, in particular Laplace's equation in two dimensions. As a first step, a solution of the form $f(x,y)=g(x)h(y)$ is sought; and then more-general solutions of the form displayed above can be generated. While it is far from obvious that all solutions to the 2-D Laplace equation must be of this form, it is also hard to find a counterexample, namely a solution that cannot be thus structured. (Indeed, is such a solution known at all?)
If we relax the conditions so as not to require even continuity of $f$ (let alone its being a solution of Laplace's equation), then there is in principle more room to find a counterexample. Thus it seems that the question posed above may be answerable. In fact, while this editor is unable to come up with a counterexample, his (admittedly unreliable) intuition says that there probably is one and that the answer is yes.
real-analysis
edited Dec 15 '18 at 19:02
John Bentin
11.3k22554
asked Dec 15 '18 at 7:33
ZeinabZeinab
563
$endgroup$
Is there any function$f:Bbb R^2toBbb R$ which has no representation of the form below?
$$f(x,y)=sum_{n=1}^{infty} g_n(x)h_n(y)quad(x,yin Bbb R).$$Editor's note: A possible source of motivation for this question lies in a trick used to find solutions to linear partial diferential equations, in particular Laplace's equation in two dimensions. As a first step, a solution of the form $f(x,y)=g(x)h(y)$ is sought; and then more-general solutions of the form displayed above can be generated. While it is far from obvious that all solutions to the 2-D Laplace equation must be of this form, it is also hard to find a counterexample, namely a solution that cannot be thus structured. (Indeed, is such a solution known at all?)
If we relax the conditions so as not to require even continuity of $f$ (let alone its being a solution of Laplace's equation), then there is in principle more room to find a counterexample. Thus it seems that the question posed above may be answerable. In fact, while this editor is unable to come up with a counterexample, his (admittedly unreliable) intuition says that there probably is one and that the answer is yes.
real-analysis
real-analysis
edited Dec 15 '18 at 19:02
John Bentin
11.3k22554
asked Dec 15 '18 at 7:33
ZeinabZeinab
563
edited Dec 15 '18 at 19:02
John Bentin
11.3k22554
asked Dec 15 '18 at 7:33
ZeinabZeinab
563
edited Dec 15 '18 at 19:02
John Bentin
11.3k22554
edited Dec 15 '18 at 19:02
John Bentin
11.3k22554
edited Dec 15 '18 at 19:02
John Bentin
11.3k22554
11.3k22554
asked Dec 15 '18 at 7:33
ZeinabZeinab
563
asked Dec 15 '18 at 7:33
ZeinabZeinab
563
asked Dec 15 '18 at 7:33
ZeinabZeinab
563
563
$begingroup$
Try $e^{xy}$.${}{}$
$endgroup$
– Seewoo Lee
Dec 15 '18 at 8:22
$begingroup$
@SeewooLee $=displaystylesum_{n=1}^{infty}frac{x^{n-1}}{(n-1)!}y^{n-1}$. But still OP goes far too broad.
$endgroup$
– metamorphy
Dec 15 '18 at 8:36
1
$begingroup$
The problem is related to (but not solved by) the Kolmogorov-Arnold representation theorem. I agree if the OP is interested in the problem, then they should add context beyond mere problem statement to their Question. I would also accept context added by @JohnBentin if so motivated.
$endgroup$
– hardmath
Dec 15 '18 at 16:37
1
$begingroup$
@jgon: For me, the question is incomparably more interesting than any details of how or why the OP came up with the question. Also, I can't see how such details would help in answering the question.
$endgroup$
– John Bentin
Dec 15 '18 at 16:46
1
$begingroup$
@jgon: In principle, this question could be stolen, dressed up with some plausible-sounding "context", and re-posted, probably to gain a good response and some upvotes. But I think that the OP should get the credit. I simply hope that enough people who dig what a lovely question it is come along and vote to restore it.
$endgroup$
– John Bentin
Dec 15 '18 at 17:54
|
show 12 more comments
$begingroup$
Try $e^{xy}$.${}{}$
$endgroup$
– Seewoo Lee
Dec 15 '18 at 8:22
$begingroup$
@SeewooLee $=displaystylesum_{n=1}^{infty}frac{x^{n-1}}{(n-1)!}y^{n-1}$. But still OP goes far too broad.
$endgroup$
– metamorphy
Dec 15 '18 at 8:36
1
$begingroup$
The problem is related to (but not solved by) the Kolmogorov-Arnold representation theorem. I agree if the OP is interested in the problem, then they should add context beyond mere problem statement to their Question. I would also accept context added by @JohnBentin if so motivated.
