Radial Basis Functions (RBFs) are commonly used for interpolating scattered data, in numerical meshfree simulation methods, and in artificial neural networks

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### Prove that modified RBF function satisfies Mercer conditions.

Suppose that I have a modified RBF kernel function. $k(\mathbf{x},\mathbf{y}) = \exp{(-||\mathbf{x}-P\mathbf{y}||^2 })$ where $\mathbf{x},\mathbf{y}$ represent $d$ dimensional inputs and $P$ is the ...
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### Ensuring Symmetry in Mixed Derivatives Using RBF-FD Method

Hello Mathematics Stack Exchange Community, I'm working on a numerical problem where I have the first-order partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ of a ...
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### Proving Divergence Free Behavior of Matrix Valued Radial Basis Function

I'm am using radial basis functions to interpolate magnetic fields, which are divergence free. I have found several research papers that state that the following takes a scalar valued Radial Basis ...
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### Showing that a function is constant on a disc D

Recall that a function $f(z)$ is called radial if it is constant along the circles of center $0$. Let $f$ be a radial holomorphic function defined on the unit disc $D$. Show that $f$ is constant. (...
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### Understanding the use of Radial Basis Function in Linear Regression

I am attempting to understand the use of Radial Basis Functions (RBFs) as used in linear regression. Building the problem: RBFs can be used as a means of separating data which is not linearly ...
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### Integral calculation of Fourier transform of a function

In this paper, an oscillatory radial basis function is introduced and its properties are considered. The RBF is as follows: \phi_d=\frac{J_{d/2-1}(\epsilon r)}{(\epsilon r)^{d/2-1}},\quad d=2,3,4,......
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### Absolute value of an RBF distance is less than the absolute value of an actual distance

I have a radial basis function with a linear kernel $f(r)=r$ in $3D.$ I constructed the surface based on this RBF and noticed that the absolute value of actual distance from any point to the ...
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### Gradient descent in $n$-dimensional space in the context of an RBF network

I am trying to implement an algorithm to perform gradient descent in a $n$-dimensional space in the context of an RBF network. My network has 5 inputs and 1 output. It has the following Gaussian ...
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### Positive weights in Radial Basis Functions

Let $\phi$ be a positive definite radial kernel in $\mathbb{R}^d$. I have a point cloud with positions $(\mathbf{x}_i)_{i\in I}$. RBF interpolation of real valued data $f = (f_i)_{i\in I}$ on this ...
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### Radial Basis Function RBF Gaussian based Interpolation

Based on short description below (an image), how do I find the highlighted f function value? I understand that it is a value associated with the vertex, sorry I am not a good math student to ...
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### Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on ...
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### Showing that a thin-plate spline RBF approximation is real analytic

I am finishing my Ph.D. dissertation in engineering and I would like to show a simple proof. I am having troubles formalizing my ideas into a proof though. I think in a mathematics paper this concept ...
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### The positive-definite-ness of RBF kernel

In Micchelli's paper Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions it mentioned that the RBF kernel $e^{-\alpha^2\|x^i-x^j\|^2/2}$ is positive ...
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