2
votes
1answer
48 views

approximation of $\log(1+z)=z$ as $z\to 0$

This is new to me and I have not done any asymptotic approximation. I don't understand how they get that $\frac{n}{N}$ stays close to $\frac{2}{3}$ as N goes to infinity. Also how do they do get ...
1
vote
1answer
17 views

can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that ...
2
votes
1answer
16 views

lower bound of modified Besselfunction

i'm looking for an lower bound for the modified Bessel function of the first kind $I_\nu(x)$ of a +ive real argument. There should be one of the form $$ce^{x^\alpha} \le I_\nu(x)$$.
2
votes
0answers
31 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
2
votes
2answers
42 views

Monotonically approximate $L^p$ function by step function

It is a classical fact that a $L^p(R^d)$ ($1\leq p<+\infty$) function can be approximated by step functions with compact support, but my question will be, can we require that the step function is ...
2
votes
1answer
24 views

Approximating error using Taylors theorem

I have used a Maclaurin series for the function $f(x) = \cos(2x)$ and have successfully produced: $\dfrac{2^n cos(\frac{n\pi}2)x^n}{n!}$ Now I want to estimate the error in approximating $\cos(2x)$ ...
2
votes
0answers
39 views

Application of Weierstrass approximation theorem

How to approximate a continuous function to a desired accuracy using a polynomial? Theorem: For any $\varepsilon > 0$ and $f \in C([a,b])$, there exists a polynomial $p$ such that $\sup_{x \in ...
0
votes
0answers
31 views

Is the following statement on the stability of the forward Euler method true or false?

My text asks whether the following statement is true or false: The forward Euler method for approximating the solution of $x'=\lambda x$ is stable for all $\lambda \in \mathbb R$ and all step ...
4
votes
1answer
124 views

Lower bound for $(x^c-1)^{1/c}$

I have been trying to find a lower bound for $x>1$, $c>0$: $$ \Large(x^c-1)^{1/c} $$ My strategy is to find a lower bound for $(x^c-1)^{1/c}$ which can hopefully get rid of some of the $c$ ...
0
votes
0answers
27 views

Show the linear span of the given collection is dense in the space

Show that space generated by $\{1, \sin x, \sin 2x, \ldots \sin nx,\ldots\}$ is dense in $C([0,\pi])$. I am having trouble to show it algebra and get explicit function that separates end points. ...
0
votes
1answer
45 views

Approximating simple functions by step functions almost uniformly

The title says it all. How can we approximate measurable simple functions by step functions almost uniformly in, say, $[0,1]$? Even with the simplest example, $\chi_{A}$, where $A$ is Lebesgue ...
2
votes
2answers
50 views

Approximation of difference of harmonic numbers

Harmonic number $H_n$ is equal to $$H_n = \sum_{i=1}^n \frac{1}{i}$$ Asymptotic expansion of harmonic humber is $$(1) H_n = \ln n + \gamma + \frac{1}{2n} - O\left(\frac{1}{n^2}\right)$$. Very popular ...
1
vote
1answer
82 views

If $\int_0^1 f(x) e^{nx} dx = 0$ for every n, then f=0

$f$ be a continuous function [0,1] to $R$. $\int_0^1 f(x)e^{nx} dx = 0$ for all $n \in N\cup\{0\}$ how to prove $f(x)= 0$ in $[0,1]$ for all $x\in[0,1]$? I solved "$\int_0^1 f(x)x^n dx = 0$ for all ...
0
votes
1answer
50 views

Approximation of $\frac{1}{2^{N}} \binom{N}{\frac{1}{2}(N + s)} $ for big N

I have the following function that I have to approximate for big N and I really have no clue what to do - I hope that anyone can help me out: $$P_N(s) = \begin{cases} \frac{1}{2^{N}} ...
2
votes
2answers
73 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
2
votes
1answer
104 views

Existence of a simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and for every ...
2
votes
1answer
127 views

Stone-Weierstrass theorem with $p(1/x)$

I am trying to prove that for a continuous $f\colon[1,\infty)\rightarrow\mathbb R$ and $f(x)\to a$ as $x\to\infty$ it could be approximated by $g(x)=p(1/x)$ where $p$ is a polynomial.
1
vote
0answers
71 views

Newton-Rhapson for reciprocal square root

I have a question about using Newton-Rhapson to refine a guess of the reciprocal square root function. The reciprocal square root of $a$ is the number $x$ which satisfies the following equation: ...
0
votes
0answers
42 views

Bounding a convolution with a maximal function

Consider a family of kernels $\{K_{\epsilon}\}_{\epsilon>0}$ such that: $\int_{\mathbb{R}^d}K_{\epsilon}\ dx=1$ $|K_{\epsilon}(x)|\leq A\delta^{-d}$ for all $\delta>0$ $|K_{\epsilon}(x)|\leq ...
6
votes
1answer
167 views

