The approximation-theory tag has no wiki summary.

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### Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...

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169 views

### Smallest degree of approximating polynomial

Let $\{0,1\}^n=S_0\cup S_1$ withh $S_0\cap S_1=\emptyset$.
Let $\epsilon\in[\frac{1}2,1)$.
Let $f:\Bbb R^n\rightarrow\Bbb R$ be a polynomial such that $$f(S_0)=0,\mbox{ ...

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**1**answer

101 views

### Maximum of a mollified/convolution function

I have a function $f:{\mathbb R}\rightarrow {\mathbb R}_+$ which has a unique maximum at $x=0$. $f$ can be symmetric or asymmetric. I am interested on the mollified-f function
...

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86 views

### Interpolation Operator Bounded in Sobolev Norm

Let $m\in \mathbb{N}$, $p\in [1,\infty]$, $W^{m,p}([0,1])$ the space of all functions $[0,1]\rightarrow \mathbb{R}$ which are $m$ times weakly differentiable and weak derivatives in $L^p$,
...

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56 views

### How can a binomial coefficient can be approximated by using Striling's formula? [closed]

I've met some difficulties with such question:
How can we approximate a binomial coefficient by using a Stirling's factorial approximation.
I've evaluate a little bit and got this
How can I ...

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57 views

### Numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if
\begin{align}
\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in ...

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62 views

### Bounds on degree from bounds on derivatives

Let $f(x)\in \Bbb R[x]$ and $r(x)\in\Bbb R(x)$. Supposing we have information about the values taken by $f(x)$ and $g(x)$ in certain intervals and also can bound their derivatives in these intervals, ...

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100 views

### Approximation of $ _2F_1((b-1)a,b;ba;x) $

Is there any (simple) approximation of this Hypergeometric function: $ _2F_1((b-1)a,b;ba;x) $, where $0<x<1$ and $b>a>1$.
Thanks!

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72 views

### Relation between Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials.
Pointwise Lagrange ...

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97 views

### Estimates of entropy of functional spaces

Let $M^n$ be a compact $n$-dimensional manifold. For $k\geq 0$ let us denote by $C^k(M)$ the Banach space of $k$ times continuously differentiable functions, and $B_{C^k}$ denote the unit ball of it.
...

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54 views

### Asymptotics of Fresnel integrals

It is known that $I(p) = \sqrt { \frac {4 \mathrm{i} p} {\pi}}\int \limits _{-\infty} ^{\infty} \mathrm{e}^{- \mathrm{i} p x^2} \varphi (x) \mathrm{d}x$ is a bounded smooth function on $(0,\infty)$ ...

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131 views

### Generalized Schwarz Lemma for near-zeros

In approximation theory, it is classical to use a result that can be considered a generalization of the Schwarz Lemma:
Let $f:[-1,1]\rightarrow\mathbb{C}$ be a function that is analytic in a domain ...

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42 views

### Approximation property of Fréchet if range is restricted to an embedded Hilbert space

Let $W$ be a separable Fréchet space, and $H\subset W$ be a separable Hilbert space that is continuously embedded (equivalently, the topology of $H$ is stronger than the subspace topology generated by ...

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629 views

### A “better” rational approximation of pi?

$355/113$ is a good fractional approximation of $\pi$, because we use six digits to produce seven correct digits of $\pi$.
$$\frac{355}{113} = 3.1415929\ldots$$
Let $R$ be the ratio of the number of ...

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62 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 ...

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143 views

### How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case

After multiple plots I noticed that function $h(x)= (1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and ...

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29 views

### Estimating overshoot in spline interpolation

Say I have a spline space $\mathcal S$ of dimension $n$ with a set of unisolvent points $(\xi_i)_{i=1}^n$, i.e., points at which I can unambiguously interpolate within the spline space. So, given ...

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153 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 ...

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123 views

### Selecting Rays for Simulated Radon Transform

I have the task of determining approximations of a 2D function $f: (x,y)\in \mathbb{R}^2\mapsto\mathbb{R}$ from integrals along lines, i.e. from its Radon transform $R(\phi,\tau)[f(x,y)]$ and, because ...

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97 views

### cohomology algebra of submanifold in euclidean space

If we write a manifold or CW-complex $X$ as a subset of $\mathbb{R}^n$, in expression of coordinates, for example, \begin{multline}
F(S^2,k+1)=\{(x_1,x_2,x_3,\cdots, ...

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26 views

### Uniform Boundedness of the Szasz-Durrmeyer Operators on Variable $L^{p(\cdot)}$ Spaces

I am stuck with the uniform boundedness problem associated with the Szasz-Durrmeyer Operators on $L^{p(\cdot)}$ spaces. The Szasz-Durrmeyer operator can be defined as
$$M_{n}(f;x)=\sum ...

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42 views

### Density of restrictions of $p$-harmonic functions on a hypersurface

Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$.
Let $1<p<\infty$.
Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...

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169 views

### Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?

Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully).
Suppose $g$ is holomorphic on ...

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32 views

### Finite elements $W^{1,\infty}$ error estimates

Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them?

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196 views

### Approximation theory on the disc

Chebyshev theory provides a very effective method for approximating continuous real valued functions on the unit interval. Is there something similar for continuous real valued functions on the ...

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158 views

### What is the advantage of the knowledge of jumps for approximating a function with trigonometric polynomials?

Let $f:(a,b)\to\mathbb{R}$ be square integrable, bounded variation and piecewise continuous function. Let the points of jump be $\{x_i/a<x_i<b,i = 1,2,3,...n\}$. The goal is to approximate the ...

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32 views

### Multidimensional Filters

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then ...

