Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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Series basic question

$$\sum^N_{n=1}\liminf_{k \to \infty} f_k(n) = \lim_{k \to \infty} \sum_{n=1}^N \inf_{j \ge k} f_j(n)$$ I am not sure that equation true. Is that equation true? Then why is it?
japee
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Want f differentiable at the origin but discontinuous everywhere else!

I am trying to get a function $f:\mathbb{R}^2 \to \mathbb{R}$ that is differentiable at the origin but discontinuous everywhere else? As a simpler case, we have that $$g\left(x\right)=\begin{cases} x^2 & \mbox{if }x \in\mathbb{Q}\\ -x^2 &…
Galois
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Limit to infinity question

Possible Duplicate: Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials Does $\lim_{x\rightarrow \infty} \frac{5x}{(1+x^2)} = 0$ or $\lim_{x\rightarrow \infty} \frac{5x}{(1+x^2)} = 1$? I am asking because I was wondering if…
Gary
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About function which Fourier coefficients satisfy $a_n=o(n^{-2}), b_n=o(n^{-2})$

Assume that a function $f: R\rightarrow R$ is $2 \pi$ -periodic and integrable on $[ -\pi,\pi] $. Let $(a_n)$, $(b_n)$ are its Fourier coefficients and $n^2 a_n, n^2 b_n \rightarrow 0$. Then by Weierstrass test $f$ is continuous. What we can say…
Alex
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boundedness of first derivative of $f$

I think this problem is kind of a famous problem... A function $f$ is real valued function from real line. Suppose that both of the absolute value of $f$ and absolute value of the second derivative of $f$ are bounded by 1. Then, show that the…
Follow me
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Dyadic rectangles in the $d$-dimensional unit cube

Let $A = \{J \in \mathcal{D}^d([0, 1]^d) \mid |J| = 2^{-n}\}$ be the set of all dyadic rectangles in the $d$-dimensional unit cube with volume $2^{-n}$, where $J = I_1 \times \cdots \times I_d$ with $I_i$, $i = 1, \ldots, d$, dyadic intervals in…
omonda nii
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Absolute continuity via maximal operator

I'm reading an article and there is a passage that is not very clear to me. The situation is as follows: $f$ is a continuous monotonically increasing function on $[a,b]$. Define: $$ G := [x \in (a,b) : (M(f))'(x) = \infty] $$ Where $M(f)$ is the…
Br09
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Rigorously proving $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx= \frac{\pi}{2}$

I want to prove the famous formula: $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx = \frac{\pi}{2}.$ There are many ways to do it, for example, by some Fourier analysis. But how about a simple method: integrating just $e^{-xy}\sin{x}$ by $y$ and…
user252478
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How to show that a sequence converges pointwise.

Hmmm...I am almost embarrassed to ask this question, but I'll ask anyway. How do I show that the sequence defined by $f_n(x) = n^{1/p}\chi_{[0,1/n]}$ ,$1\le p \lt \infty$ and $x\in [0,1]$ converges pointwise to $0$.
Linda
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Why periodic functions form a dense subset in $ C[a,b]$ with $L^2$ norm?

Let's consider the linear space $C[a,b]$ but with $L^2$ norm $$ \|f\|=(\int_a^b |f(t)|^2dt)^{\frac{1}{2}} $$ How to prove that the subspace $$ V=\{f\in C[a,b]: f(a)=f(b)\} $$ is dense in this normed space. Thanks
Richard
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Can I do this without iteration?

Here is a formula where $y$ and $z$ are known $$ x = \frac{y-0.04z}{0.27z} $$ How would I work out $z$ when only $x$ and $y$ are known?
Sunil
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How to prove that: $\exists C>1$ such that $C||g(y)||\geq ||y||$?

I'm proving the following: Let $E,F$ be two Banach spaces; let $f$ be a function $f:E\to F\ $ linear, and such that: "for every sequence $(x_n)_{n\in\mathbb{N}}\subseteq E$ which converges to $x\in E$ and such that the sequence…
Valent
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Consider $\mathbb{R}$ with the normal metric (d(x,y)=|x-y|). Exist an open cover of $[0,1] \cap \mathbb{Q} $ that haven't a finite subcover?

I don't think so, this is my argument: $[0,1] \cap \mathbb{Q} $ isn't closed, so it can't be compact. But I don't found an example.
bob
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Is $\mathbb{R}^{2}$ separable for the railmetric( or SNCF -metric or post-office metric)?

It's the follow metric: $d(x,y)= ||x|| +||y||$ if $x$ and $y$ don't lie on a line through the origin. And otherwise $d(x,y)= ||x-y||$. I think the answer is no, because I tried it with $\mathbb{Q}^{2}$ as countable and that didn't work. But I…
bob
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If $|f(x)| \leq Mx_0 sup_{y\in[0, x_0]} |f(y)|$ then why is f the zero-function?

For an analysis question I have to show that f is the zero-function on $[0,1]$. The inequality $|f(x)| \leq M x_0 sup_{y \in [0, xo]} |f(y)|$ is provided as well as the hint that M should be chosen such that $x_0 < \frac{1}{M}$. Doing this, $|f(x)|…
penguin131415
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