1,179 research outputs found
Refinement of Operator-valued Reproducing Kernels
This paper studies the construction of a refinement kernel for a given
operator-valued reproducing kernel such that the vector-valued reproducing
kernel Hilbert space of the refinement kernel contains that of the given one as
a subspace. The study is motivated from the need of updating the current
operator-valued reproducing kernel in multi-task learning when underfitting or
overfitting occurs. Numerical simulations confirm that the established
refinement kernel method is able to meet this need. Various characterizations
are provided based on feature maps and vector-valued integral representations
of operator-valued reproducing kernels. Concrete examples of refining
translation invariant and finite Hilbert-Schmidt operator-valued reproducing
kernels are provided. Other examples include refinement of Hessian of
scalar-valued translation-invariant kernels and transformation kernels.
Existence and properties of operator-valued reproducing kernels preserved
during the refinement process are also investigated
Reproducing kernels for polynomial null-solutions of Dirac operators
It is well-known that the reproducing kernel of the space of spherical
harmonics of fixed homogeneity is given by a Gegenbauer polynomial. By going
over to complex variables and restricting to suitable bihomogeneous subspaces,
one obtains a reproducing kernel expressed as a Jacobi polynomial, which leads
to Koornwinder's celebrated result on the addition formula.
In the present paper, the space of Hermitian monogenics, which is the space
of polynomial bihomogeneous null-solutions of a set of two complex conjugated
Dirac operators, is considered. The reproducing kernel for this space is
obtained and expressed in terms of sums of Jacobi polynomials. This is achieved
through use of the underlying Lie superalgebra , combined
with the equivalence between the inner product on the unit sphere and the
Fischer inner product. The latter also leads to a new proof in the standard
Dirac case related to the Lie superalgebra .Comment: 31 pages, 1 table, to appear in Constr. Appro
Nevanlinna-Pick interpolation on distinguished varieties in the bidisk
This article treats Nevanlinna-Pick interpolation in the setting of a special
class of algebraic curves called distinguished varieties. An interpolation
theorem, along with additional operator theoretic results, is given using a
family of reproducing kernels naturally associated to the variety. The examples
of the Neil parabola and doubly connected domains are discussed.Comment: 31 pages. The question left open at the end of version 1 has been
answered in the affirmative; see Theorem 1.12 and Corollary 1.13 in version
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