Questions tagged [taylor-series]

Taylor series is a method to analyze functions as polynomials.

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Weaker version of the Borel lemma for vector-valued functions

Borel's lemma for Frechét-spaces $V$ says: (i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists a smooth $f: \mathbb{R} \to V$ such that $$f^{(j)}(0) = v_j.$$ For general locally ...
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Do smooth cutoff functions analytically continue functions?

My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ...
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The Basel problem revisited?

In the Basel problem, the $sinc$ function is considered at the Wikipedia page. Let me try to make an alternative function definition: $$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
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Hadamard lemma without integration

Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero. By the product ...
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Complexity of calculating $f^{(n)}(0)$/extracting a coefficient of a functions taylor-series

Many combinatorial problems can be solved using generating functions. In such a case, we obtain a function $f(x)$, which (for usual) has a taylor-expansion: $$ f(x) = \sum_{n\ge 0 } a_n x^n $$ So ...
Sudix's user avatar
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Szegő curve for partial sum of Taylor series of Riemann $\Xi(z)$ function

I am sorry that this is long post. But it might be of interest to you. This post is related to zeros of partial sum of Taylor series of $e^x-1$. Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can ...
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Covergence of fractional Taylor series

Let $f(x)$ be a function that is continuous and infinitely smooth on entire $\mathbb R$. Let's consider Taylor-Maclaurin series for this function: $$f(x) = \sum_{0}^{∞}\frac{f^n(x_0)(x-x_0)^n}{n!}$$ ...
Мікалас Кaрыбутоў's user avatar
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Functional Taylor expansion for differential entropy

Consider an continuous distribution $F$ with density $f$. The (differential) Shannon entropy of $f$ is $h(f)=-\int f(x)\log f(x) dx$. In the literature of differential entropy estimation, ...
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Jet at a singular point or a submanifold

Let $M$ be a smooth manifold, $p\in M$ and $f\in C^\infty(M\setminus\{p\})$. We will say that $f$ has a power-law singularity at $p$ of order $\eta$ if for every smooth immersion $\gamma:(-1,1)\to M$ ...
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Convex minorants to convex functions, given partial Taylor expansion and smoothness estimate

Let $V$ denote a strictly convex function (in arbitrary dimension) whose Hessian is $L$-Lipschitz. Given only this knowledge, and the values of $\left\{ V \left( x \right), \nabla V \left( x \right), \...
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What is the meaning of big-O of a random variable?

I encountered this problem in a book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. I excerpt it below: screenshot of the book In the excerpt, the big-O notation $O(\xi^...
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Are there any zeta functions with concurrent derivative shifts in multiple variables?

Expressions for rational zeta series have been obtained by considering the Taylor series of zeta functions. For instance, one has \begin{align}\zeta(s,x+y) &= \sum_{k=0}^{\infty} \frac{y^{k}}{k!} \...
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Taylor series with less than differentiability

I have a function $f^0\colon (0;\infty) \to \mathbb R$ with the property that the following limit exists and is finite $$ F^1 := \lim_{x\to \infty} x \cdot f^0(x) $$ Then I consider $f^1(x) := x \cdot ...
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Norm of a Taylor approximation of a multivariate function

I have a function $f:\mathbb{R}^n\to\mathbb{R}^m$. My goal is to bound the first order Taylor approximation of $f$. Given $x,x'\in\mathbb{R}^n$ I have that \begin{equation} f(x)-f(x')\approx (x-x')^...
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Calculating the polynomials which are invariant under the action of a simple finite group

Let $G$ be a simple, finite group. In general, $G$ is not abelian. Let $\rho$ be a representation of this group, where each $\rho(g)$ for $g\in G$ is a unitary, complex, $d$-dimensional matrix, $\rho(...
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A certain expectation of a function of independent gammas

Suppose that $Y_1...,Y_n$ are independant gamma random variables: $Y_i \sim \Gamma(\alpha_i,\beta_i)$, with density $f_i(t) = \frac{t^{\alpha_i-1} e^{-\frac{x}{\beta}}}{\Gamma(\alpha) \beta^{\alpha}}$....
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Generalization of Carleman coefficients to multivariable functions - Carleman tensor?

Recently I learned about a matrix called Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying. Carleman linearization is a technique used to embed a finite ...
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Smoothness of coefficients of remainder term in Taylor expansion

Given a $C^{k}$ function $f:\mathbb{R}^d\to\mathbb{R},$ we can use Taylor's theorem to write it as $$f(x)=\sum_{|\alpha|\le k-1} c_\alpha x^\alpha + R(x),$$ where $R$ is $C^k$ and can be expressed ...
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Positive, Uni-modal, Log-concave Combinatorics

We define a sequence, $\{a_n\}_{n=0}^\infty$, to be a uni-modal sequence if for some $m$, $$a_0<a_1<\cdots<a_m,\ \ \ \ a_m>a_{m+1}>a_{m+2}>\cdots.$$ We define a sequence, $\{a_n\}_{...
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Exponential-like function equivalent for the Dixonian Elliptics

Is there some exponential-like function that acts as partner for the intricate Dixonian Elliptics, in a similar way that the Exponential function acts as a partner for the trigonometric functions ?
Jaime Yerbabuena's user avatar
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A problem on monotonicity rule for the ratio of two Maclaurin power series

