Questions tagged [approximation-theory]

Question 1

I am trying to understand how to approximate integrals with Bessel functions. In particular I have something like: $$I_{\ell} = \int_{0}^{\infty} j_{\ell}(pr) dr = \frac{\sqrt{\pi} \Gamma[(1+\ell)/2] ...

Question 2

I recently learned about the theory of distributions. I'm wondering if it's possible to approximate some compactly supported continuous function $f$ on $\mathbb{R^n}$ using the multivariate Taylor ...

Question 3

Let $F\subset C[0,1]$ be a linear subspace which contains the constant functions and separates points of $[0,1]$. Assume that for every $f\in C[0,1]$ and every $x\in[0,1]$ we have $$ f(x)=\sup\{g(x)\...

Question 4

Let $$f(x) = \left(x^T\cdot P \cdot x \right) ^m $$ for $x \in \mathbb{R}^n$ with $n, m \in \mathbb{N}$, and $P \in \mathbb{R}^{n\times n}$ a positive defined square real matrix. Acording to ...

Question 5

Say I have a monotonically increasing function $f : [0,1] \to {\Bbb R}$. I only know the values of $f$ for a finite set of points $x_1, \dots, x_n$. Can I use a Fourier series to approximate the ...

Question 6

Suppose we have the following function, where $s\in\mathbb{R}$ and $t_1,t_2,n\in\mathbb{N}\cup \{0\}$ are constants: $$\mathbf{P}(r)=\left(t_1+\prod_{k=1}^{r}(t_2+k^{s})\right)^n$$ Question: What is ...

Question 7

A paper I am reading [1] makes the claim that for: $$ D(\tau) := [-1 - \tau, -1 + \tau] \cup [1 - \tau, 1 + \tau] $$ Let $p(x)$ be the polynomial of a fixed degree $d$ (depending on $\tau$) minimizing ...

Question 8

Let $I=[0,1]$ be the unit interval, and $f:I\to\mathbb R$ be a $C^2$ function. For $n\ge1$, denote by $PL(n)$ the set of continuous, piecewise linear functions with $n$ pieces defined on $I$. It is a ...

Question 9

Suppose I have the following real function $$f(x) = \frac{\left[ (2 + b)^2 - x \right]^{1/2} (b^2 - x)^{3/2} \left[ (2 + b)^2 + 2x \right] (x + 2a^2) (x - 4a^2)^{1/2}}{x^{3/2} (c^2 - x)^2}$$ defined ...

Question 10

Suppose $z$ is an $N\times1$ Gaussian vector and that $X$ is an $N\times2$ ($N > 2$) matrix that contains two independent Gaussian vectors. $z$ and $X$ are independent. Matrix $X(X^{T}X)^{-1} X^{T}$...

Question 11

I am reading this proof of the Kolmogorov-Arnold representation theorem, first this sentences: By plotting out the entire grid system, one can see that every point in $[0,1]^2$ is contained in $3$ to ...

Question 12

Let $f(x)=e^{\sin(x)}$. Prove that there exist two polynomials $p(x),q(x)\in \mathbb{R}[x]$ of degree at most $2$ such that $$f(x)=\frac{p(x)}{q(x)}+o(x^4)$$ and find explicit form of them. I do not ...

Question 13

We were discussing universal approximation theorems for neural networks and showed that the triangular function $$ h(x) = \begin{cases} x+1, & x \in [-1,0] \\ 1-x, & x \in [0,1] \\ 0, & \...

Question 14

My question comes from Example 3.11 (p.184) of Applied Mathematics 4th ed by J. David Logan. The example is on finding a uniform approximation to a boundary value problem. Here's the relevant snippet ...

Question 15

Let $K$ be a constant and $x$ be a variable. What is a smooth, monotonic function that is as close to $\min(K,x)$ as possible, but never exceed $\min(K,x)$? Also f(x)>=0 for x>=0 and f(0)=0 ...

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Questions tagged [approximation-theory]

Saddle Point / Steepest Descent for Bessel Functions

Taylor series using distributional derivatives

When does the supremum over a subspace $F\subset C[0,1]$ commute with integration?

Kolmogorov's Superposition Theorem for a power of a second degree n-polynomial

Approximate unknown function with Fourier series

How do we find an asymptotic approximation of the function $\mathbf{P}(r)=\left(t_1+\prod\limits_{k=1}^{r}(t_2+k^{s})\right)^n$?

Is the error of this minmax polynomial approximation strictly increasing?

Faster approximation for piecewise linear functions in $L^2(\rho)$

How can I approximate this function to be able to integrate it?

$z^{T} X(X^{T}X)^{-1}X^{T}z$ appears exactly Chi-squared if $z$, $X$ BOTH store Gaussian entries?

Clarification about a particular step in a proof of the Kolmogorov-Arnold representation theorem

Rational approximation of $e^{\sin(x)}$ [closed]

Proving universal approximation through ReLU

Dominant balance in finding a uniform approximation to a BVP

Is there a smooth function approximating the minimum of a constant and a variable?

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