Convergence Rates of the Hilbert Uniqueness Method via Tikhonov Regularization

In this paper, we identify the Hilbert uniqueness method for a boundary control problem with the calculation of the pseudo inverse. Because of its ill-posedness, we approximate it by a regularized Hilbert uniqueness method, which we prove to be identical with Tikhonov regularization. By this equivalence, we can find sufficient conditions for convergence and convergence rates, which require approximation rates in Müntz spaces. We show that these conditions are fulfilled by an a priori bound in Sobolev norms on the exact solution.     

Generalizations of Tikhonov’s regularized method of least squares to non-Euclidean vector norms

Tikhonov’s regularized method of least squares and its generalizations to non-Euclidean norms, including polyhedral, are considered. The regularized method of least squares is reduced to mathematical programming problems obtained by “instrumental” generalizations of the Tikhonov lemma on the minimal (in a certain norm) solution of a system of linear algebraic equations with respect to an unknown matrix. Further studies are needed for problems concerning the development of methods and algorithms for solving reduced mathematical programming problems in which the objective functions and admissible domains are constructed using polyhedral vector norms.     

Learning gradients via an early stopping gradient descent method

We propose an early stopping algorithm for learning gradients. The motivation is to choose “useful” or “relevant” variables by a ranking method according to norms of partial derivatives in some function spaces. In the algorithm, we used an early stopping technique, instead of the classical Tikhonov regularization, to avoid over-fitting.After stating dimension-dependent learning rates valid for any dimension of the input space, we present a novel error bound when the dimension is large. Our novelty is the independence of power index of the learning rates on the dimension of the input space.     

Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley–Wiener spaces

We shall discuss the relations among sampling theory (Sinc method), reproducing kernels and the Tikhonov regularization. Here, we see the important difference of the Sobolev Hilbert spaces and the Paley–Wiener spaces when we use their reproducing kernel Hibert spaces as approximate spaces in the Tikhonov regularization. Further, by using the Paley–Wiener spaces, we shall illustrate numerical experiments for new inversion formulas for the Gaussian convolution as a much more powerful and improved method by using computers. In this article, we shall be able to give practical numerical and analytical inversion formulas for the Gaussian convolution that is realized by computers.     

Theory of regularizability in topological vector spaces

Two known definitions of regularizability for topological vector spaces are found to be equivalent. Regularizability in the sense of Tikhonov is considered in reflexive linear metric spaces. In particular, an example is presented of a linear continuous injective operator on a reflexive Frécnet space whose inverse cannot be regularized. The latter indicates the sharp difference between regularizability in Fréchet spaces and in Banach spaces, respectively.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 777–781, June, 1990.     

Well-posed constrained optimization problems in metric spaces

Relations between different notions of well-posedness of constrained optimization problems are studied. A characterization of the class of metric spaces in which Hadamard, strong, and Levitin-Polyak well-posedness of continuous minimization problems coincide is given. It is shown that the equivalence between the original Tikhonov well-posedness and the ones above provides a new characterization of the so-called Atsuji spaces. Generalized notions of well-posedness, not requiring uniqueness of the solution, are introduced and investigated in the above spirit.     

Solutions of Tikhonov Functional Equations and Applications to Multiplication Operators on Szegö Spaces

We consider a natural representation of solutions for Tikhonov functional equations. This will be done by applying the theory of reproducing kernels to the approximate solutions of general bounded linear operator equations (when defined from reproducing kernel Hilbert spaces into general Hilbert spaces), by using the Hilbert–Schmidt property and tensor product of Hilbert spaces. As a concrete case, we shall consider generalized fractional functions formed by the quotient of Bergman functions by Szegö functions considered from the multiplication operators on the Szegö spaces.     

Regularization method with two parameters for nonlinear ill-posed problems

This paper is devoted to the regularization of a class of nonlinear ill-posed problems in Banach spaces. The operators involved are multi-valued and the data are assumed to be known approximately. Under the assumption that the original problem is solvable, a strongly convergent approximation procedure is designed by means of the Tikhonov regularization method with two pa- rameters.     

