Articles of interpolation

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 simple manifolds (e.g. on a unit sphere) shows that computed eigenvalues of $D$ often have non-zero imaginary parts, even though they should all be real. (For simplicity […]

Interpolation inequality

Lef $u$ be at least a $C^2$ function on $\mathbb{R}^n$. Let’s denote the gradient by $D$. Also, (using the multiindex notation), define the seminorm $$||D^ku|| = \sup_{|\gamma|=k}{\sup_x{|D^{\gamma}u|}}$$ How can we prove the following : $$||Du|| \leq \epsilon||D^2u|| + C||u|| $$ where $C$ is some constant depending on $\epsilon$

How to find B-Spline represenation of an Akima spline?

Given points $t_i$ and values $y_i$, I’d like to use Akima interpolation to interpolate to a different set of locations $x_j$. This means I need to calculate the cubic polynomials $A_{3,t}(x)$. Given that these $A_{3,t}(x)$ are actually splines, it should be possible to find their B-Spline coefficients, i.e., $\alpha_l$ such that $A_{3,t}(x) = \sum_l \alpha_l\cdot […]

Finding a polynomial with a given shape

As a followup to $\qquad$Equation of a curve I’ll ask the following question . . . Does there exist a polynomial $f(x)$ with real coefficients such that $f(0) = 2,\;\;f(1) = 3,\;\;f(2)=0,\;\;f(3)=1$. $f$ is$\;$increasing on $(-\infty,1],\;$decreasing on $[1,2],\;$ increasing on $[2,\infty)$. $f$ has exactly one inflection point. Assuming such a polynomial exists, Can someone provide […]

Multivariate function interpolation

I have a (nonlinear) function which takes as input 4 parameters and produces a real number as output. It is quite complex to compute the function value given a set of parameters (as it requires a very big summation). I’d like to answer queries on this function efficiently so I was thinking of trying to […]

Is there a cubic spline interpolation with minimal curvature?

I came across the term “cubic spline with minimal curvature”. However, I am not able to find any documentations/explaination on its computation method. Can anyone help me by advising how I can go about finding more information (maybe there is another more common name)? Thanks a lot! Ryou

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal while the last one only in a single program. How can you make the 3D object smooth mathematically?

Least squares fit for an underdetermined linear system

I want fitting my data using bicubic interpolation: $$f(x,y)=\sum_{i=0}^{3}\sum_{j=0}^{3}a_{ij}x^iy^j$$ Let known $$f(0, 0)=1; f(2, 0)=1;f(1, 1)=0;f(0, 2) = 1; f(2, 2)=1$$ I used least squares method, $$min\sum_{k=1}^{5}(f(x_k, y_k)-\sum_{i=0}^{3}\sum_{j=0}^{3}a_{ij}(x_k)^i(y_k)^j)^2$$ Receiving this system: $$\forall t, s \in [0, 3]: \sum_{k=1}^{5}\sum_{i=0}^{3}\sum_{j=0}^{3}a_{ij}(x_k)^{i+t}(y_k)^{j+s} = \sum_{k=1}^{5}(x_k)^{t}(y_k)^{s}f(x_k, y_k)$$ If present as $Ax = b$: $$A = \begin{pmatrix} 1 & 1 & 1 […]

What is a cardinal basis spline?

Wikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as “Bk“. Meanwhile, Signal Reconstruction says: The basis splines Bn are shown … as the order increases, the functions approach the Gaussian function, which is exactly B∞. but then says as the order increases, the cardinal basis splines approximate the […]

Profinite and p-adic interpolation of Fibonacci numbers

On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here) For each profinite integer $s$, one can in a natural way define the $s$th Fibonacci number $F_s$, which is itself a profinite integer. Namely, given $s$, one can choose a sequence of positive integers $n_1, n_2, n_3,\dots$ that […]