In the paper [1] below, a monotonicity rule for the ratio of two Maclaurin power series was presented as follow. Let $a_k$ and $b_k$ for $k\in\{0\}\cup\mathbb{N}$ be real numbers and the power series ...
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Lower bound for variance of ratio of dependent random variables

I'm trying to find a lower bound on $\text{Var}(X / Y)$ for dependent random variables $X, Y \in [0, 1]$ with $X \leq Y$. More specifically, $X$ and $Y$ are defined as follows: Let $h, n \in \mathbb{N}...
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Showing bound $\|\nabla_t \dot{\tilde{x}}_Y(h, 0)\| \le L \|Y\|_{2, \infty}$ for smooth homotopies of geodesics

This question pertains to Lemma 3.5 of this article. Let $M$ be a smooth Riemannian manifold and $\gamma$ some geodesic with respect to the Levi-Civita connection $\nabla$. For any $C^2$ vector field $...
infinitylord's user avatar
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Is a mixture of real analytic functions again analytic?

Let $$h : \mathbb{R}^2 \to \mathbb{R}^+.$$ Suppose that for each $x$, $h(x, y)$ is a real analytic function of $y$. Let $\mu(dx)$ be a finite measure on $\mathbb{R}$, and for each $y$, suppose that $$...
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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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Estimation of parameters through multivariate Taylor expansion?

I do have a function $$f(t) = \prod\limits_{j=1}^{n} \left(1 + \sum\limits_{i=1}^{n} M_{i,j} t_i\right)^{-\alpha_{j}}$$ defined by parameters: $M_{i,j} \in \mathbb{R}_{+}, \;\forall i \in 1,...,d,\; ...
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Hadamard-like product on infinitely differentiable functions

Has the following operation $*$ on formal power series $f,g$ been studied before? $$[X^n](f*g) = n! \cdot [X^n]f \cdot [X^n]g$$ where $n$ is a nonnegative integer? This is the typical Hadamard ...
Jake Lai's user avatar
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Asymptotics to Taylor expansions?

I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments. Maybe you guys can help. https://math.stackexchange.com/questions/1440931/...
mick's user avatar
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Generating a series representation for the inverse of the operator $f(f)$

I am considering the following problem: Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...
Sidharth Ghoshal's user avatar
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Properties and name of some polynomials

I have encountered in a problem some polynomials given by $P_k(x) = \prod_{j=0}^{k-2} (kx-j)$. I need to understand if these polynomials are known, and if they have certain special properties, as ...
Geno Whirl's user avatar
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Applying the ideas of power series to certain convolutions - which identities transfer?

Let's suppose I'm working with some set of functions $f_k(n)$. $f_1(n)$ is essentially the root of my functions, and could be nearly anything, and then $f_k(n) = (f_1(n) * f_{k-1}(n))$ for some ...
Nathan McKenzie's user avatar
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Do we have tetration uniqueness by $ A = \inf \sum_n a_n^2 $?

Let $f$ be a real analytic (on at least $|x|<2$) and real solution of the functional equation $f(0) = 1,f(x+1) = \exp(f(x))$. For the existence of such $f$, see here. Then $$ f(x) = \sum_n a_n x^n ;...
mick's user avatar
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Could the patterns in the roots of the Riemann-Liouville differintegral of $(s-1)\,L(s, \chi_{q,j})$ be explained?

This question aims to extend this question to (automorphic) Dirichlet L-functions. Take: $$f(s,q,j) = (s-1)\,L(s, \chi_{q,j})$$ with $q$ the modulus and $j$ the index of a character $\chi$. A fast way ...
Agno's user avatar
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How to calculate the weights for Discrete Laplacian Operator?

I am following this paper step by step and want to build an isotropic Laplacian kernel. As shown in the following figure, I can understand until using Taylor to expand the 2D discrete Laplacian ...
Ili a's user avatar
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Linear approximation of multivariate function of bounded second order partial derivatives

I have a question about linear approximation in the multivariate case.\ Let $f:B^d_r\to \mathbb{R}$ be a real-valued $C^2$-function defined on the $d$-dimensional ball of radius $r$ centered at the ...
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Finding a square integrable dominating function for function class

problem statement For any $x \in R^d$, consider the function $$\phi(x,a) = \min_{1\leq j \leq k} \|x-a_j\|^2,$$ where $a = (a_1, \dots, a_k) \in R^{kd}$ and $\| \cdot \|$ is the $L_p$ norm for any $p ...
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Taylor approximation of $f(q) = \left(1 + q \dfrac{w_s}{w_0}\right)^{\alpha}$

I am trying to prove equations (3) given in this paper http://users.cecs.anu.edu.au/~thush/publications/vtc_final.pdf. The authors use taylor series to approximate function $f(q) = \left(1 + q \...
Emil Bjelski's user avatar
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Taylor series expansion of quantile function

Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $. We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$. Do you have any ...
NN2's user avatar
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Series expansion with remaining $log n$

Hi, I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant. $$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$ I'm trying to do a ...
ELW's user avatar
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Calculating entropy of adjacency matrix using eigenvalue decomposition?

How to calculate entropy using the eigenvalues when the eigenvalues are negative? Is there a simple relation between the entropy of a matrix and its characteristic polynomial?
kosmos's user avatar
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Jet spaces for maps with constraints

Lets be in the category $\mathbf{M}$ of smooth finite dimensional manifolds with smooth maps: Suppose we have the set of all smooth maps $Hom_\mathbf{M}(R^n,M)$ from $R^n$ to a smooth manifold $M$. ...
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