Narrow Results

A general theory of reproducing kernels and reproducing kernel Hilbert spaces on a right quaternionic Hilbert space is presented. Positive operator-valued measures and their connection to a class of generalized quaternionic coherent states are examined. A Naimark type extension theorem associated with the positive operator-valued measures is proved in a right quaternionic Hilbert space. As illustrative examples, real, complex and quaternionic reproducing kernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerre polynomials are presented. In particular, in the Laguerre case, the Naimark type extension theorem on the associated quaternionic Hilbert space is indicated.

We initiate a detailed study of two-parameter Besov spaces on the unit ball of ${ℝ}^n$ consisting of harmonic functions whose sufficiently high-order radial derivatives lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels are weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. Estimates of the growth of kernels lead to characterization of integral transformations on Lebesgue classes. The transformations allow us to conclude that the order of the radial derivative is not a characteristic of a Besov space as long as it is above a certain threshold. Using kernels, we define generalized Bergman projections and characterize those that are bounded from Lebesgue classes onto Besov spaces. The projections provide integral representations for the functions in these spaces and also lead to characterizations of the functions in the spaces using partial derivatives. Several other applications follow from the integral representations such as atomic decomposition, growth at the boundary and of Fourier coefficients, inclusions among them, duality and interpolation relations, and a solution to the Gleason problem.

This paper proposes a unified framework for the investigation of constrained learning theory in reflexive Banach spaces of features via regularized empirical risk minimization. The focus is placed on Tikhonov-like regularization with totally convex functions. This broad class of regularizers provides a flexible model for various priors on the features, including, in particular, hard constraints and powers of Banach norms. In such context, the main results establish a new general form of the representer theorem and the consistency of the corresponding learning schemes under general conditions on the loss function, the geometry of the feature space, and the modulus of total convexity of the regularizer. In addition, the proposed analysis gives new insight into basic tools such as reproducing Banach spaces, feature maps, and universality. Even when specialized to Hilbert spaces, this framework yields new results that extend the state of the art.

In this paper, we generalize the representer theorems in Banach spaces by the theory of nonsmooth analysis. The generalized representer theorems assure that the regularized learning models can be constructed by the nonconvex loss functions, the generalized training data, and the general Banach spaces which are nonreflexive, nonstrictly convex, and nonsmooth. Specially, the sparse representations of the regularized learning in 1-norm reproducing kernel Banach spaces are shown by the generalized representer theorems.

This paper extends the work of Fasshauer and Ye [Reproducing kernels of Sobolev spaces via a Green kernel approach with differential operators and boundary operators, Adv. Comput. Math. 38(4) (2011) 891921] in two different ways, namely, new kernels and associated native spaces are identified as crucial Hilbert spaces in applied mathematics. These spaces include the following spaces defined in bounded domains $\mathrm{\Omega }\subset {ℝ}^d$ with smooth boundary: homogeneous Sobolev–Slobodeckij̆ spaces, denoted by $H_0^s(\overline{\mathrm{\Omega }})$, and Sobolev–Slobodeckij̆ spaces, denoted by $H^s(\mathrm{\Omega })$, where $s>\frac{d}{2}$. Our goal is accomplished by obtaining the Green’s solutions of equations involving the fractional Laplacian and fractional differential operators defined through interpolation theory. We provide a proof that the Green’s kernels satisfying these problems are symmetric and positive definite reproducing kernels of $H_0^s(\overline{\mathrm{\Omega }})$ and $H^s(\mathrm{\Omega })$, respectively. Constructing kernels in these two ways enables the characterization of functions in native spaces based on their regularity. The Galerkin/collocation method, based on these kernels, is employed to solve various fractional problems, offering explicit or simplified calculations and efficient solutions. This method yields improved results with reduced computational costs, making it suitable for complex domains.

In this paper, Journe wavelet function is introduced as a wavelet generating function. The expression of reproducing kernel function for the image space of this wavelet transform is obtained based on the fact that the image space of the wavelet transform is a reproducing kernel Hilbert space. Then the isometric identity of Journe wavelet transform is obtained. The connections between the image space of the wavelet transform and the image space of the known reproducing kernel space are established by the theories of reproducing kernel. The properties and the structures of the image space of the wavelet transform can be characterized by the properties and the structures of the image space of the known reproducing kernel space. Using the ideas of reproducing kernel, we consider there are relations between the wavelet transform and the sampling theorem. Meanwhile, the approximations in sampling theorems is shown and the truncation error is given. This provides a theoretical basis for us to study the image space of the general wavelet transform and broadens the scope of application of theories of the reproducing kernel space.

We consider irregular sampling in shift invariant spaces V of higher dimensions. The problem that we address is: find ε so that given perturbations (λ_k) satisfying sup|λ_k| < ε, we can reconstruct an arbitrary function f of V as a Riesz basis expansions from its irregular sample values f(k + λ_k). A framework for dealing with this problem is outlined and in which one can explicitly calculate sufficient limits ε for the reconstruction. We show how it works in two concrete situations.

