Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.
0
votes
0answers
19 views
Triangle inequality: $\vert\vert a+b\vert^q-\vert a\vert^q\vert\leq \varepsilon\vert a\vert^q+C(\varepsilon)\vert b\vert^q$
I am having difficulty in proving the following inequality for $1\leq q<\infty$:
\begin{equation}
\Big|\vert a+b\vert^q-\vert a\vert^q\Big|\leq \varepsilon\vert a\vert^q+C(\varepsilon)\vert ...
2
votes
0answers
21 views
Discontinuous for rationals
Show that $f\left(x\right):=\sum_{n=1}^{\infty}\frac{\left\{nx\right\}}{n^2}$, where $\left\{nx\right\}$ is the fractional part of $nx$, is discontinuous for all rationals.
I guess it would be nice ...
1
vote
0answers
14 views
Importance of Riemann-Liouville fractional derivative from historical point of view
Why Riemann-Liouville fractional derivative is important from historical point of view than that of Caputo fractional derivative? As we know Riemann-Liouville fractional derivative is more theoretical ...
0
votes
0answers
17 views
Fractional linear transformations with given properties
I need a function of the form
$\displaystyle f(z):= \frac{az+b}{cz+d}, \qquad z\in\mathbb{C}-\{-\frac{d}{c}\}, \qquad ad-bc\neq0$
which carries the half-plane $\{z\in\mathbb{C}\ |\; ...
1
vote
1answer
25 views
Derivative of a little-o remainder
If we have a $\phi: \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$, $\phi = \phi(t, \mathbf{q},\alpha)$ one-parameter group of infinitesimal transformation which is $\mathcal{C}^2$ ...
1
vote
1answer
41 views
Why is the graph of a continuous function to a Hausdorff space closed?
Say I have two top. spaces given by $(X,\mathscr{T}_x)$ and $(Y,\mathscr{T}_Y)$ where $Y$ is Hausdorff. In addition say I have a functon $f:X\rightarrow Y$, and let it be continuous. I want to show ...
-2
votes
0answers
47 views
Trigonometric Approximation [closed]
Let f be a continuous function on $[-\pi,\pi]$ such that $f(-\pi)=f(\pi)$. The trigonometric approximation theorem asserts that for given $\varepsilon >0$, there exists a trigonometric polynomial ...
1
vote
2answers
35 views
product rule for matrix functions?
Given a real rectangular matrix $X$, and two scalar-valued matrix functions, $f(X)$ and $g(X)$, does the product rule for differentiation of a product of scalar valued functions, hold when ...
1
vote
3answers
62 views
A condition that balls have finite measure
Let $(X,d)$ be a metric space and let $\mu$ be a positive measure on $X$. I want to require that $(X,d)$ and $\mu$ have either of the following properties:
$\forall y \in X$, $\forall r \geq 0$, ...
4
votes
2answers
79 views
Find all continuous functions (did I do this correctly?)
Find all the continuous functions $f:\mathbb{R}\to\mathbb{R}$ satisfying:
$f(x+y)=f(x)+f(y)+f(x)f(y)$ for all $x,y\in\mathbb{R}$
Solution attempt:
$$\begin{align*}
f(x+y) + 1 &= f(x) + ...
1
vote
2answers
37 views
Necessary and Sufficient Condition for two metrics to have same open sets.
There are couple of independent conditions like one being scalar multiple of another, or if $$d_p(x,y)=(x^p+y^p)^{1/p}$$ then all $d_ps$ and $d_qs.$ which guarantee that open sets are same under these ...
3
votes
1answer
17 views
Calculating $d\omega$ for $\omega\in\Omega^{k}M$ explicitly for $k=2$
I am trying to explicitly calculate the exterior derivative $d\omega$ for $\omega\in\Omega^{2}M$ for a differentiable oriented manifold $M$.
I know that we can express a differential $k$-form ...
5
votes
3answers
69 views
Intermediate value-like theorem for $\mathbb{C}$?
Is there an intermediate value like theorem for $\mathbb{C}$? I know $\mathbb {C}$ isn't ordered, but if we have a function $f:\mathbb{C}\to\mathbb{C}$ that's continuous, what can we conclude about ...
1
vote
0answers
39 views
Part of proof 11.10 in Rudin's Principles of Mathematical Analysis
There is a part of proof 11.10 that I don't get in Rudin's Principles of Mathematical Analysis (3rd edition).
The whole theorem is the following two statements:
$\mathcal{M}\left(\mu\right)$ is a ...
3
votes
2answers
48 views
Showing a function is not one-to-one near the origin
Let $$f(x)=\begin{cases} x+2x^2\sin\left(\frac{1}{x}\right) \text{ if } x \neq 0 \\ 0 \text{ if } x=0 \end{cases}$$
I'm trying to show this is not one-to-one near $0$. I was given a hint to consider ...
