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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3
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0answers
56 views

Covering number of smooth functions from $\mathbb{R}^d$ to $\mathbb{R}$

Let $(\mathcal{X},d)$ be a space of function $f: \mathbb{R}^d \to \mathbb{R}$ where $d=\| \cdot \|_\infty$ (i.e., $d(f)= \sup_{x\in \mathbb{R}^d} |f(x)|$. ) Let $D_\alpha f= \frac{\partial^\alpha}{ \...
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48 views

Bounds on coefficients $c_i$ of Chebyshev expansion $f(x) = \sum_{k=0}^{n} c_kT_k(x) : [-1,1] \mapsto [-1,1]$

Let $n$ be a given positive integer and let $f(x) = \sum_{k=0}^{n} c_kT_k(x)$, where $c_i \in \mathbb{R}$, $0 \leq i \leq n$. If $|f(x)| \leq 1$, for $|x| \leq 1$, is it possible to get the maximum ...
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223 views

Maximum of the weighted binomial sum $2^{-r}\sum_{i=0}^r\binom{m}{i}$

Let $m$ be a positive integer and let $f_m(r)=2^{-r}\sum_{i=0}^r\binom{m}{i}$. Clearly $f_m(0)=f_m(m)=1$ and $f_{2r+1}(r)=2^{2r}$. Conjecture: If $m>12$, then the maximum value of $f_m(r)$ for $r\...
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90 views

Best approximation of piecewise constant function by Lipschitz functions

Let $f=\sum_{n=1}^N k_n I_{E_n}$ where $E_n$ are Borel subsets of $\mathbb{R}^n$ and $k_n\in \mathbb{R}^m$ with non-negative entries, and let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. What ...
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1answer
107 views

integral of fractional function

Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as \begin{equation} \int_b^{ + \infty } {\frac{x}{{1 + {x^a}}}}. \end{equation} Here $b$ is a ...
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51 views

Proximity operator of lower semi-continuous and convex functions pre-composed with norm

Let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a some-where finite, lower semi-continuous, increasing, and convex function. It is easy to verify that the function $\Phi:\mathbb{R}^n\...
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43 views

How to find or approximate (e.g. using method of steepest descent ) integral?

Can you give any advice on how to find or approximate the following integral $$ F(t,y) = \int_{0}^{y}\frac{i e^{-\frac{3 t^2 \left(x^2+1\right)}{2 \left(9 x^2+1\right)}-i \frac{4 t^2 x}{9 x^2+1}}}{\...
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60 views

Approximating the partial sum of remainders function

This is a question related to the one I posted here, but I have found some more interesting and general results and thought here might be a better place to ask. Let $R_{k,N}$ denote the remainder of ...
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2answers
262 views

Approximation of analytic function by a fixed number of monomials

This question seems simple but I can't manage to disprove it. Let $N\in \mathbb{N}$. We know that by its analyticity that this precise linear combination of monomials $ \sum_{n=0}^K \frac1{n!} x^n $ ...
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46 views

Can convex functions on product space be approximated by product of convex functions?

I am working on a problem where I need the following property that I guess should be true but I am not able to prove it. I have a bounded convex function $F(x, y)$ on $X\times Y$ (Think of $X=Y=\...
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1answer
53 views

Uniform approximation of indicator function of a point

Fix $x \in \mathbb{R}$ and let $I_{[x]}$ be its indicator function. Does anyone know of a sequence of (obviously) discontinuous approximations $g_n$ to $I_{[x]}$ such that $g_n$ converge uniformly ...
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72 views

Best approximation of a Lipschitz function with a piecewise polynomial Lipschitz function

Let $g : [-1, 1] \to R$ be a $1$-Lipschitz function and $f_{k,d} : [-1, 1] \to R$ a $1$-Lipschitz function whose restriction to any subinterval $[h_i, h_{i+1}] \subset [-1, 1]$, $i = 0 ... (k-1)$ with ...
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18 views

Approximate distribution for leverages of the ridge hat matrix in a randomised framework

For a couple of days, I have been struggling to extend some results of the distribution of the leverages for random Hat matrices to the case where we add a penalizing term. The setting is the ...
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307 views

Incredibly accurate recursions for the Riemann Zeta function

Last update as of Jan 27, 2021: I posted this as an article for laymen, here. It is very light mathematically speaking, but section 3 is a little more accurate than my post here. During some ...
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35 views

Convergence result on Cornish Fisher expansion of binomial distribution

Since it is known that Cornish Fisher expansion of quantiles does not have guaranteed convergence for all distribution, I wonder specifically if any convergence result is known in literature for CF ...
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1answer
75 views

Control on dimension of image

Let $f:E\rightarrow F$ be a map between Banach spaces E and F; E finite dimensional (>0) and F infinite dimensional. Let $F$ be equipped with its weak topology and suppose that $f$ is strong-weak ...
14
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3answers
736 views

Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?

