scholarly journals Stability and approximation of solutions in new reproducing kernel Hilbert spaces on a semi‐infinite domain

2021 ◽  
Author(s):  
Jabar S. Hassan ◽  
David Grow
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Spaces ◽  
Infinite Domain ◽  
Kernel Hilbert Spaces

scholarly journals Stability and approximation of solutions in new reproducing kernel Hilbert spaces on a semi-infinite domain

2021 ◽  
Author(s):  
Jabar Hassan ◽  
David E. Grow
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hilbert Spaces ◽  
Uniform Approximation ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Spaces ◽  
Hyperbolic Partial Differential Equations ◽  
Infinite Domain ◽  
Kernel Hilbert Spaces ◽  
The Stability

We introduce new reproducing kernel Hilbert spaces on a trapezoidal semi-infinite domain $B_{\infty}$ in the plane. We establish uniform approximation results in terms of the number of nodes on compact subsets of $B_{\infty}$ for solutions to nonhomogeneous hyperbolic partial differential equations in one of these spaces, $\widetilde{W}(B_{\infty})$. Furthermore, we demonstrate the stability of such solutions with respect to the driver. Finally, we give an example to illustrate the efficiency and accuracy of our results.

INDEFINITE KERNEL NETWORK WITH DEPENDENT SAMPLING

2013 ◽  
Vol 11 (05) ◽  
pp. 1350020 ◽  
Author(s):  
HONGWEI SUN ◽  
QIANG WU
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Function Class ◽  
Positive Definiteness ◽  
Reproducing Kernel Hilbert Spaces ◽  
Strong Mixing ◽  
Mixing Condition ◽  
Indefinite Kernel ◽  
Learning Rates ◽  
Kernel Hilbert Spaces

We study the asymptotical properties of indefinite kernel network with coefficient regularization and dependent sampling. The framework under investigation is different from classical kernel learning. Positive definiteness is not required by the kernel function and the samples are allowed to be weakly dependent with the dependence measured by a strong mixing condition. By a new kernel decomposition technique introduced in [27], two reproducing kernel Hilbert spaces and their associated kernel integral operators are used to characterize the properties and learnability of the hypothesis function class. Capacity independent error bounds and learning rates are deduced.

scholarly journals Scattering Systems with Several Evolutions and Formal Reproducing Kernel Hilbert Spaces

2014 ◽  
Vol 9 (4) ◽  
pp. 827-931 ◽  
Author(s):  
Joseph A. Ball ◽  
Dmitry S. Kaliuzhnyi-Verbovetskyi ◽  
Cora Sadosky ◽  
Victor Vinnikov
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Spaces ◽  
Kernel Hilbert Spaces ◽  
Scattering Systems

scholarly journals THE DECAY OF THE WALSH COEFFICIENTS OF SMOOTH FUNCTIONS

2009 ◽  
Vol 80 (3) ◽  
pp. 430-453 ◽  
Author(s):  
JOSEF DICK
Keyword(s):  
Lower Bound ◽  
Fractional Order ◽  
Bounded Variation ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Upper Bounds ◽  
Reproducing Kernel Hilbert Spaces ◽  
Smooth Functions ◽  
Kernel Hilbert Spaces

AbstractWe give upper bounds on the Walsh coefficients of functions for which the derivative of order at least one has bounded variation of fractional order. Further, we also consider the Walsh coefficients of functions in periodic and nonperiodic reproducing kernel Hilbert spaces. A lower bound which shows that our results are best possible is also shown.

scholarly journals Factorizations of Kernels and Reproducing Kernel Hilbert Spaces

2017 ◽  
Vol 87 (2) ◽  
pp. 225-244 ◽  
Author(s):  
Rani Kumari ◽  
Jaydeb Sarkar ◽  
Srijan Sarkar ◽  
Dan Timotin
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Spaces ◽  
Kernel Hilbert Spaces
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