Circular operators on minimal norm ideals ofℬ(ℋ)

In this paperwesolve Sylvester matrix equation with infinitely-many solutions and conduct their classification. If the conditions for their existence are not met, we provide a way for their approximation by least-squares minimal-norm method.

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On weighted Strand’s iteration

Carpathian Journal of Mathematics ◽

10.37193/cjm.2018.02.06 ◽

2018 ◽

Vol 34 (2) ◽

pp. 183-190

Author(s):

D. CARP ◽

◽

C. POPA ◽

T. PRECLIK ◽

U. RUDE ◽

...

Keyword(s):

Iterative Method ◽

Least Squares ◽

Numerical Approximation ◽

Iterative Algorithms ◽

Linear Least Squares ◽

Minimal Norm ◽

Least Squares Problem ◽

Minimal Norm Solution ◽

Linear Least Squares Problem

In this paper we present a generalization of Strand’s iterative method for numerical approximation of the weighted minimal norm solution of a linear least squares problem. We prove convergence of the extended algorithm, and show that previous iterative algorithms proposed by L. Landweber, J. D. Riley and G. H. Golub are particular cases of it.

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Interpolating Functions of Minimal Norm, Star-Invariant Subspaces, and Kernels of Toeplitz Operators

Proceedings of the American Mathematical Society ◽

10.2307/2159482 ◽

1992 ◽

Vol 116 (4) ◽

pp. 1007

Author(s):

Konstantin M. Dyakonov

Keyword(s):

Toeplitz Operators ◽

Invariant Subspaces ◽

Minimal Norm

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On Learning Vector-Valued Functions

Neural Computation ◽

10.1162/0899766052530802 ◽

2005 ◽

Vol 17 (1) ◽

pp. 177-204 ◽

Cited By ~ 191

Author(s):

Charles A. Micchelli ◽

Massimiliano Pontil

Keyword(s):

Hilbert Space ◽

Learning Theory ◽

Reproducing Kernel ◽

Reproducing Kernel Hilbert Space ◽

Support Vector ◽

Translation Invariant ◽

Minimal Norm ◽

Finite Set ◽

Vector Valued Functions ◽

Vector Valued

In this letter, we provide a study of learning in a Hilbert space of vector-valued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a reproducing kernel Hilbert space of functions whose values lie in Y. In this setting, we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory. We consider specific examples of such functionals corresponding to multiple-output regularization networks and support vector machines, for both regression and classification. Finally, we provide classes of operator-valued kernels of the dot product and translation-invariant type.

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Functions of Minimal Norm with the Given Set of Fourier Coefficients

Mathematics ◽

10.3390/math7070651 ◽

2019 ◽

Vol 7 (7) ◽

pp. 651

Author(s):

Pyotr Ivanshin

Keyword(s):

Fourier Series ◽

Existence And Uniqueness ◽

Fourier Coefficients ◽

Nonnegative Function ◽

Trigonometric Fourier Series ◽

Minimum Norm ◽

Minimal Norm ◽

Uniqueness Of The Solution ◽

The Given

We prove the existence and uniqueness of the solution of the problem of the minimum norm function ∥ · ∥ ∞ with a given set of initial coefficients of the trigonometric Fourier series c j , j = 0 , 1 , … , 2 n . Then, we prove the existence and uniqueness of the solution of the nonnegative function problem with a given set of coefficients of the trigonometric Fourier series c j , j = 1 , … , 2 n for the norm ∥ · ∥ 1 .

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