arXiv subjects
Publications and source records attributed to Thaís Jordão.
A characterization of the generalized Lipschitz and Besov spaces in terms of decay of Fourier transforms is given. In particular, necessary and sufficient conditions of Titchmarsh type are obtained. The method is based on two-sided estimate for the rate of approximation of a $β$-admissible family of multipliers operators in terms of decay properties of Fourier transform.
We present estimates for the covering numbers of the unit ball of Reproducing Kernel Hilbert Spaces (RKHSs) of functions on $M^d$ a d-dimensional compact two-point homogeneous space. The RKHS is generated by a continuous zonal/isotropic positive definite kernel. We employ the representation in terms of the Schoenberg/Fourier series expansion for continuous isotropic positive definite kernels, given in terms of a family of orthogonal polynomials on $M^d$. The bounds we present carry accurate information about the asymptotic constants depending on the dimension of the manifold and the decay or growth rate of the coefficients of the kernel. The results we present extend the estimates previously known for continuous isotropic positive definite kernels on the d-dimensional unit sphere. We present the weak asymptotic equivalence for the order of the growth of covering numbers associated to kernels on $M^d$ with a convergent geometric sequence of coefficients. We apply our estimates in order to present a bound for the covering numbers of the spherical Gaussian kernel, and to present bounds for formal examples on $M^d$.
Decay rates for the sequence of eigenvalues of positive and compact integral operators has been largely investigated for a long time in the literature. In this paper, the focus will be on positive integral operators acting on square integrable functions on the unit sphere and generated by a kernel satisfying a Hölder type assumption defined via average operators. In the approach to be presented here, the decay rates will be reached from convenient estimations on the eigenvalues of the operator themselves, with the help of specific properties of a generic approximation operator defined through the so-called generalized Jackson kernels. The decay rates have the same structure of those known to hold in the cases in which the Hölder condition is the classical one. Therefore, within the spherical setting, the abstract approach to be introduced here extends some classical results on the topic.
We provide estimates for weighted Fourier sums of integrable functions defined on the sphere when the weights originate from a multiplier operator acting on the space where the function belongs. That implies refined estimates for weighted Fourier sums of integrable kernels on the sphere that satisfy an abstract Hölder condition based on a parameterized family of multiplier operators defining an approximate identity. This general estimation approach includes an important class of multipliers operators, namely, that defined by convolutions with zonal measures. The estimates are used to obtain decay rates for the eigenvalues of positive integral operators on L^2(Sm) and generated by a kernel satisfying the Hölder condition based on multiplier operators on L^2(Sm).