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We consider a regular indefinite Krein–Feller differential expression of Stieltjes type \(-D_mD_x\). This can be regarded as an indefinite generalization of a vibrating Stieltjes string wearing only concentrated (now positive or negative) “masses” which accumulate at a finite right endpoint. From the general theory of indefinite Krein–Feller operators we conclude a number of spectral properties. In particular, we obtain a spectral function \(\sigma \) which is non-increasing on \((-\infty ,0)\) and non-decreasing on \((0,\infty )\) and which allows the existence of all moments \(\int \lambda ^n \; d\sigma \) for \(n \in {\mathbb N}\). The main result of the present paper is an inverse statement: Starting from a function \(\tau \) with properties like \(\sigma \), the (unique) “masses” of an indefinite Krein–Feller operator of Stieltjes type are reconstructed such that \(\tau \) belongs to the same so-called spectral class like the associated spectral function. All elements of this class are identified as the spectral functions of similar operators with a generally “heavy” right endpoint (and one additional function).

A long series of previous papers have been devoted to the (one-dimensional) moment problem with nonnegative rational measure. The rationality assumption is a complexity constraint motivated by applications where a parameterization of the solution set in terms of a bounded finite number of parameters is required. In this paper we provide a complete solution of the multidimensional moment problem with a complexity constraint also allowing for solutions that require a singular measure added to the rational, absolutely continuous one. Such solutions occur on the boundary of a certain convex cone of solutions. In this paper we provide complete parameterizations of all such solutions. We also provide errata for a previous paper in this journal coauthored by one of the authors of the present paper., 25 pages. Revision: minor corrections

Let \({s = \{s_{jk}\}_{0 \leq j+k \leq 3}}\) be a given complex-valued sequence. The cubic complex moment problem involves determining necessary and sufficient conditions for the existence of a positive Borel measure \({\sigma}\) on \({\mathbb{C}}\) (called a representing measure for s) such that \({s_{jk} = \int_{\mathbb{C}}\bar{z}^j z^k d\sigma(z)}\) for \({0 \leq j + k \leq 3}\). Put $$\Phi = \left(\begin{array}{lll} s_{00} & s_{01} & s_{10} \\s_{10} & s_{11} & s_{20} \\s_{01} & s_{02} & s_{11}\end{array}\right), \quad \Phi_z = \left(\begin{array}{lll}s_{01} & s_{02} & s_{11} \\s_{10} & s_{12} & s_{21} \\s_{02} & s_{03} & s_{12}\end{array} \right)\quad {\rm and}\quad\Phi_{\bar{z}} = (\Phi_z)^*.$$ If \({\Phi \succ 0}\), then the commutativity of \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) is necessary and sufficient for the existence a 3-atomic representing measure for s. If \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) do not commute, then we show that s has a 4-atomic representing measure. The proof is constructive in nature and yields a concrete parametrization of all 4-atomic representing measures of s. Consequently, given a set \({K \subseteq \mathbb{C}}\) necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure \({\sigma}\) which satisfies \({{\rm supp} \sigma \cap K \neq \emptyset}\) or \({{\rm supp} \sigma \subseteq K}\). The cases when \({K = \overline{\mathbb{D}}}\) and \({K = \mathbb{T}}\) are considered in detail.

The complex moment problem for a sequence γ(2n) = {γij}0≤i,j≤n has been studied by Curto-Fialkow, where positivity and extension properties of the moment matrix M(n) = M(n)(γ) (γ ≡ γ(2n)) are involved, for guaranteeing the existence of representing measure. But it was showed that positivity and recursiveness are not sufficient in order to have a representing measure for γ. Here we combine our techniques based on the Fibonacci sequences’s properties with some Curto-Fialkow’s results to obtain sufficient conditions for insuring that γ is a truncated moment sequence. We focus ourself on the case when rank M(n) ≤ n + 1, and finally we stretch our exploration to the finite-rank infinite positive moment matrix.

In this paper we consider the truncated complex moment problem suggested by Curto and Fialkow. First, we give another computing proof for a solution of the nonsingular quadratic moment problem. Then we consider the nonsingular quartic moment problem and give some partial solutions.

