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.
1,142 questions
0 votes
0 answers
52 views
Saddle Point / Steepest Descent for Bessel Functions
I am trying to understand how to approximate integrals with Bessel functions. In particular I have something like: $$I_{\ell} = \int_{0}^{\infty} j_{\ell}(pr) dr = \frac{\sqrt{\pi} \Gamma[(1+\ell)/2] ...
6 votes
3 answers
191 views
When does the supremum over a subspace $F\subset C[0,1]$ commute with integration?
Let $F\subset C[0,1]$ be a linear subspace which contains the constant functions and separates points of $[0,1]$. Assume that for every $f\in C[0,1]$ and every $x\in[0,1]$ we have $$ f(x)=\sup\{g(x)\...
- 608
-1 votes
1 answer
99 views
Approximate unknown function with Fourier series
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 ...
- 526
1 vote
1 answer
83 views
Is the error of this minmax polynomial approximation strictly increasing?
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 ...
- 451
2 votes
1 answer
116 views
How can I approximate this function to be able to integrate it?
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 ...
- 23
1 vote
0 answers
70 views
Clarification about a particular step in a proof of the Kolmogorov-Arnold representation theorem
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 ...
- 9,770
2 votes
0 answers
42 views
Proving universal approximation through ReLU
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, & \...
4 votes
3 answers
601 views
Is there a smooth function approximating the minimum of a constant and a variable?
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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