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# Questions tagged [interpolation]

Interpolation is the theory of constructing smooth functions, usually polynomials or trigonometric polynomials, whose graph passes through a number of given points in the plane. Splines and Bézier curves, piecewise linear or cubic interpolation, Lagrange and Hermite interpolation are example topics.

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### Polynomial-preserving boundary conditions for spline interpolation

Spline interpolation requires the definition of boundary condition because the smoothness requirements do not yield enough conditions for a unique solution. Question: which kind of boundary ...
0answers
47 views

### History of Underdetermined Interpolation

Are there any examples, earlier than spline-interpolation, of mathematical investigations of interpolation problems with more unknowns than conditions or are (polynomial) splines the earliest? ...
1answer
66 views

### Interpolating Maximum function with symmetric polynomials

Let $n$ and $p$ be two positive integers. Consider the function $$\max_{n,p}:\{0,\dots,n\}^p\to\{0,\dots,n\}$$ that computes the maximum of a $p$-tuple of integers in the range $\{0,\dots,n\}$. Are ...
2answers
100 views

### Comparison of methods to define a matrix function (Jordan canonical form, Hermite interpolation and Cauchy integral)? [closed]

There are many equivalent ways of defining a function $f(A)$ of a matrix $A$. We focus on Jordan canonical form, Hermite interpolation and Cauchy integral. What is the difference between methods for ...
1answer
67 views

### On a case of real-analytic interpolation

Given strictly increasing sequence $x_n$ of rational numbers with $\sup x_n = x$. In which case (sufficient condition on $x_n$) there exists real-analytic function $f:U_\epsilon(0)\to\mathbb{R}$ ...
1answer
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### Interpolation of a trilinear functional

Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If ...
1answer
155 views

### Marsden's Identity and B-splines

Marsden's Identity states that for every $\tau$ in $\mathbb{R }$: $$(\cdot -\tau)^{k-1}=\sum_j\Psi_{j,k}(\tau)B_{j,k,t} \, ,$$ with $\Psi_{j,k}=(t_j-\tau)\times...\times(t_{j+k-1}-\tau)$. Following ...
0answers
42 views

Consider the interpolation space $Z=(X,Y)_{\theta,p}$, in the case $Y\subseteq X$ do we have that the following norm: $x\longrightarrow\left(\int_{0}^{a} \vert t^{-\theta}k(t,x)\vert^p \frac{dt}{t}\... 0answers 202 views ### An open problem in Sobolev spaces Let$\Omega\subset\mathbb{R}^nbe a bounded domain. Suppose that there there is a bounded extension operator $$E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n) \quad \text{and} \quad E:W^{1,q}(\Omega)\to ... 1answer 159 views ### For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often Fix an interval [a,b]. Is it true that for every table of interpolating nodes \{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}, there exists a continuous function f:[a,b]\to (0,\infty) such that ... 1answer 121 views ### For which n, can we find a sequence of n+1 distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve Fix an interval [a,b]. For which integers n>1, does there exist n+1 distinct points \{x_0,x_1,...,x_n\} in [a,b] such that for every continuous function f:[a,b] \to (0,\infty), the ... 2answers 417 views ### Does every positive continuous function have a non-negative interpolating polynomial of every degree? Let f:[a,b] \to (0,\infty) be a continuous function. Then is it necessarily true that for every n\ge 1, we can find n+1 distinct points \{x_0,x_1,...,x_n\} in [a,b] such that the ... 1answer 280 views ### Asymptotic behavior of sum linked with Lagrange interpolation I already asked this a few weeks ago with no answer, so let me formulate differently. In performing Lagrange interpolation with nodes 1/n, one encounters the sum$$S(f)=\dfrac{1}{N!}\sum_{n=0}^N(-1)^{... 0answers 42 views ### Defining Boundary Conditions for Spline Interpolation via the Euler–Maclaurin Formula The Euler–Maclaurin formula states an interdependency between \begin{align} I\quad:=&\quad\int_m^nf(x) \, dx;\ m,n\in\mathbb{Z}\\[6pt] S\quad:=&\quad\sum_{k=m}^n f(k) \\[6pt] D\quad:=&\... 1answer 88 views ### Chebyshev interpolation [closed] Let's define the n-th degree Chebyshev polynomials by $$T_{n} (x)=\cos(n\arccos(x)).$$ Find a polynomialP\$ such that $$\mid y- P (x) \mid$$ is minimal, using the first three Chebyshev ...

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