# Tagged Questions

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### interpolation and approximation [closed]

Given a function $f$ in C^k[a,b], can we always construct function $g \neq f$ such that $g(x) \ge f(x)$ for all $x \in [a,b]$, $f^{(m)}(a)=g^{(m)}(a)$ and $f^{(m)}(b)=g^{(m)}(b)$ for $m=0,1,\dots, k$ ...
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### Is there an example where the error of Gauss-Laguerre quadrature does not vanish?

The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum $$\sum_{i=1}^n f(x_i) w_i$$ where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
Let $f_0,f_1,\ldots$ be a sequence of functions $f_n : [0,1] \rightarrow R$ defined as follows: $$f_0(x) =1+2x$$ $$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } ... 1answer 154 views ### Approximation of the sum involving binary entropy function Given the following sum: S(n) = \sum_{i=1}^{n} \frac{1}{(1-\operatorname{H}(p))^i} where H is the binary entropy function defined as: \operatorname{H}(p) = -p\log p - (1-p)\log (1-p) . Let ... 1answer 113 views ### Do interpolation nodes have to be dense? Let f(x) = \exp(x) and (\xi_i)_{i=0}^\infty, \, \xi_i \in (0,1) be a sequence of points from the unit interval. For n \in \mathbb{N} let P_n be a polynomial of degree n that interpolates ... 0answers 242 views ### Approximation by polynomials The following is a well-known theorem (see e.g. The Chebyshev Polynomial by Rivlin): If p(x) = x^n + a_{n_1} x^{n-1} + \ldots + a_0, then \max_{-1\leq x \leq 1} |p(x)| \geq 2^{1-n} for n \geq 1 ... 1answer 2k views ### The unreasonable effectiveness of Pade approximation I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ... 1answer 217 views ### Approximation theory under L_1-error Is there a reference for results in approximation theory of bounded functions of one (and multiple) variables under L_1-error? Formal definitions for functions of one variable are below. Let C ... 1answer 286 views ### Lebesgue constant as condition number of polynomial interpolation Let T = \{ x_0,\ldots,x_n \} be a set of n+1 different points in the real interval [a,b]. Let X_T be the associated interpolation operator on C[a,b]: it takes a function f \in C[a,b] into ... 2answers 719 views ### Approximating erf by tanh It appears to be well-known that \tanh(x)\le \mathrm{erf}(x) on [0,\infty). It's off-handedly mentioned here, for example. Where can I find a formal proof? On the one hand, it's hard to imagine ... 4answers 797 views ### When we use Bernstein polynomials in application When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple ... 2answers 218 views ### Approximation by polynom 1) with respect to supremum-norm 2) I need F_{approx} > F_{exact} Given a function F, how to find polynom which is best/good approximate with respect supreremum-norm, i.e. minimize over P_{approx} sup|F-P_{approx}| ? I am intersted in polynoms in two variables of ... 0answers 346 views ### Padé approximations of e The following question came up in the analysis of some algorithm. Let R_{s,t}(z) be the PadÃ© approximants of e^z, and define r_{s,t} = R_{s,t}(1). Using the explicit expression for the error ... 0answers 297 views ### Approximations of negative Sobolev norms Consider the standard Cahn-Hilliard free energy, augmented by a nonlocal interaction term which measures the H^{-1} norm of a zero-mean function. Could someone point me to a reference where this ... 6answers 581 views ### best approximation to the LambertW(x) or exp(LambertW(x)) what is the best available approximation ( say up to 10 digits ) for LambertW(x) or exp(LambertW(x)) for x > 2000 3answers 371 views ### Approximating derivatives between gridpoints Hi, Suppose we have a grid (possibly irregular) of N function/value pairs, (x_i, f_i), i=1...N. The function is differentiable everywhere at least twice (perhaps more). What would be a good way ... 1answer 280 views ### Schrodinger's equation over a randomized grid I am interested in solutions to$$ \frac{d}{dt} \Psi = -iH \Psi $$for H hermitian and time independent. This boils down to evaluating$$ \Psi(t) = e^{-iHt}\Psi_0  at points of interest $t_n$. I ...
Problem: We would like to calculate $S=\sum_{i=1}^{k} c_i x_i$, where $k$ is a constant, $x_i$ are some fixed algebraic numbers, $c_i=\frac{p_i}{q_i}$ are rational numbers such that integers $p_i$ ...