These notes give an introduction to the theory of reproducing kernel Hilbert spaces and their multipliers. We begin with the material that is contained in Aronszajn’s classic paper on the subject. We take a somewhat algebraic view of some of his results and discuss them in the context of pull-back and push-out constructions. We then prove Schoenberg’s results on negative definite functions and ...

The problem of finding the pattern that deviates from other observation is termed as outlier. detection outlier getting importance in research area nowadays due to reason technique has been used various mission critical applications such military, health care, fault recovery, and many. analysis functional data its depth function plays a crucial role statistical model for detecting values alone ...

We establish the following Hilbert-space analogue of Gleason-Kahane-\.Zelazko theorem. If $\mathcal{H}$ is a reproducing kernel Hilbert space with normalized complete Pick kernel, and if $\Lambda$ linear functional on such that $\Lambda(1)=1$ $\Lambda(f)\ne0$ for all cyclic functions $f\in\mathcal{H}$, then multiplicative, in sense $\Lambda(fg)=\Lambda(f)\Lambda(g)$ $f,g\in\mathcal{H}$ $fg\in\m...

Important information on the structure of complex systems can be obtained by measuring to what extent the individual components exchange information among each other. The linear Granger approach, to detect cause-effect relationships between time series, has emerged in recent years as a leading statistical technique to accomplish this task. Here we generalize Granger causality to the nonlinear c...