237 views

Your TeX didn't render on my screen and it looks a bit off. I believe you're asking about the topology that consists of sets $\tau = \{O : O \subset X, p \notin O \text{ or } O = X\}$. This is ...

166 views

Given data sites (centers) $\{x_i\}_{i=1}^N$ and data values to interpolate $\{y_i\}_{i=1}^N$, the thin plate spline interpolant is the function $$s(x) = \sum_{i=1}^n c_i \varphi(\|x-x_i\|) + \sum_{l=... View answer 1 answers 2 votes 199 views 3 votes In a reproducing kernel Hilbert space K, the reproducing kernel \phi "reproduces" functions pointwise by$$(f, \phi(\cdot, x)_K = f(x)$$You can actually build your own reproducing kernel ... View answer 2 answers 1 votes 87 views 3 votes For a general scattered data interpolation problem with N known data points in \mathbb{R}^d, I recommend trying a radial basis function approach. Let the data sites be denoted \{x_1,...,x_N\} ... View answer 1 answers 3 votes 205 views Accepted answer 3 votes I believe that since T is a positive operator, it has a positive square root. A positive operator is self-adjoint, so T^{\frac{1}{2}} = (T^{\frac{1}{2}})^*. Therefore, you can decompose T into ... View answer 1 answers 1 votes 83 views Accepted answer 3 votes What have you tried so far? I'll give some hints to help you get started and more details if you need. For 1: To show that W_1 \cap W_2 is a subspace, use your definitions! Remember that a subset ... View answer 1 answers 2 votes 436 views Accepted answer 2 votes I might be misunderstanding the question here, but it seems that you're a bit confused about what \langle f,k(x_i,\cdot)\rangle  means. Yes, f is a vector, but it's not necessarily a column vector ... View answer 2 answers 2 votes 208 views 2 votes For multivariate/spatial interpolation (I'm interested in RBFs and meshfree methods), I see things published in SIAM Journal of Numerical Analysis, Mathematics of Computation (Math. Comp), Foundations ... View answer 3 answers -1 votes 59 views 2 votes Hint: You can re-write this as 4 u^{-\frac{1}{2}}. Now, use the rule for integrating functions of the form \int u^n du. I'll give more details if you need. View answer 2 answers 0 votes 122 views 2 votes Hint: \int x^\alpha dx = \frac{1}{\alpha + 1} x^{\alpha + 1} + C, as long as \alpha \neq -1. Also, note that \int 8 t^{-\frac{1}{2}} dt = 8 \int t^{-\frac{1}{2}} dt. View answer 1 answers 0 votes 196 views Accepted answer 2 votes I think you might want a slight change. In your construction, you can't say E^*_2 = E^*_2 - (E_1 \cup E_2), because you have E_2^* on both sides. But, you're close. Define E_k^* = E_k - [\... View answer 2 answers 0 votes 3k views 2 votes There are a few ways to do this. I'm going to assume you know the chain rule and how to differentiate sine and cosine. Then,$$\frac{d}{dx} \csc(x) = \frac{d}{dx} \frac{1}{\sin(x)} = \frac{d}{dx} (\...

326 views

I might be misinterpreting your question, but here is one approach (which Leonid Kovalev discussed a bit above). Let's say we have some samples of a function $f: \mathbb{R}^n \to \mathbb{R}$ at some ...

565 views
Here is a hint to help you get started. First, recall the definition of a Cauchy sequence. We need to show that for each $\epsilon > 0$, there exists an $N$ such that for all $m,n > N$, we have $... View answer 1 answers 3 votes 215 views 1 votes I've ben informed that on the sphere, the Laplacian is rotation invariant. Therefore, the claim that was given to me in the original question appears to be true. I also found the following .pdf that ... View answer 3 answers 3 votes 21k views 1 votes The two vectors you list are linearly dependent, as one is just a scalar multiple of the other. This matrix has one eigenvector corresponding to$\lambda = 1$, given by the vector$(1 \; 0)^T$and ... View answer 1 answers 1 votes 168 views Accepted answer 1 votes I can't remember Egorov's Theorem exactly to be honest, so here's another approach if you're curious. I'll give some hints and more details if you want. With the additional assumption that the$f_n$... View answer 1 answers 1 votes 247 views Accepted answer 1 votes This may not be exactly what you're looking for, but I'm suspicious of your function definition of a subsequence that you mention. One can view a sequence$(a_n)$as a function$f: \mathbb{N} \to \...