Bernstein polynomials preserves nicely several global properties of the function to be approximated: if e.g. $f:[0,1]\to\mathbb R$ is non-negative, or monotone, or convex; or if it has, say, non-...
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139 views

Lower bound of the modulus $|\eta(s)|$ of the Dirichlet Eta function if $0.6 < \Re(s) < 0.9$

Let $s=\sigma + it$, with $0.6 < \sigma < 1$ and $\sigma=\Re(s)$. I am trying to get good enough approximations for $\eta(s)$, hoping something useful might come out of it. I stumbled upon a ...
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1answer
91 views

Polynomial Markov versus Chernoff Bound for random variables

Suppose that $X\geq0$, and that the moment generating function of $X$ exists in an interval around 0. Given some $\delta>0$ and integer $k=1,2,...$, show that $$\inf_{k=0,1,...}\frac{E(|X|^k)}{\...
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61 views

Relationship between Wasserstein projections and metric projections in a linear space

Let $(X,d)$ be a metric space, $x\in X$, and $Y\subset X$ is a closed set. Assume that $X$ is also a real vector space, so that we can form linear combinations over $\mathbb{R}$, but I do not assume ...
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1answer
567 views

What are ways to compute polynomials that converge from above and below to a continuous and bounded function in $[0,1]$?

My interest is to take a coin of unknown bias $\lambda$ and use it to produce a coin of bias $f(\lambda)$. This is called the Bernoulli Factory problem, and only certain functions $f$ can be simulated ...
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1answer
98 views

Existence of uniform approximator that also approximates derivative

Let $S$ be a subset of $C^1([0, 1], \mathbb{R})$. It is a well-known fact that given a function $f\in C^1([0, 1], \mathbb{R})$ and a sequence $\{f_n\}\subset C^1([0,1], \mathbb{R})$ such that $f_n\to ...
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61 views

Can we improve the error bounds for spline interpolation if the interpolated function is smooth?

Let me first state the original problem I want to solve: Given a closed curve $C:[a,b]\to\mathbb R^2$ that is smooth ($C^\infty$), a partition in the parameter space $a=t_0<t_1<\cdots<t_n=b$,...
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50 views

Approximation Rates for Multivariate Taylor Series

Let $k,n,m$ be positive integers and suppose that $f$ is $C^{k}(\mathbb{R}^n,\mathbb{R}^m)$ functions. For any given $\epsilon>0$ and $x_0\in \mathbb{R}^n$, are there known sharp approximation ...
53
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9answers
4k views

Examples of back of envelope calculations leading to good intuition?

Some time ago, I read about an "approximate approach" to the Stirling's formula in M.Sanjoy's Street Fighting Mathematics. In summary, the book used a integral estimation heuristic from ...
1
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1answer
77 views

Saddle point approximation of terms in a sum

(asked in MSE, but received no attention) Suppose I need to compute a sum, $$ \sum_{n=0}^N a_n,$$ each term of which involves an integral, $$a_n=\int e^{Nf(x)+ng(x)}dx.$$ I am interested in the large-$...
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64 views

Reverse Inequality

I was doing some numerical integration when I figured the function I was dealing with (i.e., the function I was integrating) evaluated to big numbers on a tiny portion of the interval (over which I ...
1
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1answer
189 views

Checking the uniform denseness of a set in $C([0, 1], \mathbb{R}^2)$

Let $\lambda:[0, 1]\to \mathbb{R}$, and $b_{1j}, b_{2j}:[0, 1] \to \mathbb{R}$, $j = 1, \ldots, m$ be smooth functions. Consider the following two sets $$\begin{align*} S_1 &= \left\{ \begin{...
3
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1answer
132 views

Stone-Weierstrass theorem for modules of non-self-adjoint subalgebras

In "Weierstrass-Stone, the Theorem" by Joao Prolla, there is a Stone-Weierstrass theorem for modules, stated as the following: Let $\mathcal{A}$ be a subalegebra of $C(X, \mathbb{R})$ and $...
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104 views

A method for extracting a condition to check whether a feature is related to an object

Let we have the object $\bf S$. This object has some properties such as length, temperature and other features. Assume that for the object $\bf S$ we selected $n$ features.For example the vector ${\bf ...
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0answers
66 views

Approximating matrix multiplication with integer arithmetic

The following question is inspired with approximation of matrix multiplication computations occurring in numerical simulations and machine learning algorithms with a use of efficient integer ...
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3answers
239 views

Uniformly approximating a function and its derivative using polynomials

I'm struggling either proving or disproving the following statement: Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If ...
4
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1answer
81 views

Family of Pettis integrals functions “uniformly approximated” by sums

In this book (proof of $4.1.3.$ Lemma. exactly), one can find this passage, that I tried to rephrase here: Let $f:I\times E\rightarrow E$ a Pettis integrable function, where $I:=[0,T]\subset \mathbb{...
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29 views

Eigenvalues associated to differential opertator over natural spline spaces

Let $S_n^k$ denote de linear space of natural splines of degree $k$ and uniform grid $a < t_1 < \ldots < t_n < b$ over the bounded interval $[a,b] \subseteq \mathbb R$. For any $m \in \...
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1answer
115 views

How do you make an accurate, integrable approximation of $a \operatorname{mod} \left(\frac xb,1 \right)$ with a scaling constant $N$?

