All questions with tag [math: approximation-theory]


Compressed sensing, approximately sparse, Power law

An x in $\mathbb{R}^n$ is said to be sparse if many of it's coefficients are zeroes. x is said to be compressible(approximately sparse) if many of its coefficients are close to Let $x=(x_1,x_2,....x_n)$. Sort the absolute values of the coefficients in decreasing order with new indices as $|x_{(1)}|\geq|x_{(2)}|\geq,..,|x_{(n)}|$. x is ...
2022-07-14 01:43:01

Books on Function approximation and Regression

Can you recommend books/articles on Function approximation Let me price estimate from the above wiki: Second, the target function, call it g, might be unidentified ; as opposed to a specific formula, just a set of factors of the kind (x, g (x)) is given . Relying on the framework of the domain name and also codomain of g, numerous strategies f...
2022-07-13 22:52:04

Distance from $x^n$ to lesser polynomials

I am interested in the $L_1$ distance of $x^n$ to the $\mathbb R$-span of $\{1,x,\ldots,x^{n-1}\}$ over some interval. We can WLOG consider the interval $[0,1]$ (say) because scaling and shifting only affects the distance by a constant factor. So far we have these basic results: Theorem $x^n$ is not in the $\mathbb R$-span of $\{1,x,\ldots,x^{n...
2022-07-02 23:21:07

Approximation in the Plane of Constant Curvature

In Euclidean space, There is a classic Theorem claims that: The length of every rectifiable curve can be approximated by sufficiently small straight line segment with ends on the curve. Now, The question is that: Is there a similar thing happen on plane of constant curvature? I.e., Can the length of every rectifiable curve on the plane of cons...
2022-06-28 22:30:43

Interpolation, Extrapolation and Approximations rigorously

An international publication stated that "when the Lagrange is interpolation formula falls short (as an example with huge example as a result of Runge is sensation), you need to make use of approximation approaches such as Least - squares - method." I am perplexed due to the fact that I have actually constantly assumed that inter...
2022-06-10 15:42:16

Error term when Lagrange interpolating continuous non-differentiable functions

Suppose I recognize the values of a continual function on $[a,b]$ in some limited variety of factors $x_0,x_1 \ldots x_n$. I can create the Lagrange inserting polynomial, $p$. I wonder if there is any kind of intriguing price quote of the expression $|f(x)-p(x)|$ for an approximate factor $x$ that does not think $f$ is smooth? (Perhaps it would ...
2019-12-02 21:10:20

What is "Approximation Theory"?

Just what is "Approximation Theory"? If I read the wikipedia - write-up I does not get much more clear. Why are "pure" mathematicians curious about it? I see a great deal of individuals that do harmonic evaluation additionally do approximation theory.
2019-12-01 23:49:45

Approximation theorems

The Weierstrass' approximation theory for continual features on a portable room by utilizing polynomials is well - recognized. Regarding I recognize, there are some versions of this theory, as an example Stone - Weierstrass that refers not just to polynomials as approximator features. Where could I locate these Weierstrass - like approximation t...
2019-05-10 01:31:54