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In [[mathematics]], the '''Radó–Kneser–Choquet theorem''', named after [[Tibor Radó]], [[Hellmuth Kneser]] and [[Gustave Choquet]], states that the [[Poisson integral]] of a homeomorphism of the [[unit circle]] is a [[harmonic function|harmonic]] diffeomorphism of the open [[unit disk]]. The result was stated as a problem by Radó and solved shortly afterwards by Kneser in 1926. Choquet, unaware of the work of Radó and Kneser, rediscovered the result with a different proof in 1945. Choquet also generalized the result to the Poisson integral of a homeomorphism from the unit circle to a simple Jordan curve bounding a convex region. | |||
==Statement== | |||
Let ''f'' be an orientation-preserving homeomorphism of the unit circle |''z''| = 1 in '''C''' and define the Poisson integral of ''f'' by | |||
:<math>\displaystyle{F_f(re^{i\theta}) ={1\over 2\pi} \int_0^{2\pi} f(\varphi)\cdot {1- r^2\over 1 - 2r\cos (\theta -\varphi) + r^2}\,d\varphi,}</math> | |||
for ''r'' < 1. Standard properties of the Poisson integral show that ''F''<sub>''f''</sub> is a [[harmonic function]] on |''z''| < 1 which extends by continuity to ''f'' on |''z''| = 1. With the additional assumption that ''f'' is orientation-preserving homeomorphism of this circle, ''F''<sub>''f''</sub> is an orientation preserving diffeomorphism of the open unit disk. | |||
==Proof== | |||
To prove that ''F''<sub>''f''</sub> is locally an orientation-preserving diffeomorphism, it suffices to show that the Jacobian at a point ''a'' in the unit disk is positive. This Jacobian is given by | |||
:<math>\displaystyle{J_f(a)=|\partial_zF_f(a)|^2 - |\partial_{\overline{z}} F_f(a)|^2.}</math> | |||
On the other hand that ''g'' is a Möbius transformation preserving the unit circle and the unit disk, | |||
:<math>\displaystyle{F_{f\circ g}=F_f \circ g.}</math> | |||
Taking ''g'' so that ''g''(''a'') = 0 and taking the change of variable ζ = ''g''(''z''), the chain rule gives | |||
:<math>\displaystyle{(F_f\circ g)_z= [(F_f)_\zeta \circ g]\cdot g_z, \,\, (F_f\circ g)_{\overline{z}}=[(F_{f})_{\overline{\zeta}} \circ g] \cdot\overline{g_z}.}</math> | |||
It follows that | |||
:<math>\displaystyle{J_{f\circ g}(a)= J_f(0)\cdot |g_z(a)|^2.}</math> | |||
It is therefore enough to prove positivity of the Jacobian when ''a'' = 0. In that case | |||
:<math>\displaystyle{J_f(0)=|a_{1}|^2 - |a_{-1}|^2,}</math> | |||
where the ''a''<sub>''n''</sub> are the Fourier coefficients of ''f'': | |||
:<math>\displaystyle{a_n={1\over 2\pi}\int_0^{2\pi} f(e^{i\theta}) e^{-in\theta}\, d\theta.}</math> | |||
Following {{harvtxt|Douady|Earle|1986}}, the Jacobian at 0 can be expressed as a double integral | |||
:<math>\displaystyle{|a_{1}|^2 - |a_{-1}|^2={1\over 4\pi^2}\int_0^{2\pi}\int_0^{2\pi} \Re [f(e^{i\theta})\overline{f(e^{i\varphi})} (e^{i(\theta-\varphi)}-e^{-i(\theta-\varphi)})]\,d\theta\, d\varphi.}</math> | |||
Writing | |||
:<math>\displaystyle{f(e^{i\theta})=e^{ih(\theta)},}</math> | |||
where ''h'' is a strictly increasing continuous function satisfying | |||
:<math>\displaystyle{h(\theta+2\pi)=h(\theta) +2\pi},</math> | |||
the double integral can be rewritten as | |||
:<math>\displaystyle{|a_{1}|^2 - |a_{-1}|^2={1\over 2 \pi^2} \int_0^{2\pi}\int_0^{2\pi} \sin (\theta-\varphi) \sin(h(\theta)-h(\varphi))\,d\theta\,d\varphi={1\over 2 \pi^2} \int_0^{2\pi}\int_0^{2\pi} \sin (\theta) \sin(h(\theta+\varphi)-h(\varphi))\,d\theta\,d\varphi.}</math> | |||
Hence | |||
:<math>\displaystyle{|a_{1}|^2 - |a_{-1}|^2={1\over 2 \pi^2} \int_0^{\pi}\sin \theta \int_0^\pi R(\theta,\varphi)\,d\varphi\,d\theta,}</math> | |||
where | |||
:<math>R(\theta,\varphi)= \sin(h(\varphi+\theta) - h(\varphi)) + \sin(h(\varphi+2\pi)-h(\varphi+\theta +\pi)) + \sin(h(\varphi+\theta +\pi) -h(\varphi +\pi)) + \sin(h(\varphi+\pi) - h(\varphi+\theta)).</math> | |||
This formula gives ''R'' as the sum of the sines of four non-negative angles with sum 2π, so it is always non-negative.<ref>This elementary fact holds more generally for any number of non-negative angles with sum 2π. If all the angles are less than or equal to π, all the sines are non-negative. If one is greater than π, the result states that the sine of the sum of the other angles is less than the sine of their sum. This follows by induction from the result for two angles, itself a direct consequence of the trigonometric formula for the sine of the sum.</ref> But then the Jacobian at 0 is strictly positive and ''F''<sub>''f''</sub> is therefore locally a diffeomorphism. | |||
It remains to deduce ''F''<sub>''f''</sub> is a homeomorphism. By continuity its image is compact so closed. The non-vanishing of the Jacobian, implies that ''F''<sub>''f''</sub> is an open mapping on the unit disk, so that the image of the open disk is open. Hence the image of the closed disk is an open and closed subset of the closed disk. By connectivity, it must be the whole disk. For |''w''| < 1, the inverse image of ''w'' is closed, so compact, and entirely contained in the open disk. Since ''F''<sub>''f''</sub> is locally a homeomorphism, it must be a finite set. The set of points ''w'' in the open disk with exactly ''n'' preimages is open. By connectivity every point has the same number ''N'' of preimages. Since the open disk is [[simply connected]], ''N'' = 1. In fact taking any preimage of the origin, every radial line has a unique lifting to a preimage, and so there is an open subset of the unit disk mapping homeomorphically onto the open disk. If ''N'' > 1, its complement would also have to be open, contradicting connectivity. | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