The fusion of wavelet technique and support vector machines (SVMs) has become an intensive study in recent years. Considering that the wavelet technique is the theoretical foundation of multiresolution analysis (MRA), it is valuable for us to investigate the problem of whether a good performance could be obtained if we combine the MRA with SVMs for signal approximation. Based on the fact that the feature space of SVM and the scale subspace in MRA can be viewed as the same Reproducing Kernel Hilbert Spaces (RKHS), a new algorithm named multiresolution signal decomposition and approximation based on SVM is proposed. The proposed algorithm which approximates the signals hierarchically at different resolutions, possesses better approximation of smoothness for signal than conventional MRA due to using the approximation criterion of the SVM. Experiments illustrate that our algorithm has better approximation of performance than the MRA when being applied to stationary and non-stationary signals.

In this paper, we show that the space of continuous wavelet transform is a reproducing kernel Hilbert space based on the fundamental theorem of linear transform. An admissible wavelet is got by convolution computation which is made into continuous wavelet transform. By the theory of reproducing kernel we can discuss correlative properties of image space of wavelet transform, which provide theoretic frame for us to study image space of the general wavelet transform.

The reproducing kernel Hilbert space construction is a bijection or transform theory which associates a positive definite kernel with a Hilbert space of functions. Recently, reproducing kernel Hilbert space (RKHS) has come wildly alive in the pattern recognition and machine learning community. In this paper, we propose a novel method named multiple kernel learning with reproducing property (MKLRP) to achieve some classification tasks. The MKLRP consists of two major steps. First, we find the basic solution of a generalized differential operator by delta function, and prove this basic solution is a new specific reproducing kernel called H²-reproducing kernel (HRK) in RKHS. Second, in RKHS, we prove that the HRK satisfies the condition of Mercer kernel. Furthermore, a novel specific multiple kernel learning (MKL) called MKLRP, which is based on reproducing kernel is proposed. We perform an extensive experimental evaluation on synthetic and real-world data, which shows the effectiveness of the proposed approach.

In this paper, we classify the family of entire functions $\{\rho ,\sigma \}$ such that the induced weighted composition map $A_{\rho ,\sigma }$, is a conjugation namely $J_{b,\alpha }$, where $b$ is an arbitrary complex number and $\alpha$ is a real number such that $0<\alpha <1$. In the last section, we characterize the $J_{b,\alpha }$-symmetric maximal differential operators on the Fock space ${𝔽}_{\alpha }^2$.

By a new concept and method we shall give practical and numerical solutions of linear singular integral equations by combining the two theories of the Tikhonov regularization and reproducing kernels.

We show how to use Guilbart's embedding of signed measures into a R.K.H.S. to study some limit theorems for random measures and stochastic processes.

We shall give very natural, analytical, numerical and approximate real inversion formulas of the Laplace transform for natural reproducing kernel Hilbert spaces by using the ideas of best approximations, generalized inverses and the theory of reproducing kernels having a good connection with the Tikhonov regularization. These approximate real inversion formulas may be expected to be practical to calculate the inverses of the Laplace transform by computers when the real data contain noises or errors. We shall illustrate examples, by using computers.

We give a new algorithm constructing harmonic functions from data on a part of a boundary. Our approach is based on a general concept and we can apply our methodology to many problems, but here we numerically deal with an ill-posed Cauchy problem which appears in many applications. On some part Γ of a boundary, for suitably given functions f and g, we look for a constructing formula of an approximate harmonic function u satisfying u = f and , the outer normal derivative. Our method is based on the Dirichlet principle by combinations with generalized inverses, Tikhonov's regularization and the theory of reproducing kernels.

In our previous papers we have proved a theorem on harmonic extension of (complex) harmonic functions on "N_p-balls" by using the "harmonic Bergman kernel". Here we consider the "Cauchy integral representation" and give another proof of the theorem on harmonic extension by using the "Cauchy integral kernel".

We introduce our three types of numerical inversion formulas of the Laplace transform based on a Fredholm integral equation of the second kind, the Sinc functions (the Sinc method) and the numerical singular value decomposition of the Laplace transform in a certain reproducing kernel Hilbert space. However, we will need the power of the high-accuracy numerical method with multiple-precision arithmetic for difficult situations. We shall give computational experiments for some very difficult situations for the real inversion.

We shall give a new inversion formula for a linear system based on physical experimental data and by using reproducing kernels and Tikhonov regularization. In particular, we will not make any analytical assumption on the linear system, but will use physical experimental data for obtaining an approximate inversion formula for the linear system.

A dictionary is presented listing concepts from sampling theory in reproducing kernel spaces and their counterparts in harmonic analysis. These concepts help us to discuss concrete and discrete reproducing formulae in the setting of operators on Paley–Wiener spaces. The Riesz transforms provide an example.