Approximate Holder continuous functions by smooth functions

Let $g \in C^{\alpha} (B_1)$ be given. Can we find a sequence $(f_n) \subset C^{\infty} (B_1)$ such that $f_n \rightarrow g$ in $C^{\alpha}(\overline{B_1})$? If so, how can it be done? I have tried ...
3
votes
1answer
203 views

On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)

The Stone-Weierstrass Theorem tells us that we can approximate any continuous $f:\mathbb{R}^n\to\mathbb{R}$ arbitrary well on a compact subset of $\mathbb{R}^n$ by some polynomial. Suppose that $f$ is ...
11
votes
1answer
159 views

Is $e = \sum_n 1/n!$ the most efficient sequence of denominators for rational series for $e$?

The classical series $e = \lim_{n \to \infty} X_n$ where $X_n = \sum_{k=0}^n 1/k!$ is incredibly efficient. But is it known to be the most efficient series in terms of denominators for using fractions ...
3
votes
2answers
60 views

Prove approximation given by the physicist Max Born

In an old book about optics, I have found a nice approximation, that for large l one has: $$P_l(\cos(\theta)) \sim \sqrt{\frac{2}{l \pi \sin(\theta)}} \sin \left((l+\frac{1}{2}) \theta + ...
1
vote
0answers
91 views

Turn ugly series into a nice approximation

I am currently struggeling with a series(actually two). The problem is, that I can do nothing with them, since this expression is so ugly. I would love to hear about any kind of approximations that ...
1
vote
1answer
24 views

The value of $w$ also has a max error of $p\%$

Suppose $\frac{1}{w}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ where each variable $x,y,z$ can be measured with a max error of $p\%$ Prove that the calculated value of $w$ also has a max error of $p\%$ ...
0
votes
1answer
54 views

I know what I need to do but dont know how to apply: the question related to The first order approximation theorem

$\mathbf{Question:}$ Prove that $\displaystyle \lim_{(x,y)\to (0,0)} \dfrac{\sin(2x+2y)-2x-2y}{\sqrt{x^{2}+y^{2}}}=0$ $\mathbf{My\ ideas:}$ I will use the First Order Approximation Theorem. But ...
2
votes
1answer
42 views

Verifing the solution

$\mathbf{Question:}$ Let $f(x,y)=e^{\sin(x-y)}$ for $(x,y)\in \Bbb R^2$ Find the affine function that is a first order approximation to the function $f$ at the point $(0,0)$ $\mathbf{Answer:}$ ...
2
votes
1answer
134 views

Approximating sum by Gaussian integral - how big is the error?

I have the following infinite sum: $$S=\sum_{n=1}^{\infty}e^{-an^2}$$ Where $a$ is a positive constant. Is there a simple way to estimate the error when approximating $S$ by: $$S \approx \int_0^ ...
2
votes
1answer
65 views

Given the first n derivatives of a function at two points, is it possible to approximate the function between these points?

That is, the function is on an interval $f:[a,b]\rightarrow\mathbb{R}$ and smooth; and at the boundaries of the interval $(a,b) \in\mathbb{R}$, all $f^{(m)}(a)$ and $f^{(m)}(b)$ are known for ...
2
votes
2answers
111 views

Show that for a finitely differentiable function there exists a polynomial that both their $k$-th derivatives converge

We need to show that for all $f: [a,b] \rightarrow \Bbb R $ which is differentiable n times, and for all $\epsilon>0$ there exists a polynomial $p\colon[a,b] \rightarrow R$ s.t. $\forall n ...
1
vote
1answer
53 views

Properties of Lebesgue functions

If $f\in \mathcal {L}$ then there exists a sequence $\{f_k\}$ of step functions s.t. $\lim_{k\to\infty} f_k(x)=f(x)$ for almost all $x$ and $$\lim_{k\to\infty} \int|f(x)-f_k(x)|\,dx=0.$$ If I have ...
1
vote
2answers
540 views

Weierstrass Approximation Theorem for continuous functions on open interval

I am studying for my introductory real analysis final exam, and here is a problem I am somewhat stuck on. It is Question 2, in page 3 of the following past exam (no answer key unfortunately!): ...
0
votes
2answers
289 views

How to show that a measurable function on $R^d$ can be approximated by step functions?

In Stein's Book: Real Analysis, Theorem 4.3 says that any measurable function $f$ on $R^d$ can be approximated by step functions. But the proof provided there only show that when $f=\chi_E$, with ...
1
vote
2answers
90 views

Asymptotic expansion for $\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du$?

