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.
0
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0answers
14 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 $...
2
votes
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 ...
3
votes
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 ...
0
votes
0answers
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 ...
1
vote
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 ,...
0
votes
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,...
0
votes
0answers
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]$...
0
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0answers
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 ...
1
vote
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 ...
1
vote
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}^...
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0answers
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 ...
0
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0answers
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, ...
0
votes
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}{...
0
votes
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 ...
5
votes
3answers
110 views
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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$ ...
0
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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 ...
2
votes
0answers
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$ ...
1
vote
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}\...
0
votes
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 ...
2
votes
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$ ...
0
votes
0answers
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 $(...
4
votes
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 ...
1
vote
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) & ...
0
votes
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 ...
0
votes
0answers
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 ...
0
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0answers
29 views
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 ...
4
votes
2answers
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}{...
3
votes
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'...
3
votes
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^{\...
0
votes
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 - $(x_1,y_1), ..... , (x_n,y_n)$. ...
1
vote
0answers
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 ...
0
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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...
0
votes
1answer
39 views
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......
3
votes
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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0answers
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}-...
1
vote
0answers
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 ...
0
votes
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?
2
votes
0answers
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
$$|T(f_1 + f_2)(x)| \leq |T(f_1)(x)| + |T(f_2)(x)|$...
1
vote
0answers
13 views
Reference for complex interpolation for the product of two normed space
Let $X=X_1\times X_2$ and $Y=Y_1\times Y_2$ and
$(X_i,Y_i)$ are interpolation couple. Then it seems obvious that
$$
[X,Y]_{\theta}=[X_1,Y_1]_\theta \times [X_2,Y_2]_\theta
$$
for every
$\theta\in (...
1
vote
1answer
90 views
What is this Linear Interpolation Method, Brain Teaser!!
Hey geniuses out there!
Definitely got a brain teaser here (for me anyway!).
I need some help. I'm developing some calculation software that needs to match all calculation results with an existing ...
3
votes
0answers
23 views
How to find the minimun of an interpolation polynomial by the Brent's Method?
If we determinate the second-degree interpolation polynomial that pass through the points $(a,f(a)),(b,f(b))$ and $(c,f(c))$ as:
$$P_2(x)=f(a)+f[a,b](x-a) +f[a,b,c](x-a)(x-b)$$
Where the values of $...
0
votes
1answer
26 views
What is the general descriptor of this class of numerical interpolation schemes?
I have been working on developing some n-dimensional interpolation code for
a project of mine, and have developed something that works well. But I am
wondering how it would be categorized according to ...
1
vote
0answers
66 views
Approximation with stability constant 1
I am looking for an approximation method for a typical 1-dimensional approximation problem, that has stability constant 1.
The setting
We have
an interval $[a,b]$
an unknown function $f$ on $[a,b]$ ...
1
vote
0answers
41 views
$(X,Y)$ interpolation couple, then $(X^\prime,Y^\prime)$ interpolation couple
Let $(X,Y)$ be an interpolation couple, i.e. there exists a Hausdorff topological vector space $U$ s.t. both $X$ and $Y$ embed continuously into $U$. We define $X\cap Y$ and $X+Y$ in $U$ and turn them ...
0
votes
1answer
45 views
Let $Y\subset X$ be Banach spaces, then is $X+Y\subset X$ a continuous embedding?
Define the norm $$\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?
7
votes
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 ...
1
vote
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 "...
