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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 ...
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        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? ...
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        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 ...
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        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 ...
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        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 $f:U_\epsilon(0)\to\mathbb{R}$ ...
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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 ...
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        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 ...
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        Interpolation theory

        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}\...
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        An open problem in Sobolev spaces

        Let $\Omega\subset\mathbb{R}^n$ be 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 ...
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        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 ...
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        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 ...
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        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 ...
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        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)^{...
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        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:=&\...
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        Chebyshev interpolation [closed]

        Let's define the n-th degree Chebyshev polynomials by $$ T_{n} (x)=\cos(n\arccos(x)).$$ Find a polynomial $P$ such that $$\mid y- P (x) \mid$$ is minimal, using the first three Chebyshev ...

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