# Tagged Questions

The tag has no usage guidance.

318 views

### Can I approximate Schwartz functions which integrate to zero by $C_0^\infty$ functions which integrate to zero?

Let $X$ be the closed subspace of Schwartz space $\mathcal{S}(\mathbb{R}^N)$ defined by \begin{equation*} X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}. \end{equation*} My ...
290 views

### Approximation of curves

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 ...
699 views

### State of the Art in Approximating Fresnel Integrals

Background of my question is, that I need to calculate Clothoids and I found an AMS article "Chebyhev Approximations for Fresnel Integrals" by W.J. Cody from 1968 (http://www.ams.org/journals/mcom/...
108 views

### How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where Y is Binomial(n,p)

How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where $Y$ is Binomial(n,p)? If it is not exactly computable, then are their ways to approximate this qty?
84 views

### The proximality of low rank function approximation

The paper "Best $n$-Dimensional approximation to sets of functions" by A. L. Brown in 1964 gave a negative answer to the following question: Q1: Is there for a given integer $n$ always a best ...
514 views

### Empirical estimator for total variation distance between two product distributions

Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...
405 views

### 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 ...
186 views

### Bivariate Function Approximation

I am working on a nonlinear control design and having difficulty in finding approximation of bivariate functions. Are there papers or methods discussing the following question: For any bivariate ...
126 views

### Polynomials are dense in $A_{B(0,1)}$

Let $D(0,1)$ be the disk of center 0 and radius 1 and call $A_{D(0,1)}= \{ f:\overline{D(0,1)} \rightarrow \mathbb{C} : f \text{ is continuous and } f|_{D(0,1)} \text{ is holomorphic} \}$. Can ...
686 views

### Approximation of continuous functions by Lipschitz functions in the topology of uniform convergence on compact sets

I was involved into this subject when I answered this question from MSE. Trying to generalize my answer, I am thinking about a following Question. Let $X$ and $Y$ be metric spaces. When each ...
83 views