$endgroup$
– hardmath
Dec 15 '18 at 16:37
1
$begingroup$
@jgon: For me, the question is incomparably more interesting than any details of how or why the OP came up with the question. Also, I can't see how such details would help in answering the question.
$endgroup$
– John Bentin
Dec 15 '18 at 16:46
1
$begingroup$
@jgon: In principle, this question could be stolen, dressed up with some plausible-sounding "context", and re-posted, probably to gain a good response and some upvotes. But I think that the OP should get the credit. I simply hope that enough people who dig what a lovely question it is come along and vote to restore it.
$endgroup$
– John Bentin
Dec 15 '18 at 17:54
$begingroup$
Try $e^{xy}$.${}{}$
$endgroup$
– Seewoo Lee
Dec 15 '18 at 8:22
$begingroup$
Try $e^{xy}$.${}{}$
$endgroup$
– Seewoo Lee
Dec 15 '18 at 8:22
$begingroup$
@SeewooLee $=displaystylesum_{n=1}^{infty}frac{x^{n-1}}{(n-1)!}y^{n-1}$. But still OP goes far too broad.
$endgroup$
– metamorphy
Dec 15 '18 at 8:36
$begingroup$
@SeewooLee $=displaystylesum_{n=1}^{infty}frac{x^{n-1}}{(n-1)!}y^{n-1}$. But still OP goes far too broad.
$endgroup$
– metamorphy
Dec 15 '18 at 8:36
1
1
$begingroup$
The problem is related to (but not solved by) the Kolmogorov-Arnold representation theorem. I agree if the OP is interested in the problem, then they should add context beyond mere problem statement to their Question. I would also accept context added by @JohnBentin if so motivated.
$endgroup$
– hardmath
Dec 15 '18 at 16:37
$begingroup$
The problem is related to (but not solved by) the Kolmogorov-Arnold representation theorem. I agree if the OP is interested in the problem, then they should add context beyond mere problem statement to their Question. I would also accept context added by @JohnBentin if so motivated.
$endgroup$
– hardmath
Dec 15 '18 at 16:37
1
1
$begingroup$
@jgon: For me, the question is incomparably more interesting than any details of how or why the OP came up with the question. Also, I can't see how such details would help in answering the question.
$endgroup$
– John Bentin
Dec 15 '18 at 16:46
$begingroup$
@jgon: For me, the question is incomparably more interesting than any details of how or why the OP came up with the question. Also, I can't see how such details would help in answering the question.
$endgroup$
– John Bentin
Dec 15 '18 at 16:46
1
1
$begingroup$
@jgon: In principle, this question could be stolen, dressed up with some plausible-sounding "context", and re-posted, probably to gain a good response and some upvotes. But I think that the OP should get the credit. I simply hope that enough people who dig what a lovely question it is come along and vote to restore it.
$endgroup$
– John Bentin
Dec 15 '18 at 17:54
$begingroup$
@jgon: In principle, this question could be stolen, dressed up with some plausible-sounding "context", and re-posted, probably to gain a good response and some upvotes. But I think that the OP should get the credit. I simply hope that enough people who dig what a lovely question it is come along and vote to restore it.
$endgroup$
– John Bentin
Dec 15 '18 at 17:54
|
show 12 more comments
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$begingroup$
Try $e^{xy}$.${}{}$
$endgroup$
– Seewoo Lee
Dec 15 '18 at 8:22
$begingroup$
@SeewooLee $=displaystylesum_{n=1}^{infty}frac{x^{n-1}}{(n-1)!}y^{n-1}$. But still OP goes far too broad.
$endgroup$
– metamorphy
Dec 15 '18 at 8:36
1
$begingroup$
The problem is related to (but not solved by) the Kolmogorov-Arnold representation theorem. I agree if the OP is interested in the problem, then they should add context beyond mere problem statement to their Question. I would also accept context added by @JohnBentin if so motivated.
$endgroup$
– hardmath
Dec 15 '18 at 16:37
1
$begingroup$
@jgon: For me, the question is incomparably more interesting than any details of how or why the OP came up with the question. Also, I can't see how such details would help in answering the question.
$endgroup$
– John Bentin
Dec 15 '18 at 16:46
1
$begingroup$
@jgon: In principle, this question could be stolen, dressed up with some plausible-sounding "context", and re-posted, probably to gain a good response and some upvotes. But I think that the OP should get the credit. I simply hope that enough people who dig what a lovely question it is come along and vote to restore it.
$endgroup$
– John Bentin
Dec 15 '18 at 17:54