Questions tagged [interpolation]
Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points. It is necessary because in science and engineering we often need to deal with discrete experimental data.
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Lemma 6.33 in Gilbarg & Trudinger's book
Lemma 6.33. Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ and let $S$ be a bounded subset of the Banach space
$$C_*^{k, \alpha} = \left\{ u \in C^{k, \alpha}(\Omega) \mid |u|_{k, \alpha; \Omega}...
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Birkhoff interpolation at complex points
Does anyone know any results regarding existence of a solution to a Birkhoff interpolation problem at complex points? In particular, are there conditions for when there exists a real polynomial $p(x)$ ...
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Why does linear interpolation always underestimate square roots?
If we estimate a square root using the so-called Babylonian method, the result is always overestimated and the reason obvious: we are ignoring the quadratic component of the solution. However, if we ...
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Proving the existence of a fixed-point for the interpolation problem
This is a rephrasing of the original post in (Interpolation problem with varying nodes)
Let $\{f_i\}^{M}_{i=0}$ be a set of real numbers satisfying either $$f_0>f_1<f_2>f_3 \dots$$ or $$f_0&...
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Interpolation problem with varying nodes
I want to prove the following theorem on the existence of a low-order interpolating polynomial and interpolation nodes:
Let $\{f_i\}_{i=0}^{M+1}$ be a set of real numbers satisfying either
$$f_0 > ...
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The simplest formula for stable, monotonic C2 interpolation
I want to interpolate $n+1$ data points $D=\{(x_0,y_0), \cdots, (x_n,y_n)\}$ by a smooth C2 function $f:\mathbb R\to\mathbb R$. Here the data is guaranteed to be strictly monotonic, that is, $x_i<...
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How to bound a periodic function's first derivative by its $H^3$ norm and the $L^2$ norm of $\Delta u$?
Precisely, I want to give more details on the following inequalities in here ( page 19) :
\begin{align*}
\|\partial_y u\|_{L_{x,y}^{\infty}}&\lesssim\sum_{\alpha\in \mathbb{Z}}\|\widehat{\...
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Chebyshev interpolation using DCT with multiple terms interpolation polynomial
I need to evaluate:
$u(x_j) = \sum_{j=0}^N \hat{u}_j \phi_j(x_j) $
where the coefficients $\hat{u}_j$ are known and the interpolation polynomial is:
$\phi_j(x)=T_k(x)+a_kT_{k+1}(x)+b_kT_{k+2}(x)$
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Doubts on a proof of Mittag-Leffler theorem
I am having a hard time understanding the proof of Mittag-Leffler theorem as a consequence of Runge's theorem in the book "Complex Made Simple" by David Ullrich. The first part is similar to ...
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Is there a known interpolation method using exponentials instead of Newton polynomials?
I found what looks like an exponential analogue of Newton’s forward difference interpolation.
The n-th basis function is
$$B_n(x) \;=\; \frac{1}{n!}\sum_{k=0}^{n}(-1)^{\,n-k}\binom{n}{k}\,(k+1)^x$$,
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If the integral of a function is decreasing, does it imply that the summation decreasing? [duplicate]
I'm back with the subsequent problem to this which has been answered using Polynomial Interpolation. Now, I want to prove that the class of functions:
$\displaystyle B(x,n)=\sum_{k=0}^{n-1} \sqrt{1-c^...
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How to get an infinitely differentiable curve given a set of points
When working in $2D$, I can have a polynomial function
$$P(t) = c_0 t^0 + c_1 t^1 +c_2 t^2 + ... + c_n t^n$$
Which is infinitely differentiable and I can use it to fit a set of points $(t_0,x_0), ...,...
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How to derive analytic formula or efficient computational scheme for function underlying pseudoinverse to linear interpolation matrix?
Background: I am resizing image dimensions to powers of two for practical purposes in an image compression scheme. Then I need to switch back to the previous size. For example an image of dimensions ...
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Is the derivative bounded below?
Let $p_n$ be the $n$th prime, and define $\DeclareMathOperator{sinc}{sinc}\DeclareMathOperator{sinhc}{sinhc}$
$$p(x)=\sum_{k=1}^{\infty}\frac{p_k\sinc(\pi(x-k))}{\sinhc(x-k)}$$
Where
$$\sinc(x)=\begin{...
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Analytic extension of the $n$th prime and the prime counting function to the complex plane
Question: Is it possible to extend the $n$th prime and the prime counting function analytically to the complex plane?
Thoughts:
This paper contain the formula
$$\pi(x)=-8\sum_{h=1}^{\infty}H_{x,-2h}\...
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