*{{citation|first=Hellmuth |last=Kneser |title=Lösung der Aufgabe 41 |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume=35|pages=123–124|year= 1926}} | |||
*{{citation|last=Choquet|first= Gustave|title= Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques|journal= Bull. Sci. Math.|volume= 69|year=1945|pages=156–165}} | |||
*{{citation|last=Douady|first= Adrien|last2= Earle|first2 =Clifford J.|title=Conformally natural extension of homeomorphisms of the circle|journal=Acta Math.|volume= 157|year=1986|pages=23–48}} | |||
*{{citation|last=Duren|first= Peter|title= Harmonic mappings in the plane|series= Cambridge Tracts in Mathematics|volume=156|publisher= Cambridge University Press|year= 2004|isbn= 0-521-64121-7}} | |||
*{{citation|last=Sheil-Small|first= T.|title= On the Fourier series of a finitely described convex curve and a conjecture of H. S. Shapiro|series= Math. Proc. Cambridge Philos. Soc.|volume= 98 |year=1985|pages= 513–527}} | |||
{{DEFAULTSORT:Rado-Kneser-Choquet theorem}} | |||
[[Category:Mathematical theorems]] | |||
[[Category:Harmonic analysis]] | |||
Latest revision as of 16:39, 10 December 2013
In mathematics, the Radó–Kneser–Choquet theorem, named after Tibor Radó, Hellmuth Kneser and Gustave Choquet, states that the Poisson integral of a homeomorphism of the unit circle is a harmonic diffeomorphism of the open unit disk. The result was stated as a problem by Radó and solved shortly afterwards by Kneser in 1926. Choquet, unaware of the work of Radó and Kneser, rediscovered the result with a different proof in 1945. Choquet also generalized the result to the Poisson integral of a homeomorphism from the unit circle to a simple Jordan curve bounding a convex region.
Statement
Let f be an orientation-preserving homeomorphism of the unit circle |z| = 1 in C and define the Poisson integral of f by
for r < 1. Standard properties of the Poisson integral show that Ff is a harmonic function on |z| < 1 which extends by continuity to f on |z| = 1. With the additional assumption that f is orientation-preserving homeomorphism of this circle, Ff is an orientation preserving diffeomorphism of the open unit disk.
Proof
To prove that Ff is locally an orientation-preserving diffeomorphism, it suffices to show that the Jacobian at a point a in the unit disk is positive. This Jacobian is given by
On the other hand that g is a Möbius transformation preserving the unit circle and the unit disk,
Taking g so that g(a) = 0 and taking the change of variable ζ = g(z), the chain rule gives
It follows that
It is therefore enough to prove positivity of the Jacobian when a = 0. In that case
where the an are the Fourier coefficients of f:
Following Template:Harvtxt, the Jacobian at 0 can be expressed as a double integral
Writing
where h is a strictly increasing continuous function satisfying
the double integral can be rewritten as
Hence
where
This formula gives R as the sum of the sines of four non-negative angles with sum 2π, so it is always non-negative.[1] But then the Jacobian at 0 is strictly positive and Ff is therefore locally a diffeomorphism.
It remains to deduce Ff is a homeomorphism. By continuity its image is compact so closed. The non-vanishing of the Jacobian, implies that Ff is an open mapping on the unit disk, so that the image of the open disk is open. Hence the image of the closed disk is an open and closed subset of the closed disk. By connectivity, it must be the whole disk. For |w| < 1, the inverse image of w is closed, so compact, and entirely contained in the open disk. Since Ff is locally a homeomorphism, it must be a finite set. The set of points w in the open disk with exactly n preimages is open. By connectivity every point has the same number N of preimages. Since the open disk is simply connected, N = 1. In fact taking any preimage of the origin, every radial line has a unique lifting to a preimage, and so there is an open subset of the unit disk mapping homeomorphically onto the open disk. If N > 1, its complement would also have to be open, contradicting connectivity.
Notes
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References
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
- ↑ This elementary fact holds more generally for any number of non-negative angles with sum 2π. If all the angles are less than or equal to π, all the sines are non-negative. If one is greater than π, the result states that the sine of the sum of the other angles is less than the sine of their sum. This follows by induction from the result for two angles, itself a direct consequence of the trigonometric formula for the sine of the sum.