Questions tagged [interpolation-theory]

For questions about interpolation of operators. This includes: real and complex interpolation, interpolation estimates, interpolation spaces. Questions about the estimation of a function from a given input should be asked under the [interpolation] tag instead.

42 questions
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If we determinate the second-degree interpolation polynomial that pass through the points $(a,f(a)),(b,f(b))$ and $(c,f(c))$ as: $$P_2(x)=f(a)+f[a,b](x-a) +f[a,b,c](x-a)(x-b)$$ Where the values of $... 0answers 39 views Understanding multilinear interpolation Do I understand the interpolation of bounded multilinear operators on Banach spaces correct ? For simplicity, let us Consider bilinear operators on$L^p$spaces. Suppose the bilinear operator$T$... 0answers 23 views Counterexample in real interpolation Can we find a linear operator$T$that maps$L^1$to$L^{2,\infty}$and$L^2$to$L^{2,\infty}$such that$T$does not map any$L^p$,$1<p<2$to$L^{2,\infty}$? Can we find a linear operator$T$... 0answers 119 views Proving the Marcinkiewicz Interpolation Theorem I am trying to prove a form of the Marcinkiewicz Interpolation Theorem.$T$mapping a measurable function to a measurable function is sublinear if $$|T(f_1 + f_2)(x)| \leq |T(f_1)(x)| + |T(f_2)(x)|... 0answers 82 views What is the smallest polynomial with almost linear condition? What is the smallest degree univariate polynomial with f(0)=0, f(1)\in(0,0.5) and f(x)={x-1}+\epsilon where \epsilon\in(-0.5,0.5) at every x\in\{2,\dots,a-1,a\}? Can we have O((\log a)^c) ... 0answers 105 views Weighted vector norms and interpolation Let \mu be a probability vector of \mathbb{R}^n. Then we can define the weighted \ell^p (1 \leq p < \infty) norms by$$ \|x\|_{p,\mu} = \left(\sum_{i=1}^n |x_i|^p \mu_i\right)^{1/p},$$... 0answers 34 views Complex interpolation between H^1 and L^1 We have the complex interpolation [H^1,L^p]_\theta=L^q for 1<p<\infty where H^1 is the Hardy space and \frac{1}{q}=1-\theta+\frac{\theta}{p}. This raises the obvious question as to ... 0answers 19 views Interpolation between log and polynomials using Riesz-Thorin I consider two L^1 weighted spaces, with m_1(x) = e + x, m_0 = \ln(e+x). It is known that the Riesz-Thorin interpolation theorem holds for L^p-weighted space. I have an operator T: L^1(m_1) \to ... 0answers 25 views Reference for compact embedding for Holder space on \mathbb{R}^n Suppose 0<\alpha<\beta, and \Omega is a bounded subset of \mathbb{R}^n. Then the Holder space C^{\beta}(\Omega) is compactly embedded into C^{\alpha}(\Omega). But if \Omega=\mathbb{R}^... 0answers 32 views 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}\... 0answers 59 views Choose support for polynomial interpolation I have the following table, where t is defined in minutes and v(t) in m / s: \begin{array}{|c|c|c|c|} \hline t& 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\ \hline v(t) & ... 0answers 81 views Use 1-degree Chebyshev polynom to approximate \cos(x) and calculate the error The task is to give for \cos(x) the nodes of the interpolation polynom of degree 1 that approximates the function on [-\pi,\pi] the best as well as the related error. I want to solve this task ... 0answers 367 views Linear interpolation of over time of standard deviation measurements I have backscatter (radar image value) measurements of corn fields taken at multiple points along the growing season. I can estimate the expected backscatter value of corn by plotting the mean ... 0answers 13 views Reference for complex interpolation for the product of two normed space Let X=X_1\times X_2 and Y=Y_1\times Y_2 and (X_i,Y_i) are interpolation couple. Then it seems obvious that$$ [X,Y]_{\theta}=[X_1,Y_1]_\theta \times [X_2,Y_2]_\theta $$for every \theta\in (... 0answers 68 views Approximation with stability constant 1 I am looking for an approximation method for a typical 1-dimensional approximation problem, that has stability constant 1. The setting We have an interval [a,b] an unknown function f on [a,b] ... 0answers 41 views (X,Y) interpolation couple, then (X^\prime,Y^\prime) interpolation couple Let (X,Y) be an interpolation couple, i.e. there exists a Hausdorff topological vector space U s.t. both X and Y embed continuously into U. We define X\cap Y and X+Y in U and turn them ... 0answers 155 views How to prove that a quadratic function is equal to quadratic spline? I need to prove that a quadratic spline S and a quadratic function f are equal. We are given points ([X_1,f(X_1)],.....,[X_n,f(X_n)]) and the boundary condition that the first derivatives of ... 