# Questions tagged [approximation-theory]

Approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.

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### How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
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### Polynomial approximation of circle or ellipse

Trying again, with a somewhat simpler sounding question, since my previous one (Generalizations of equi-oscillation criterion) got zero response: Let $F:[0,1] \to R^2$ be a parametric polynomial ...
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### Generalizations of equi-oscillation criterion

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
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### How to justify unicyclic connected graphs formula?

I have read this formula in a book for $c(k,k)$, the number of unicyclic connected graphs on $k$ vertices A057500, i.e. $n=m=k$. How can one prove it? It seems that it is iterating on the length of ...
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### Natural cubic spline interpolation error estimate

I am looking for an error estimation for natural (one with $s''(a) = s''(b) = 0$ boundary conditions) cubic spline interpolation on an evenly spaced grid. The best result I've found was $O(h^2)$ ...
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### Taylor polynomial approximation

How do you determine if adding more terms to the Taylor polynomial will improve its approximation of $f(p)$ or in other words, how do you determine if a Taylor series converges for a particular value ...
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### How to approximate the integral $\int_{-b}^{\infty}\log(t+b)e^{-t}e^{-e^{-t}}dt$

Suppose we have the following integral $$\int_{-b}^{\infty}\log(t+b)e^{-t}e^{-e^{-t}}dt,$$ where $b$ is a positive constant. It seems very difficult to derive the exact ...
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### An error formula for linearization

Question: Can anyone shed some light on this formula? I can't find any information on it. It has three corollaries that I also need to understand:
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### Approximation of $L^1$ function with compactly supported smooth function with same mass and same uniform bounds

Recently, I have asked this question. Now, I even want to make this better. Given $f\in L^1(\mathbb{R})$ with $0\leq f\leq 1$, I can find for any $\epsilon>0$ a $g\in C_c^\infty(\mathbb{R})$ such ...