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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.

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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 $...
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1answer
113 views

Interpolation inequality for Holder continuous functions.

Let $\Omega$ be a bounded open connected set in $\mathbb{R}^n$ with $C^1$ boundary and let $0<\alpha<1$. Then there exists a real number $\sigma_0>0$ and a dimensional constant $C>0$ such ...
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1answer
104 views

What is the geometric meaning of this null-determinant?

While reading about interpolation I came across the following (linear algebra) equation in Norlund. It involves determinants and I don't understand it in full yet. I do know how Lagrange and Newton ...
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58 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 ...
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1answer
55 views

How to derive the Newton interpolation polynomial

Consider the interpolation problem: find the polynomial through a given set of points $(x_0,y_0),...,(x_n,y_n)$. Suppose we want the polynomial in newtonian form: $$N(x)=\sum _{j=0}^k[y_0,\ldots ,...
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1answer
87 views

Formula for coefficients of interpolation polynomial

(Question) What is the (conventional) formula for coefficients of interpolation polynomial? Consider the interpolation problem: find the polynomial through a given set of points $(x_0,y_0),...,(x_n,...
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41 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 $p_0, p_1, q_0, q_1 \in (0, \infty]$...
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40 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 ...
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1answer
35 views

Newton-Gregory interpolation with divided differences calculations for new extra interpolation point

So lets suppose that we have the following $x_0=-1, y_0=2$ $x_1= 0, y_1=1$ $x_2=1, y_2=2$ $x_3=3, y_3=10$ and we know that all the above $x_i,y_i$ belong to $p_2(x)=x^2+1$ , and we want to add ...
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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}^...
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62 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 ...
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68 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, ...
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1answer
38 views

For $f(x) = \tan(\pi \cdot x)$, find the Interpolation $Q(x) = b_0 + b_1 x + b_2 \frac{1}{x-\frac{1}{2}}$

Progress so far: In a previous task, I determined a polynomial interpolation using a system of linear equations. The data points to be used were $(0, f(0)), (\frac{1}{6}, f(\frac{1}{6})), (\frac{1}{...
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1answer
25 views

Interpolation- Lagrange polynomial

Let $x_0,x_1,...,x_n$ will be different real numbers. Show, that: $f[x_0,x_1,...,x_n]=\sum_{i=0}^m\frac{f(x_i)}{\Phi '(x)}$ where $\Phi (x)=(x-x_0)(x-x_1)...(x-x_m)$ So, I have some problems.How to ...
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Interpolation with a new point of $f'$

Background: (Lagrange Interpolation) Let $f\in C^{n+1}([a,b])$ and $x_0,...,x_n\in[a,b]$. If they are different there is a unique $p_n\in\mathcal{P}_n$ such that $p_n(x_i)=f(x_i)$. Also, we have ...
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2answers
42 views

Approximation using Lagrange Interpolation

I am aware of the formula of this method. However, is it true that the method produces more accurate polynomial when the $x$ points are closer to each other? if so or not, why? Moreover, If I am ...
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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 ...
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1answer
30 views

Construction of real interpolation Banach space

Let $(A,\| \cdot \|_A)$, $(B,\| \cdot \|_B)$ and $(E,\| \cdot \|_E)$ real Banach spaces such that $A\subseteq E$ and $B\subseteq E$ with continuous injections. Let $0 < \theta < 1$. For $x\in E$ ...
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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 ...
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33 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 $T$ ...
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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}\...
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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 ...
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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 $T$ ...
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43 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 $(...
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2answers
254 views

Who knows this formula for polynomial interpolation?

For my high school math project I studied polynomial interpolation: given a set of points $(x_0,y_0),...,(x_n,y_n)$, find the polynomial of degree $n$ that passes through all points. The solutions by ...
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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) & ...
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1answer
105 views

Polynomial interpolation of Bessel function

I used polynomial interpolation of degree 5 to approximate one 0 of the Bessel function $J_0$. I used the following data: With the data above, I computed a polynomial using the images of the Bessel ...
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29 views

Interpolation space between $L^1$ and $L^1\cap L^2$

Let $T$ be a linear operator from $L^1+L^2$ to $L^2+L^\infty$ such that \begin{align} \|Tf\|_{L^\infty}&\leq \|f\|_{L^1}, \\ \|Tf\|_{L^2}&\leq \|f\|_{L^2}+\|f\|_{L^1}. \end{align} Prove that ...
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Practical applications of intermediate spaces and interpolation theorems

