The apolar bilinear form in geometric modeling

Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the de Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives

On the mixed Cauchy problem with data on singular conics

We consider a problem of mixed Cauchy type for certain holomorphic partial differential operators whose principal part $Q_{2p}(D)$ essentially is the (complex) Laplace operator to a power, $\Delta^p$. We pose inital data on a singular conic divisor given by P=0, where $P$ is a homogeneous polynomial of degree $2p$. We show that this problem is uniquely solvable if the polynomial $P$ is elliptic, in a certain sense, with respect to the principal part $Q_{2p}(D)$

On the Matricial Truncated Moment Problem. II

We continue the study of truncated matrix-valued moment problems begun in arXiv:2310.00957. Let $q\in\mathbb{N}$. Suppose that $(\mathcal{X},\mathfrak{X})$ is a measurable space and $\mathcal{E}$ is a finite-dimensional vector space of measurable mappings of $\mathscr{X}$ into $\mathcal{H}_q$, the Hermitian $q\times q$ matrices. A linear functional $\Lambda$ on $\mathcal{E}$ is called a moment functional if there exists a positive $\mathcal{H}_q$-valued measure $\mu$ on $(\mathcal{X},\mathfrak{X})$ such that $\Lambda(F)=\int_\mathcal{X} \langle F, \mathrm{d}\mu\rangle$ for $F\in \mathcal{E}$. In this paper a number of special topics on the truncated matricial moment problem are treated. We restate a result from (Mourrain and Schm\"udgen, 2016) to obtain a matricial version of the flat extension theorem. Assuming that $\mathcal{X}$ is a compact space and all elements of $\mathcal{E}$ are continuous on $\mathcal{X}$ we characterize moment functionals in terms of positivity and obtain an ordered maximal mass representing measure for each moment functional. The set of masses of representing measures at a fixed point and some related sets are studied. The class of commutative matrix moment functionals is investigated. We generalize the apolar scalar product for homogeneous polynomials to the matrix case and apply this to the matricial truncated moment problem

The apolar bilinear form in geometric modeling

Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the de Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives

The Apolar Bilinear Form in Geometric Modeling

Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the De Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives. Keywords: Apolar bilinear form, polarization, homogeneous polynomials, lineal polynomials, dual basis, Euler's identity, Marsden's identity, Bernstein-B'ezier patches, B-patches, De Casteljau, De Boor, recurrence relations, algorithm, basis conversion. 1991 Mathematics Subject Classification. Primary 41A15, 65D17; Secondary 65D07, 41A63. 1 Introduction A common problem in Computer Aided Geometric Design (CAGD) and Approximation Theory is the construction of suitable bases for the space o..