# 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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(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-$... 0answers 64 views ### Reverse Inequality I was doing some numerical integration when I figured the function I was dealing with (i.e., the function I was integrating) evaluated to big numbers on a tiny portion of the interval (over which I ... 1answer 189 views ### Checking the uniform denseness of a set in$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, mbe smooth functions. Consider the following two sets \begin{align*} S_1 &= \left\{ \begin{... 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 ... 0answers 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 ... 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 ... 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 ... 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{... 0answers 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 \... 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 ... 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 ... 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$$The exponential term shall dominate, yet I ... 2answers 785 views ### Vector-Valued Stone-Weierstrass Theorem? The standard statement of the Stone-Weierstrass theorem is: Let X be compact Hausdorff topological space, and \mathcal{A} a subalgebra of the continuous functions from X to \mathbb{R} which ... 0answers 80 views ### Smooth function approximating pi(x) We can define the prime number function as$$\pi(x) = \int_{-\infty}^x \sum_{p}\delta(p-x).dx$$That is, we include each prime p as a delta function \delta_p(x) = \delta(p-x), set P(x) = \sum_{p}\... 0answers 56 views ### Solving \int_0^\infty N(1-F(t))^{N-1}tf(t)dt when the expected value is known Suppose that f:\mathbb R_{\geq 0} \rightarrow \mathbb R_{\geq 0} is a probability density function, and F is a cumulative distribution function (i.e. F(t)=\int_0^t kf(k)dk). Also, assume that ... 0answers 20 views ### Mismatching degrees and # derivatives in polynomial interpolation error formula It is well known that if f : [a,b] \to \mathbb{R} is n+1 times differentiable and p(x) denotes the polynomial interpolant to f(x) in the n+1 points \bigl(x_k \in [a,b]\bigr)_{k = 1}^{n+1}, ... 2answers 143 views ### What is the approximation of \log(|\zeta'(\frac{1}{2}+it)|) in Dirichlet polynomial if it is exists? I have done some search many times on web to find any approximation of \log|(\zeta'(s))| in Dirichlet polynomial but I didn't got it, Probably that \log(|\zeta'(s)| dosn't have a Dirichlet ... 3answers 426 views ### Approximating by incrementing positive integers Let \mathbb{N} denote the set of positive integers. For \alpha\in \; ]0,1[\;, let$$\mu(n,\alpha) = \min\big\{|\alpha-\frac{b}{n}|: b\in\mathbb{N}\cup\{0\}\big\}.$$(Note that we could have ... 1answer 194 views ### Time of peak of an SIR epidemic I've learned some classical results on the peak and the attack rate of an idealized epidemic which evolves according to a SIR model \dot{s} = -\beta\cdot i \cdot s \dot{i} = +\beta\cdot i \cdot s -... 2answers 185 views ### Approximated solutions of SEIR models Numerical solutions of the SEIR equations (describing the spreading of an epidemic disease) – or variations thereof – \dot{S} = - N \dot{E} = + N - E/\lambda \dot{I} = + E/\lambda - I/\delta ... 0answers 44 views ### Image restoration quality general lower bounds A typical image restoration model posits that, starting from a true image f = f(x,y), we observe$$ \tilde f = f \star h + n $$where \star is convolution, h is the point spread function (caused,... 1answer 413 views ### Approximating power series coefficients — Why does a clearly illegitimate method (sometimes) work so well? For reasons that don't matter here, I want to estimate the power series coefficients t_{ij} for the rational function$$T(x,y)= {(1+x)(1+y)\over 1- x y(2+x+y+x y)}=\sum_{i,j} t_{ij}x^iy^j$Using a ... 2answers 513 views ### Approximation of smooth diffeomorphisms by polynomial diffeomorphisms? Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a polynomial diffeomorphism? More precisely: Let$f:\mathbb{R}^d\rightarrow\mathbb{R}^d$be a smooth diffeomorphism for$d&...
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 ...
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 ...