Is there an asymptotic expansion for the function: \begin{equation} g(x)=\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du, \end{equation} over the domain $x\in [0,\infty)$ in terms of ...
11
votes
1answer
289 views

Approximation of Semicontinuous Functions

Assume that $k \in \mathbb{N}$ and $f : \mathbb{R}^d \rightarrow [0,\infty)$ is lower semicontinuous, i.e. $f(x) \leq \liminf_{y \rightarrow x} f(y)$ for all $x \in \mathbb{R}^d$. Does there exist a ...
2
votes
1answer
175 views

Approximate continous function with linear growth condition by Lipschitz function

Suppose a continuous function $f(u)<K(1+|u|)$ for some positive number $K$. How can we find a sequence of Lipschitz functions $f_{n}$ that converge to $f$ uniformly on $\mathbb{R}$. If we require ...
2
votes
1answer
48 views

Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that ...
2
votes
1answer
70 views

If f is in LipK[a, b], show that f can be uniformly approximated by polynomials in LipK[ a, b].

Question: If f $\in$ LipK[a, b], show that f can be uniformly approximated by polynomials in LipK[ a, b]. Context: f $\in$ LipK[a,b] then it is Lipschitz with constant K. The text I am currently ...
2
votes
1answer
116 views

approximate error between integral an sum

I am new here. My problem: There is an integral $I:=\int_0^1 f(x)\,dx$ for $f\colon [0,1]\to\mathbb{R}$ and I want to compute it by ...
12
votes
1answer
241 views

Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$.

This integral is very closely related to the sophmores dream that states $$ \int_0^1 x^{-x}\mathrm{d}x = \sum_{n=1}^\infty n^{-n} = 1.27\ldots $$ For example here ...
10
votes
2answers
360 views

Summation of divergent series of Euler: $0!-1!+2!-3!+\cdots$

Consider the series $$\sum\limits_{k=0}^\infty (-1)^kk!x^k\in\mathbb{R}[[x]]$$ Let $s_n(x)$ be partial sum. And let $\omega_{k,n}=(k!^2(n-k)!)^{-1}$. My question is: Prove that ...
2
votes
1answer
878 views

Approximations for the partial sums of exponential series

Though the question here (Partial sums of exponential series - Stack Exchange) is similar, it is more specialized and I rather need a general approximation for an arbitrary partial sum. Essentially, ...
7
votes
2answers
554 views

Approximate $\int_a^b \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-(x-\mu)^2/2 \sigma^2}\log(1+e^{-x}) \ \ dx $

I am trying to find an approximation to $$ I = \int_a^b \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-(x-\mu)^2/2 \sigma^2}\log(1+e^{-x}) \ \ dx. $$ My attempt is as follows: $$ \begin{align} I &= \int_a^b ...
3
votes
1answer
97 views

A problem on $C^k$

Can anyone help me solve the following problem ? Let $a,k >0$, $a$ is real and $k$ is integer. Consider the set $S$ of all function $f\in C^k([0,a])$ such that 1) $f(0)=0$ and $f(a)=1$ 2) ...
5
votes
1answer
139 views

$C^1$ approximation of a continuous curve.

Suppose I have two points $\alpha,\beta \in \Bbb{R}^n$. Define $$ X=\{\gamma \in C^1([0,1] , \Bbb{R}^n),\ \gamma(0)=\alpha,\gamma(1)=\beta ,0 <|\gamma'|<K\}$$ parametrized curves joining ...
3
votes
1answer
1k views

Approximation of a bounded measurable function with step functions?

I'm having trouble judging whether this statement is correct: For an arbitrary bounded measurable function $f$ defined on $[0,1]$, $\exists{}\ $a sequence of step functions $\{\phi_n\}$, such that ...
0
votes
0answers
215 views

Comparing norms of a vector

Let $a$ be a vector in $\mathbb R^m$, such that $\sum_{i=1}^{m}a_i=0.$ I would like to compare $\sqrt{2m(2m−1)}\|a\|_{\infty}$ and $\sqrt{2m}\|a\|_2$, in the case when the vector $a$ satisfies the ...
-1
votes
1answer
119 views

inequality and equivalence for norms

Let $x \in R^m$. It is known that $\|x\|_{\infty}\leq \|x\|_2\leq \sqrt m\|x\|_{\infty}$. What the difference between above inequality and if we are saying that $\|x\|_2\sim \sqrt m\|x\|_{\infty}$? ...
2
votes
0answers
100 views

explicit error bounds for Multivariate interpolation

I want to interpolate a function of $d$ variables over a Cartesian grid, using multivariate interpolation, while characterizing interpolation error in terms of bounds on partial derivatives of the ...
1
vote
1answer
462 views

Picard's method application

Hello guys here is my question thank you for all the help. I need to determine On which integral the Picard's Method is applicable for $y'=xy^2$, $y(0)=0$ and need to calculate the first ...