Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus...
19
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0answers
1k views
Undergraduate Research Topics in Analysis
I am thinking of doing some research (Real Analysis preferably) under a professor this year and I need ideas for interesting topics that might be accessible to an undergraduate.
Does anyone have any ...
10
votes
0answers
131 views
Abel limit theorem
I would like to know if the Abel limit theorem works if the limit is infinite.
Let the series $\sum_{k=0}^\infty a_k x^k$ have radius of convergence 1. Assume further that $\sum_{k=0}^\infty a_k = ...
10
votes
0answers
149 views
Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality
By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent.
Is is true that ...
9
votes
0answers
267 views
Osgood condition
Let $h$ and $g$ be continuous, non-decreasing and concave functions in the interval $[0,\infty)$ with $h(0)=g(0)=0$ and $h(x)>0$ and $g(x)>0$ for $x>0$ such that both satisfy the Osgood ...
9
votes
0answers
194 views
A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals
Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$.
We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
8
votes
0answers
124 views
n'th derivative does not vanish, but $\lim_{n\to \infty} f^{(n)}=0$.
Let $f\,$$\in$$\,C^\infty[\mathbb{R},\mathbb{R}]$ . Apparently the only functions $f$ for which there exists $n\in\mathbb{N}$ such that $f^{(n)}=0$ are polynomials in $\mathbb{R}[x]$.
Is it ...
7
votes
0answers
85 views
All nested partial sums of a sequence tend to $0$. Is the sequence constant?
$S^0:\mathbb N\to \Bbb R$ is a function. For any $m\in \Bbb N$, we define
$$S^m:\Bbb N\to \Bbb R$$
$$S^m(n)=\sum_{k=1}^n S^{m-1}(k)$$
For each $m\in \Bbb N$, we have:
$$\lim_{n \to \infty}S^m(n)=0$$
...
7
votes
0answers
165 views
Difference of differentiation under integral sign between Lebesgue and Riemann
Here is a consequence of Lebesgue dominated convergence theorem on differentiation under integral sign.
Function $f(x, t)$ is differentiable at $x_0$ for almost all $t \in A$, and $t \to f(x, t)$ ...
7
votes
0answers
154 views
Can all subseries of an infinite series be pairwise independent over $\mathbb{Q}$?
I'm wondering about a simple question that has multiple possible variants depending on a few parameters. The prototypical one would be:
Does there exist an infinite series such that any two ...
7
votes
0answers
167 views
Invariant functions on the space of finite sequences of reals
Let $S$ be a space of all finite sequences of real numbers (we don't endow it with metric or topology in general). Before asking the main question, some notation.
For each $\mathbf s\in S$ we define ...
6
votes
0answers
54 views
Asymptotic Expansion of an Oscillating Integral
Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$.
What is the leading order in $\lambda$ as $\lambda\rightarrow 0$ of ...
6
votes
0answers
133 views
Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?
I've written the question first, then the motivation behind it and lastly some background. Note that the question makes references to definitions and theorems written in the background bit at the end. ...
6
votes
0answers
61 views
Monotonic version of Weierstrass approximation theorem
Let $f\in\mathcal{C}^1([0,1])$ be an increasing function over $[0,1]$.
Prove or disprove the existence of a sequence of real polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ with the properties:
$p_n(x)$ ...
6
votes
0answers
96 views
Convergence of indicator functions in $L^2[0,1]$ when $m(\limsup(E_n)\setminus \liminf(E_n)) = 0$
I am trying to solve a qualifying exam problem.
I would like to know what can be said about the convergence of the indicator functions $I_{E_k}$ in $L^2[0, 1]$ when it's known that $m(\limsup ...
6
votes
0answers
218 views
an infinite series expansion in terms of the polylogarithm function
we have the complex valued function :
$$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$
we wish to recover the coefficients $a_{n}$ . the only thing i though ...