I am reading the book of Lions and Magenes about intermediate spaces (see Def. 2.1 in p. 10) and interpolation theorems (see p. 27) and I wish to know what are the practical applications of these ...
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83 views

Inequality of p-norm for Hardy-Littlewood maximum function

I have a question that let $1<p<\infty$, and f $\in L^{p}(\mathbb{R}^{n})$ . I want to prove that $\|Mf\|_{p}\leq2(3^{n}p')^{\frac{1}{p}}\|f\|_{p}$ whereas p' is given by $\frac{1}{p'}+\frac{1}{...
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1answer
59 views

An elementary (interpolation) inequality?

I see the following inequality from a paper $\| f\|_{L^1(\mathbb{R})}^2 \leq C\| xf\|_{L^2(\mathbb{R})} \| f\|_{L^2(\mathbb{R})} $. Although the authors say it's elementary. I try some hours and can'...
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1answer
58 views

How to prove $\|P|X|^\theta\|_{p/\theta} \le \|P|X|\|_p^{\theta}$

Let $ 0<p\le \theta<1$. Let $X$ be a self-adjoint bounded linear operator on a Hilbert space $H$ and $P$ is a projection on $H$. Why do we have $$\|P|X|^\theta\|_{p/\theta} \le \|P|X|\|_p^{\...
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1answer
18 views

Is step-wise interpolation possible for a curve which cannot be represented by a function?

According to definition of Interpolation - A function $y=P(x)$ can interpolate a set of data points if $y_i = P(x_i) | 1\le i\le n$ for the set of data points being - $(x_1,y_1), ..... , (x_n,y_n)$. ...
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75 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 ...
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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...
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Can I decompose the Lagrange interpolating polynomial of the sum of 2 functions into 2 separate Lagrange polynomials? [closed]

For example:L[x0,x1,...,xn;f+g]=?=L[x0,x1,...,xn;f] + L[x0,x1,...,xn;g] where f and g are ordinary functions, not neccessarily polynomials......
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2answers
310 views

Lagrange linear, quadratic, and cubic interpolations maximum interpolation error functions comparison

Using Theorem 1, calculate that the maximum interpolation error that is bounded for linear, quadratic, and cubic interpolations. Then compare the found error to the bounds given by Theorem 2. The ...
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330 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}-...
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345 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 ...
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1answer
33 views

Approximation of poly of degree 4 by degree 2

Let $(x)=x^4$ be approximated by a polynomial of degree less or equal to 2, which interpolates $x^4$ at x = -1,0,1then the maximum absolute interpolation error over the interval[-1,1] is equal to?
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112 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)|$...
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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 (...
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1answer
90 views

What is this Linear Interpolation Method, Brain Teaser!!

Hey geniuses out there! Definitely got a brain teaser here (for me anyway!). I need some help. I'm developing some calculation software that needs to match all calculation results with an existing ...
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How to find the minimun of an interpolation polynomial by the Brent's Method?

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 $...
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1answer
26 views

What is the general descriptor of this class of numerical interpolation schemes?

I have been working on developing some n-dimensional interpolation code for a project of mine, and have developed something that works well. But I am wondering how it would be categorized according to ...
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66 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 $[a,b]$ ...
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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 ...
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1answer
45 views

Let $Y\subset X$ be Banach spaces, then is $X+Y\subset X$ a continuous embedding?

Define the norm $$\lVert x\rVert_{X+Y}=\inf_{a+b=x}(\lVert a\rVert_X+\lVert b\rVert_Y)$$ on $X+Y$, then is it true that $X+Y\subset X$ embeds continuously?
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2answers
384 views

Analytic “Lagrange” interpolation for a countably infinite set of points?

Suppose I have a finite set of points on the real plane, and I want to find the univariate polynomial interpolating all of them. Lagrange interpolation gives me the least-degree polynomial going ...
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1answer
49 views

Boundedness of functions in complex interpolation method

In the method of complex interpolation one evaluates traces of suitable holomorphic functions on the strip. I have looked in the book of Lunardi and the one of Bergh/Löfström and in both this "...