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121 views

### Divergence of the Lagrange interpolation on the Chebyshev nodes

Faber theorem states that for every $\lbrace x_k^{(n)} \rbrace$ there exists a continuous $f$ function such that $\| f - L_n \|_{\infty} \not\rightarrow 0$, where $L_n$ is interpolation polynomial on ...

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175 views

### Approximating Lie groups by finite groups

How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to ...

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259 views

### Construct the best piece-wise linear continuous function fitting given curve

How to construct the optimal piece-wise linear continuous function fitting given curve and given number of knots (optimal knots positions also must be determined by this method)?

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125 views

### Approximate $F(\theta)=\sin(\theta)\int_{-l}^{l} e^{-ikz\cos \theta} h(z)\,dz$

$$F(\theta)=\sin(\theta)\int_{-l}^{l} e^{-ikz\cos \theta} h(z)\,dz$$
We know that $F(\theta)$ is defined on $0\le \theta \le \pi$ and $h(z)$ is defined on $|z|\le l$ and $z$ is real in this case, but ...

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432 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, ...

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1k views

### Matrix approximation

Let A be an $m\times n$ matrix and $k$ be an integer. Assume that $A$ is non-negative. We want to find a scalar $\epsilon$ and an $n\times n$ matrix $B$ such that $A\leq A(\epsilon I + B)$ (where ...

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60 views

### Best s-term approximation and unit balls in weak $\ell^p$ norm

In the book "A Mathematical Introduction to Compressive Sensing" by Foucart and Rauhut there is the following asymptotic estimate at page 332, equation (11.1):
$$\sup_{\mathbf{x}\in B^N_{r,\infty}} ...

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55 views

### Estimating polynomial approximation error in high dimension

Question
Let $x \in [-1, 1]^d \subset \mathbb{R}^d$ be a $d$-dimensional variable and assume that -- given $n$ -- I have a way of computing a polynomial $p_n(x)$ of degree $n$ that approximates a ...

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94 views

### About the maximum degree of multivariate polynomial interpolation

It is well known that in the univariate case, to interpolate $k$ points in $\mathbb{R}$, we need to use a polynomial of degree $k-1$.
My question is about multivariate polynomial interpolation in ...

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147 views

### Jackson's theorem for partial sum of Fourier series

There is a classical theorem of Jackson stating that the $N$-th partial sum $S_N f$ of the Fourier series of a Lipschitz continuous function $f$ (which is periodic with period 1) satisfies
$$
|f(x) - ...

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72 views

### Questions related to a previous question about interpolation based on non-decreasing polynomials

Let $n$ be a positive integer greater than $2$. Let $X(1),X(2),\cdots,X(n)$ and $ Y(1),Y(2),\cdots,Y(n)$ be two strictly increasing sequences of n real numbers each, listed in order of increasing ...

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89 views

### Uniform approximation of increasing function in $C^{\infty}$

I have an increasing continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$ which is not differentiable everywhere, and I would like to approximate it with an infinitely differentiable function ...

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158 views

### regarding Uniform Algebra on a compact set [closed]

Suppose $K\subset\mathbb{C}$ is compact and let $R(K)$ be the uniform algebra of rational functions on $K.$ Moreover assume that $R(K)$ contains $\overline{z}.$ How to show that if a rational function ...

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95 views

### How to formulate approximation from above?

(This is perhaps a stupid question. If so, please give me a hint and a down vote.)
I have a sequence of Banach spaces $X_{1}\supset X_2\supset ...$ and a sequence of elements $x_j\in X_j$ ...

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46 views

### Partial Constraint of Low Rank Matrix

Suppose $X \in \mathbb{R}^{m \times n}$ is a rank $r$ matrix. Let $\Omega$ be a generic subset of $\in \{1, \ldots, m\} \times \{1, \ldots, n\}$ of cardinality $r(m + n - r)$. Denote by $X_{\Omega}$ ...

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71 views

### on lifting elements in a tangent space

Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$
Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and ...

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113 views

### Multivariate polynomial interpolations

I have a multi-variate, continuous function from $R^n$ to $R$, which I can query for its output for any input. I would like to create an interpolation of that function by sampling a subset of the ...

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101 views

### using polynomials as lower / upper bound?

I'm interested in the question of given a differentiable and bounded function $f(\vec{x})$ (over a single variable or multiple variables, over a bounded domain $D$), finding a pair of polynomials ...

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48 views

### Error estimates for $L^2$ approximation of multivariate $C^k$ functions in terms of norms on derivatives

Let $\Omega \subset \mathbb{R}^d$ be compact and convex, and let $f \in C^{k+1}(\Omega)$. Let $P_k$ be the set of multivariate polynomials up to degree $k$ on $\Omega$. I am looking for any results in ...

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118 views

### $\epsilon$-covering number of a set of rank-2 matrices

Suppose that two unit-norm vectors $\boldsymbol{a}\in \mathbb{R}^m$ and $\boldsymbol{b}\in\mathbb{R}^n$ are given with $m\leq n$. Furthermore, let $\boldsymbol{F}_{m,n}$ denote the first $m$ rows of ...

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268 views

### Approximation of the form $\frac{1}{u}\pm\frac{1}{v}$

Given positive integers $m<n\in\mathbb{N}$ is there an algorithm to find integers $z_1, z_2\in \mathbb{Z}\setminus\{0\}$ such that $\frac{m}{n}$ is best approximated by ...

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175 views

### Approximating rational values in ]0,1[ by a sum or difference of unit fractions

Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions.
Are there positive integers $m<n \in \mathbb{N}$, such that for ...

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271 views

### Are inclusions from spaces of $C^\infty$ sections into spaces of $C^k$ sections homotopy equivalences?

[EDIT: The answer to my original question was obviously no, as user56365 pointed out. Here is what I should have asked.]
For finite-dimensional smooth manifolds $E,M$, let $E\to M$ be a smooth fibre ...