I'm working on a project where I'm working with modulo functions. However, to continue, I need to integrate integral powers of a weighted sum of them (e.g of the form $\left(c+\operatorname{weighted ...
3
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1answer
158 views

Invertibility of neural network as operator on Wasserstein space

Question statement: Consider the space of probability measures with finite second moments $P_2(\mathbb{R}^d)$, which is equipped with the Wasserstein-2 distance $W_2$, and the square integrable ...
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1answer
71 views

Asymptotic expansion / analysis of this integral

As $M \to +\infty$, how could I make a good asymptotic analysis of this integral? $$\int_0^1 \dfrac{\cos(M x)}{1 + x^2} e^{-M (x^2 - 1/9)}\ \text{d}x$$ The exponential term shall dominate, yet I ...
4
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2answers
785 views

Vector-Valued Stone-Weierstrass Theorem?

The standard statement of the Stone-Weierstrass theorem is: Let $X$ be compact Hausdorff topological space, and $\mathcal{A}$ a subalgebra of the continuous functions from $X$ to $\mathbb{R}$ which ...
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80 views

Smooth function approximating pi(x)

We can define the prime number function as $$\pi(x) = \int_{-\infty}^x \sum_{p}\delta(p-x).dx$$ That is, we include each prime p as a delta function $\delta_p(x) = \delta(p-x)$, set $P(x) = \sum_{p}\...
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56 views

Solving $\int_0^\infty N(1-F(t))^{N-1}tf(t)dt$ when the expected value is known

Suppose that $f:\mathbb R_{\geq 0} \rightarrow \mathbb R_{\geq 0}$ is a probability density function, and $F$ is a cumulative distribution function (i.e. $F(t)=\int_0^t kf(k)dk$). Also, assume that ...
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0answers
20 views

Mismatching degrees and # derivatives in polynomial interpolation error formula

It is well known that if $f : [a,b] \to \mathbb{R}$ is $n+1$ times differentiable and $p(x)$ denotes the polynomial interpolant to $f(x)$ in the $n+1$ points $\bigl(x_k \in [a,b]\bigr)_{k = 1}^{n+1}$, ...
1
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2answers
143 views

What is the approximation of $\log(|\zeta'(\frac{1}{2}+it)|)$ in Dirichlet polynomial if it is exists?

I have done some search many times on web to find any approximation of $\log|(\zeta'(s))|$ in Dirichlet polynomial but I didn't got it, Probably that $\log(|\zeta'(s)|$ dosn't have a Dirichlet ...
6
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3answers
426 views

Approximating by incrementing positive integers

Let $\mathbb{N}$ denote the set of positive integers. For $\alpha\in \; ]0,1[\;$, let $$\mu(n,\alpha) = \min\big\{|\alpha-\frac{b}{n}|: b\in\mathbb{N}\cup\{0\}\big\}.$$ (Note that we could have ...
6
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1answer
194 views

Time of peak of an SIR epidemic

I've learned some classical results on the peak and the attack rate of an idealized epidemic which evolves according to a SIR model $\dot{s} = -\beta\cdot i \cdot s$ $\dot{i} = +\beta\cdot i \cdot s -...
4
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2answers
185 views

Approximated solutions of SEIR models

Numerical solutions of the SEIR equations (describing the spreading of an epidemic disease) – or variations thereof – $\dot{S} = - N$ $\dot{E} = + N - E/\lambda$ $\dot{I} = + E/\lambda - I/\delta$ ...
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0answers
44 views

Image restoration quality general lower bounds

A typical image restoration model posits that, starting from a true image $f = f(x,y)$, we observe $$ \tilde f = f \star h + n $$ where $\star$ is convolution, $h$ is the point spread function (caused,...
9
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1answer
413 views

Approximating power series coefficients — Why does a clearly illegitimate method (sometimes) work so well?

For reasons that don't matter here, I want to estimate the power series coefficients $t_{ij}$ for the rational function $$T(x,y)= {(1+x)(1+y)\over 1- x y(2+x+y+x y)}=\sum_{i,j} t_{ij}x^iy^j$$ Using a ...
17
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2answers
513 views

Approximation of smooth diffeomorphisms by polynomial diffeomorphisms?

Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a polynomial diffeomorphism? More precisely: Let $f:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be a smooth diffeomorphism for $d&...
5
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0answers
205 views

Can we construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the torus?

Consider any continuous function $f$ on an $m$-dimensional torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
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1answer
153 views

Can we construct a sequence of trigonometric polynomials that converges pointwise to a given continuous function on the torus?

Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric polynomials), with the band width (degree of the ...

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