I'd like to have some informations about Markov-type functions (or Cauchy-type): $f(z)=\int_{\Gamma} \frac{\mathrm{d}\gamma(\xi)}{\xi-z}.$ $\gamma$ is a positive measure with compact support $\... 2answers 139 views ### Successive Inner or Outer Approximation of Simple Polygons with Hierarchies of Implicit Functions The problem I want to solve, is to quickly decide, whether a point$p=(x^*,y^*)$is inside or outside of a polygon$P := (p_1, p_2,..., p_n=p_1), p_i := (x_i,y_i)$, with$n$potentially very large. ... 1answer 194 views ### Approximating an iteratively defined function Let$f_0,f_1,\ldots$be a sequence of functions$f_n : [0,1] \rightarrow R$defined as follows: $$f_0(x) =1+2x$$ $$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } f_{n-1}\left(\frac{x}{t}... 1answer 379 views ### Approximation of the sum involving binary entropy function Given the following sum: S(n) = \sum_{i=1}^{n} \frac{1}{(1-\operatorname{H}(p))^i} where H is the binary entropy function defined as: \operatorname{H}(p) = -p\log p - (1-p)\log (1-p) . Let ... 1answer 320 views ### How to prove that the odd continued fraction approximants of ln(1+X) are upper bounds? The odd order continued fraction approximants for \ln(1+X) are$$X,\quad \frac{X^2+6X}{4X+6,}\quad \frac{X^3+21X^2+30X}{9X^2+36X+30,}\quad \dots.$$In "Some bounds for the logarithmic function", ... 0answers 91 views ### Greedy interpolation of functions Let f:[-1,1]\rightarrow \mathbb{R} be a continuous function. Consider the following greedy algorithm for interpolation: Set r_0 = f. for k = 0,1,\ldots, Find the location of the global ... 1answer 144 views ### Do interpolation nodes have to be dense? Let f(x) = \exp(x) and (\xi_i)_{i=0}^\infty, \, \xi_i \in (0,1) be a sequence of points from the unit interval. For n \in \mathbb{N} let P_n be a polynomial of degree n that interpolates f... 0answers 203 views ### Is there an absolutely continuous function f Is there an absolutely continuous function f satisfying$$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \mbox{const}\frac{|\delta|}{\log \frac{1}{|\delta|}},\,\,\, |\delta|<1, $$which is not C^{1}? 0answers 125 views ### On derivatives of polynomials majorized by \max(1,|x|^d) In the course of generalizing the Bernstein-Markov theorem to normed space, Harris came up with the following question. Suppose that p is a real polynomial satisfying |p(x)| \leq (1+|x|)^d. ... 1answer 100 views ### Norms of B-spline coefficients In Shumaker's book (Spline Functions: Basic Theory), we know that the l^\infty-norm of B-spline coefficients is bounded above and below by the L^\infty-norm of the spline itself. Are there similar ... 1answer 190 views ### What's the asymptotic behavior of this function at large distance? [closed] This question is based on some Physics motivation. Define a distance function f(\mathbf{r})=\int_{\Omega }d^2k\int_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}], where \mathbf{r},\... 1answer 260 views ### Approximation of a given function by rational functions Given a function 1/\sqrt{x^2 -k^2} where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the L_p (p=2 or \infty) norm of ... 1answer 569 views ### Is there a continuous function f satisfying the following Zygmund condition but not differentiable. Suppose that a continuous function f on the line and satisfies$$ |f(x+2h)−2f(x+h)+f(x)|\leq const \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, 1] $$... 0answers 85 views ### The d-dimension extension of Bernoulli Polynomial It is known that Bernoulli polynomial has the following Fourier expansion: \begin{equation*} B_{2n}(x) = \frac{(-1)^{n-1}2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos(2k\pi x)}{k^{2n}}. \end{... 0answers 167 views ### Rational interpolation: Error bounds for coefficients The following question was asked on MSE, but might be more suitable here. Assume there is a rational function$$ f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}} $$of type (m,n) with ... 2answers 149 views ### Non-global oscillation of banded Fourier transform Can we say something like monotonicity, growth rate and oscillation of the Fourier transform of a banded function f with support [0, N]$$\mathcal{F}f(\xi) = \int_{0}^N f(x)e^{-ix\xi}dx.$$Of ... 1answer 186 views ### Approximation Runge's Theorem Let X be a Riemann Surface and K a compact subset of X. Every holomorphic function in K be uniformly approximable on K by holomorphic functions on X if X-K have no connected component ... 0answers 113 views ### On Artin-Hironaka lemma and Galois theory Let A=k[[t]] Let B a flat A-finite algebra which is etale and Galois at the generic point. Then by Artin lemma 3.12 (ii) in his IHES paper on approximation, we know that there exists an integer ... 3answers 337 views ### Approximating higher dimension step function Let s \in R^{n} (meaning s is n \times 1 vector), where n is the dimension of the vector. The ideal sliding term, \nu is taken to be: \nu = \frac{s}{\|s\|} \end{... 1answer 719 views ### Multivariate polynomial approximation of smooth functions Let f be a function defined on [-1,1]^d. Assume that all partial derivatives of f up to order r are continuous; and the \infty-norm of these partial derivatives are uniformly upper bounded ... 1answer 743 views ### Estimate on sum of squares of multinomial coefficients I am interested in approximating the sum of the squares of the multinomial coefficients, i.e. a_\ell^p := \sum_{k_0+\ldots+k_p = \ell} (\frac{\ell!}{k_0! \ldots k_p!})^2 or more general, a_\... 1answer 104 views ### Approximating rational generating functions Suppose we have a initial segment x_1,\ldots,x_N (for reasonably large N) of a sequence of natural numbers (x_i). We have reason to believe the generating function \sum_{i=0}^\infty x_iX^i is ... 0answers 71 views ### Jackson inequality for a nonpolynomial basis Hi everybody, this is my first question.L^2,H^p are the standard Lebesgue,Sobolev spaces here, and I am deliberately omitting the domains because I'll accept an answer if it's on an interval or a ... 0answers 134 views ### Spectral norm for a truncated Hilbert matrix Let T_{N} be the (Hilbert) matrix defined by T_{N}(m,n)=\frac{1}{m-n} if 1\leq m,n \leq N and m\neq n , and T_{N}(n,n)=0 if 1\leq n \leq N . It's well known that \Vert T_{N}\Vert < \... 1answer 200 views ### Is the Binomial Expectation of a Multivariate Convex Function Convex in the Vector p? Let \mathbf{p}=(p_1,\dots,p_m) be a vector in [0,1]^m and let \mathbf{X}=(X_1,\dots,X_m) be a vector of independently-distributed binomial random variables such that X_i\sim \text{Binom}(n,p_i)... 0answers 109 views ### Two Different Representations of Multivariate Bernstein Polynomials In the literature the multivariate Bernstein polynomial of a function f:[0,1]^m\rightarrow\mathbb{R} is often defined as the following:$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}... 0answers 279 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$n \geq 1$... 2answers 5k 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 538 views ### Hilbert's Theorem on$L_2$norm of polynomials in$\mathbb{Z}[X]$- Explicit construction and a converse? Consider the set of polynomials with real coefficients as a vector space with the following inner-product:$\langle f, g \rangle = \int_{a}^{b} f(x)g(x) dx$. Hilbert showed, in a paper from 1894, ... 1answer 266 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 417 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 458 views ### Approximating a subspace by sampling a base without replacement Let$X$be a$p \times n$matrix, with$p > n$. Now, suppose I sample$m < n$columns from$X$at random, without replacement. I would like to characterize the distance between the subspace ... 2answers 1k 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 1k views ### Using Quotient of Prime Numbers to Approximation Reals We know a positive rational number can be uniquely written as$m/n$where$m$and$n$are coprime positive integers. Particularly, we can pick out those numbers with$m$and$n$both prime. Question ... 2answers 564 views ### Convex upper bound on a linear-fractional function I have a function of the form$f(x,y) = \frac{x}{c+y}$where$c$is a positive constant,$c \ge x \ge 0$, and$y \ge 0$. I would like to find a convex upper-bound for this function. Is there a ... 4answers 1k 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 ... 1answer 315 views ### Approximation power of wavelets The Wikipedia article on Wavelet Transform states that: Wavelet compression is not good for all kinds of data: transient signal characteristics mean good wavelet compression, while smooth, ... 0answers 66 views ### approximation in Lie algebras Let$x_{1}$,$x_{2}$,$x_{3}$three disctinct closed points of a curve$X$over an algebraically closed field k. Let G a connected reductive group and$\mathfrak{g}$his Lie algebra. I fix a Borel$...
Suppose a function $f:[-1,1]^2\rightarrow \mathbb{C}$ has a singular value decomposition: $$f(x,y) = \sum_{k=1}^\infty \sigma_k u_k(y) v_k(x), \qquad \sum_{k=1}^\infty \sigma_k^2 <\infty,$$ ...