In this paper, by using some classical Mulholland type inequality, Berezin symbols and reproducing kernel technique, we prove the power inequalities for number $ber(A)$ self-adjoint operators $A$ on ${H}(\Omega )$. Namely, inequality Hilbert space are established. By applying that $(ber(A))^{n}\leq C_{1}ber(A^{n})$ any positive operator

The nature of the classical canonical phase-space variables for gravity suggests that the associated quantum field operators should obey affine commutation relations rather than canonical commutation relations. Prior to the introduction of constraints, a primary kinematical representation is derived in the form of a reproducing kernel and its associated reproducing kernel Hilbert space. Constra...

We analyze the regularized least square algorithm in learning theory with Reproducing Kernel Hilbert Spaces (RKHS). Explicit convergence rates for the regression and binary classification problems are obtained in particular for the polynomial and Gaussian kernels on the n-dimensional sphere and the hypercube. There are two major ingredients in our approach: (i) a law of large numbers for Hilber...

The aim of this paper is to construct a generalized Fourier analysis for certain Hermitian operators. When A, B are entire functions, then H(A,B) will be the associated reproducing kernel Hilbert spaces of Cn×n-valued functions. In that case, we will construct the expansion theorem forH(A,B) in a comprehensive manner. The spectral functions for the reproducing kernel Hilbert spaces will also be...

β-mixing sequence Reproducing kernel Hilbert spaces ℓ 2-empirical covering number Capacity dependent error bounds a b s t r a c t We study learning algorithms for classification generated by regularization schemes in reproducing kernel Hilbert spaces associated with a general convex loss function in a non-i.i.d. process. Error analysis is studied and our main purpose is to provide an elaborate ...

Sufficient conditions are established in order that, for a fixed infinite set of sampling points on the full line, a function satisfies a sampling theorem on a suitable closed subspace of a unitarily translation invariant reproducing kernel Hilbert space. A number of examples of such reproducing kernel Hilbert spaces and the corresponding sampling expansions are given. Sampling theorems for fun...

In this article, we implement a relatively new analytical technique, the reproducing kernel Hilbert space method (RKHSM), for solving integro-differential equations of fractional order. The solution obtained by using the method takes the form of a convergent series with easily computable components. Two numerical examples are studied to demonstrate the accuracy of the present method. The presen...

Reproducing Kernel Hilbert Space (RKHS) and Reproducing Transformation Methods for Series Summation that allow analytically obtaining alternative representations for series in the finite form are developed.

Conditionally positive definite kernels provide a powerful tool for scattered data approximation. Many nice properties of such methods follow from an underlying reproducing kernel structure. While the connection between positive definite kernels and reproducing kernel Hilbert spaces is well understood, the analog relation between conditionally positive definite kernels and reproducing kernel Po...

Let κ be an U-invariant reproducing kernel and let H (κ) denote the reproducing kernel Hilbert C[z1, . . . , zd]-module associated with the kernel κ. Let Mz denote the d-tuple of multiplication operators Mz1 , . . . ,Mzd on H (κ). For a positive integer ν and d-tuple T = (T1, . . . , Td), consider the defect operator