For a degree 2n real d-dimensional multisequence \(\beta \equiv \beta^{(2n)} = \{\beta_i\}_{i\in{Z}^{d}_{+},|i|\leq 2n}\) to have a representing measure μ, it is necessary for the associated moment matrix \({\mathcal{M}}(n)(\beta)\) to be positive semidefinite and for the algebraic variety associated to β, \({\mathcal{V}} \equiv {\mathcal{V}}_{\beta}\), to satisfy rank \({\mathcal{M}}(n) \leq\) card \({\mathcal{V}}\) as well as the following consistency condition: if a polynomial \(p(x) \equiv \sum_{|i|\leq 2n} a_{i}x^{i}\) vanishes on \({\mathcal{V}}\), then \(\sum_{|i|\leq 2n} a_{i}{\beta_i} = 0\). We prove that for the extremal case \((\rm{rank}\,{\mathcal{M}}(n) = \rm{card}\,{\mathcal{V}})\), positivity of \({\mathcal{M}}(n)\) and consistency are sufficient for the existence of a (unique, rank \({\mathcal{M}}(n)\)-atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of \({\mathcal{M}}(n)\) .

In this paper, we present a synthesis of our differentiable approach to the generalized moment problem, an approach which begins with a reformulation in terms of differential forms and which ultimately ends up with a canonically derived, strictly convex optimization problem. Engineering applications typically demand a solution that is the ratio of functions in certain finite dimensional vector space of functions, usually the same vector space that is prescribed in the generalized moment problem. Solutions of this type are hinted at in the classical text by Krein and Nudelman and stated in the vast generalization of interpolation problems by Sarason. In this paper, formulated as generalized moment problems with complexity constraint, we give a complete parameterization of such solutions, in harmony with the above mentioned results and the engineering applications. While our previously announced results required some differentiability hypotheses, this paper uses a weak form involving integrability and measurability hypotheses that are more in the spirit of the classical treatment of the generalized moment problem. Because of this generality, we can extend the existence and well-posedness of solutions to this problem to nonnegative, rather than positive, initial data in the complexity constraint. This has nontrivial implications in the engineering applications of this theory. We also extend this more general result to the case where the numerator can be an arbitrary positive absolutely integrable function that determines a unique denominator in this finite-dimensional vector space. Finally, we conclude with four examples illustrating our results.

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.

We solve the truncated complex moment problem for measures supported on the variety $ \mathcal{K}\equiv $ { z $ \in $ C: z $\widetilde{z}$ =A+Bz+C $\widetilde{z}$ +Dz 2 ,D $ \neq $ 0}. Given a doubly indexed finite sequence of complex numbers $ \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} $ , there exists a positive Borel measure $\mu$ supported in $ \mathcal{K} $ such that $ \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) $ if and only if the moment matrix M(n)( $\gamma$ ) is positive, recursively generated, with a column dependence relation Z $\widetilde{Z}$ = A1+BZ +C $\widetilde{Z}$ +DZ 2 , and card $\mathcal{V}(\gamma)\geq$ rank M(n), where $\mathcal{V}(\gamma)$ is the variety associated to $ \gamma $ . The last condition may be replaced by the condition that there exists a complex number $ \gamma_{n,n+1} $ satisfying $ \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} $ . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for $ \mathcal{K} $ , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z k = p(z, $ \widetilde{Z}$ ), deg p < k.

The article is devoted to two generalizations of the classical power moment problem, namely: 1) instead of representing the moment sequence by λn, a representation by polynomialsPn(λ), ℝ1, connected with a Jacobi matrix, appears; 2) in the representation, instead of λn, the expression λ⊗n figures, where λ is a real generalized function (i.e., we investigate some infinite-dimensional moment problem).

For the quadratic complex moment problem $$\gamma _{ij} = \int {\bar z^i } z^j d\mu \left( {0 \leqslant i + j \leqslant 2} \right)$$ , we obtain necessary and sufficient conditions for the existence of representing measures μ supported in the unit circleT or in the closed unit disk $$\bar D$$ . We explicitly construct all finitely atomic representing measures supported inT or $$\bar D$$ which have the fewest atoms possible. For the quadratic $$\bar D$$ -moment problem in which the moment matrixM(1) is positive and invertible, there exists an ellipse eɛD such that the minimal (3-atomic) representing measures are supported in the complement of the interior region of e. Finally, we apply these results to obtain information on the location of the zeros of certain cubic polynomials.

This paper contains the proofs and some development of results that were published without proofs in [KN2]. It is completed with comments added by the second author explaining how these results became the basis of statements and the solutions of problems in the theory of entire functions, in the moment problem, in direct and inverse problems of the spectral theory of nonhomogeneous strings and in other problems.

Hermitian kernels are introduced with the property that their Kolmogorov decompositions admit a Schur-type description. The main technical tool is the solution of an extension problem for indefinite factorizations and applications are indicated to some recent Kreįn space versions of the trigonometric moment problem and the Caratheodory-Schur problem.

Due to a technical problem, we accidentally omitted a paragraph from the proof of one of the main results in [1]. In this Addendum we provide the portion of the proof that did not appear in [1].

The Friedrichs extension and the Krein extension of a positive operator in a Krein space are characterized in terms of their spectral functions in a Krein space.