Solution of the Truncated Parabolic Moment Problem

Raúl Curto; Lawrence Fialkow

doi:10.1007/s00020-003-1275-3

Given real numbers % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaey % yyIORaeqOSdi2aaWbaaSqabeaacaGGOaGaaGOmaiaad6gacaGGPaaa % aOGaeyypa0Jaai4Eaiabek7aInaaBaaaleaacaWGPbGaamOAaaqaba % GccaGG9bWaaSbaaSqaaiaadMgacaGGSaGaamOAaiabgwMiZkaaicda % caGGSaGaamyAaiabgUcaRiaadQgacqGHKjYOcaaIYaGaamOBaaqaba % GccaGGSaaaaa!51B8! $$\beta \equiv \beta ^{(2n)} = \{ \beta _{ij} \} _{i,j \geq 0,i + j \leq 2n} ,$$ with γ00 >0 , the truncated parabolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in the parabola p(x, y) = 0, such that % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS % baaSqaaiaadMgacaWGQbaabeaakiabg2da9maapeaabaGaamyEamaa % CaaaleqabaGaamyAaaaakiaadIhadaahaaWcbeqaaiaadQgaaaaabe % qab0Gaey4kIipakiaadsgacqaH8oqBcaaMf8UaaiikaiaaicdacaaM % c8UaeyizImQaaGPaVlaadMgacaaMc8Uaey4kaSIaaGPaVlaadQgaca % aMc8UaeyizImQaaGPaVlaaikdacaWGUbGaaiykaaaa!5838! $$\beta _{ij} = \int {y^i x^j } d\mu \quad (0\, \leq \,i\, + \,j\, \leq \,2n)$$ We prove that β admits a representing measure μ (as above) if and only if the associated moment matrix % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFZestcaGGOaGaamOB % aiaacMcacaaMc8Uaaiikaiabek7aIjaacMcaaaa!476F! $$\mathcal{M}(n)\,(\beta )$$ is positive semidefinite, recursively generated and has a column relation p(X, Y) = 0, and the algebraic variety ν(β) associated to β satisfies card % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maai % ikaiabek7aIjaacMcacaaMe8UaeyyzImRaaGjbVlaabkhacaqGHbGa % aeOBaiaabUgacaaMc8+efv3ySLgznfgDOfdaryqr1ngBPrginfgDOb % YtUvgaiuaacqWFZestcaGGOaGaamOBaiaacMcacaaMc8Uaaiikaiab % ek7aIjaacMcacaGGUaaaaa!56F6! $$\nu (\beta )\; \geq \;{\text{rank}}\,\mathcal{M}(n)\,(\beta ).$$ In this case, β admits a rank % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFZestcaaMc8Uaaiik % aiaad6gacaGGPaaaaa!4475! $$\mathcal{M}\,(n)$$ -atomic (minimal) representing measure.

Solution of the Truncated Parabolic Moment Problem

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