0answers 800 views Tricubic Interpolation I am currently writing a plugin for 3D analysis software and I am working with a data grid where certain values are stored at XYZ coordinates, and I need to find an estimated value of a point that ... 0answers 42 views Example of a “abrupt function” I need example of a simple function to show that cubic spline gives better result than Lagrange's interpolation in case of some special functions. Thank you 0answers 60 views Property of intersections of Bochner spaces My question: Assume I have a function u \in H^2(0,T;L^2(\Omega)) \cap L^2(0,T;H^2(\Omega)). Now I want to bound the gradient of u. Can I deduce that u \in H^1(0,T;H^1(\Omega)) and under which ... 0answers 27 views Monotonic linear interpolation of monotonic data on a scattered grid Let x_1,\dots,x_n\in[0,1]^d and y_1,\dots,y_n\in[0,1] satisfy:$$\forall i,j\in {1,\dots,n},\hspace{10pt} x_i \le x_j \rightarrow y_i \le y_j.$$I seek to find a continuous function f:[0,1]^d\... 0answers 15 views Hello. How to prove theorem about relationship between the interpolation operator norm and the constant of Lebesgue: A norm of the linear operator P_n can be defined by ||P_n|| = max_{f \in [a,b]} \frac{||P_n f||}{||f||} where on the right-hand side one takes any convenient norm for functions. Taking the ... 0answers 60 views Question about linear splines. I have a question about linear splines. A computer package I am using has the option to use linear splines, but there is no user manual I can find. I am having problems with the kind of inputs the ... 0answers 45 views Marcinkiewicz Interpolation for exponents p_0, q_0, p_1, q_1 \in (0, \infty]. I would like to try and extend the Marcinkiewicz Theorem in Folland (Theorem 6.28) using a similar argument in the text. The theorem I am after is as follows: Let p_0, p_1, q_0, q_1 \in (0, \infty]... 0answers 45 views Prove | u(x) - u_h(x) | \leq h \ \underset{0 \leq y \leq 1}{\max} |u''(y)| Prove: | u(x) - u_h(x) | \leq h \ \underset{0 \leq y \leq 1}{\max} |u''(y)| for 0 \leq x \leq 1 Using the fact that || (u-u_h)' || \leq \underset{0 \leq y \leq 1}{\max} |u''(y)| and ... 0answers 66 views What is the basic difference between interpolation & inference? In mathematics we've studied interpolation as predicting the structure of a function from it's given finitely many values from which consequently we use to construct certain function which nearly ... 0answers 90 views Linear Interpolation Error estimation Suppose we have applied trilinear interpolation technique over grid points in the space A\times B \times C \times D={(x,y,z)|x\in A, y\in B,z \in C}} . The interpolant, which is piecewise function, ... 0answers 11 views how to estimate maximum of Lebesgue function of arbitrary nodes? Denote S_n=\left\{x_{n}=(x_{n0},x_{n1},...,x_{nn})|a\leq x_{n0}<x_{n1}<\cdots<x_{nn}\leq b\right\} with -\infty<a<b<+\infty and n\geq 1. For any x_n\in S_n, define the ... 0answers 23 views Romberg's algorithm for integration what does the row and the column in Romberg's algorithm for integration indicate? I assume that the column represents the integration value with smaller intervals considered for calculating the ... 0answers 39 views Interpolation polynomial for f=|x|. Why the interpolation polynomials for f=|x|,x\in[-1,1] will oscillate near the endpoint of [-1,1] as n increases? I know that one explanation of the Runge's phenomenon is that the interpolation ... 0answers 47 views Question above proof of Marcinkiewicz Interpolation Theorem I am reading through the proof in Tao's lecture notes and I have a question about justifying a "without loss of generality" statement. Statement: Let T be a sublinear operator taking functions on (... 0answers 40 views What is the Lagrange Interpolation polynomial of 1/{(x-1)}? So what is the value of$$L[x_0,x_1,\ldots,x_n; \; 1/{(x-1)}]=\text{?}$$I tried writing a Mathematica program to compute it, but I couldn't figure it out... 0answers 342 views How can Newton's Divided Difference Interpolation accuracy be improved? Are there any modifications to Newton's divided difference interpolation that can be made in order to improve the accuracy of the value given, specifically for exponential functions such as$y=e^{-x}-...
Determine the nodes of the first degree interpolating polynomial which approximate $f(x) = cos(x)$ the best on $[-\pi,\pi]$ in the uniform norm. What is the interpolation error? I think I need a ...
I am a little bit lost in interpolation theory. Let $A$ be a linear operator on $\mathbb{R}^{n}.$ Denote $V_0$ a subspace of $\mathbb{R}^n$ of codimension one. Suppose that $AV_0 \subseteq V_0$ and we ...