Tensor norms and operators in the category of Banach spaces

The theory of dual functors in the category of Banach spaces is applied to the study of tensor norms in the sense of Grothendieck. The dual functors of the tensor norms arising from the projective and inductive tensor product as well as from more general tensor norms, such as the norms d_p of Saphar, are identified as various spaces of operators, which include p-integral and absolutely p-summing operators. Properties of these operators are then easily derived by categorical means. Applications of the methods provide simplified proofs of composition theorems and the characterization of dual spaces of type (L).The author acknowledges the hospitality of the University of Massachusetts at Amherst during the year when the first draft of this paper was written as well as support from the Natural Sciences and Engineering Research Council of Canada.     

Source inversion of heat conduction from a finite number of observation data

This article deals with a backward heat conduction type problem. Namely, the Gaussian convolution is here analysed in a new way so that inverse source formulae to the heat conduction problem are obtained from a finite number of observation data at time and space points. In view of obtaining this main goal, different reproducing kernel Hilbert spaces, iteration schemes and Tikhonov regularization procedures are used and combined in an unified way.     

Lifting functors to Eilenberg-Moore category of monad generated by functor CpCp

The second iteration of the contravariant functor of spaces of continuous functions in the pointwise convergence topology is a functorial part of a monad (triple) on the category of Tikhonov spaces. The problem of lifting functors to the Eilenberg-Moore category of this monad is investigated.Published in Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1289–1291, September, 1992.     

General Inhomogeneous Discrete Linear Partial Differential Equations with Constant Coefficients on the Whole Spaces

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.     

Extremal Functions on Sturm–Liouville Hypergroups

We give an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev spaces associated with a singular second-order differential operator. Next, we come up with some results regarding the multiplier operators for the Sturm–Liouville transform.     

Product of Volterra Type Integral and Composition Operators on Weighted Fock Spaces

We characterize the bounded, compact, and Schatten class product of Volterra type integral and composition operators acting between weighted Fock spaces. Our results are expressed in terms of certain Berezin type integral transforms on the complex plane ?. We also estimate the norms and essential norms of these operators in terms of the integral transforms. All our results are valid for weighted composition operators when acting between the class of weighted Fock spaces considered.     

On Function Spaces with Mixed Norms — A Survey

The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is to provide a detailed proof for a useful inequality about mixed Lebesgue norms and the Hardy–Littlewood maximal operator and also to improve some known results on the maximal function characterizations of anisotropic mixed-norm Hardy spaces and the boundedness of Calderón–Zygmund operators from these anisotropic mixed-norm Hardy spaces to themselves or to mixed Lebesgue spaces. The last one is to correct some errors and seal some gaps existing in the known articles.     

Hardy–Steklov Operators and Duality Principle in Weighted Sobolev Spaces of the First Order

Boundedness criteria for the Hardy–Steklov operator in Lebesgue spaces on the real axis are presented. As applications, two-sided estimates for the norms of spaces associated with weighted Sobolev spaces of the first order with various weight functions and summation parameters are established.     

On Weighted α-Besov Spaces and α-Bloch Spaces of Quaternion-Valued Functions

In this paper, we give the definitions of weighted α-Besov-type spaces and α-Bloch spaces of quaternion-valued functions, then we obtain characterizations of these quaternion α-Bloch spaces by quaternion α-Besov-type spaces. Relations between Q _p norms and weighted α-Besov norms are also considered. The role of ρ?^α sequences in securing non-Bloch functions is highlighted in quaternion sense.     

Tikhonov regularization method for a system of equilibrium problems in Banach spaces

The purpose of this paper is to investigate the Tikhonov regularization method for solving a system of ill-posed equilibrium problems in Banach spaces with a posteriori regularization-parameter choice. An application to convex minimization problems with coupled constraints is also given.     

Dunkl Multiplier Operators on a Class of Reproducing Kernel Hilbert Spaces

We study some class of Dunkl multiplier operators; and we establish for them the Heisenberg-Pauli-Weyl uncertainty principle and the Donoho-Stark''s uncertainty principle. For these operators we give also an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev-Dunkl spaces.