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        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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        2answers
        154 views

        Probability of at least two of $n$ independent events occurring subject to some conditions

        Given a set of independent Bernoulli random variables $\{x_1, \dots, x_n\}$, let $p = \sum_{0<i\leq n}\Pr[x_i = 1]$ and $X=\sum_{0<i\leq n} x_i$. We know that for any $i$, we have $\Pr[x_i = 1]\...
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        1answer
        76 views
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        For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often

        Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...
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        vote
        1answer
        107 views

        For which $n$, can we find a sequence of $n+1$ distinct points s.t. the interpolating polynomial of every +ve continuous function is itself +ve

        Fix an interval $[a,b]$. For which integers $n>1$, does there exist $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that for every continuous function $f:[a,b] \to (0,\infty)$, the ...
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        1answer
        302 views

        Does every positive continuous function have a non-negative interpolating polynomial of every degree?

        Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
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        0answers
        19 views

        Approximate sum of log terms from sum of terms?

        Suppose I have a program that processes time series data and calculates the discounted cumulative sum of future reward, like so $$ X_{t} = r_{t} + \gamma r_{t+1} + \gamma^2 r_{t+2} + ... + \gamma^N ...
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        vote
        1answer
        79 views

        Accurate estimate/lower bound for the l2 approximation error of trigonometric polynomial approximation

        This is a restated version of my original very broad question. Let $P$ be probability a measure on an interval $[a,b]$ ($-\infty<a<b<\infty$) that's dominated by Lebesgue measure. Let $\...
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        1answer
        126 views

        Approximate the following series on the euclidean grid

        I had trouble in finding a closed form solution for the following series, so now I am trying to find a good approximation for it. The $\sqrt{i^2 + j^2}$ in the exponent comes from distances on the ...
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        2answers
        152 views

        A min-max approximation

        Let $n\ge 1$ be an integer, $\mathcal P_n$ be the vector space of all polynomial functions over $[a,b]$, of degree at most $n$. My question is : Is it true that $$\inf_{x_0,x_1,...,x_n\in[a,b], x_0&...
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        1answer
        80 views

        Chebyshev interpolation [closed]

        Let's define the n-th degree Chebyshev polynomials by $$ T_{n} (x)=\cos(n\arccos(x)).$$ Find a polynomial $P$ such that $$\mid y- P (x) \mid$$ is minimal, using the first three Chebyshev ...
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        0answers
        38 views

        Representability of smooth invertible Lipschitz functions by a finite composition of near-identity functions

        Theorem 1 of this paper shows that For every positve integer $K$ and for every nonempty bounded domain $\mathcal X \subseteq \mathbb R^d$, the restriction on $\mathcal X$ of any function $h: \...
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        0answers
        36 views

        On different norms of the interpolating operator

        Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
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        1answer
        66 views

        Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm

        Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
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        votes
        1answer
        85 views

        Minimax Approximation to Sine Function on interval [-K, K]

        Let $p_{n,K}(x)$ be the polynomial of degree $n$ that minimizes $\epsilon_{n,K} = ||p_{n,K}(x) - sin(x)||_\infty$ on the interval $[-K, K]$. Question: what is the asymptotic behavior of $\epsilon_{n,K}...
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        1answer
        111 views

        Does log-concave approximable distribution satisfy transportation-cost inequality?

        Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition $$ \...
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        1answer
        162 views

        Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise

        Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...

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