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).
44,438 questions
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Does uniformly continuous functions apply to something like "sandwich theorem"?
Suppose $f,g$ are two uniformly continuous functions on $\mathbb R$, and $h$ is a continuous function on $\mathbb R$ that satisfies:$$f(x)\le h(x) \le g(x)$$Does that mean $h$ is a uniformly ...
- 1
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Formal Justification of Alternating Harmonic Series: $1-\frac{1}{2}+\frac{1}{3} - \cdots = \ln(2)$ [duplicate]
It's well known that $\sum_{n=1}^\infty \frac{(-1)^n}{n}=\ln(2)$, which can most easily be seen by the following derivation:
$$\frac{1}{1-x} = \sum_{n=0}^\infty x^n \tag{Geometric Series}$$
$$\frac{1}{...
- 4,992
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Find the best approximant in $L^1$ space to the $0$ function in $[1,-1]$ with weight $w(x) = \frac{1}{\sqrt{1-x^2}}$ [closed]
How can I prove that the best approximant in $L^1$ space to the $0$ function in $[1,-1]$ with weight $w(x) = \frac{1}{\sqrt{1-x^2}}$ is $T_n(x)$, the Chebyshev polynomials of the first kind?
I know ...
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Representation of this indicator function.
Let $(S,\Sigma)$ be a measurable space, $f_1,\cdots,f_n:S\to\Bbb R$ be measurable functions, and $f:=(f_1,\cdots,f_n)$. For $A\in \mathcal F:=\{f^{-1}(B)\mid B\in\mathcal B(\Bbb R^n)\}$, I want to ...
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The intuition behind defining trigonometric functions in the complex plane as special combinations of exponential functions [closed]
I was studying Complex Analysis from "A First Course of Complex Analysis" and the authors stated directly that sine and cosine are defined as follows (without any intuition):
$$ \sin\left(z\...
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On computing the infimum of a functional [closed]
Let $P, Q \in \mathbb{R} \to \mathbb{R}$ be monic polynomials with real coefficients such that $\deg(Q) > \deg(P)$, $\min P > 0$, $\min Q > 0$. For $a<b$ and $A,B\in\mathbb{R}$, define
$$ ...
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upper bound of the modulus of the complex-valued function $F(z)=\frac{1+i\theta(z)}{1-i\theta(z)}$
This little problem arises in a proof of a generalization of the Hermite-Biehler theorem. Let
$$
F(z):=\frac{1+i\theta(z)}{1-i\theta(z)},
$$
where $\theta(z)$ maps the upper half-plane onto the upper ...
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Control of an ODE to attain target
Consider the following ODE system:
$$
\begin{aligned}
dR_1(t) &= -\lambda_1 R_1(t)\,dt + \lambda_1 \left( \beta_0 - \beta_1 R_1(t) + \beta_2 R_2(t) \right) C\,dt, \\
dR_2(t) &= -\lambda_2 R_2(...
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measurability definitions in time-dependent Lebesgue spaces
I have some questions regarding the notion of measurability for functions $f : (0,T) \to L^q(\mathbb{R})$ that belong to spaces like $L^p(0,T; L^q(\mathbb{R}))$.
How is measurability of such a ...
- 1,371
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$p$-adic completeness of convergent sequences in $\mathbb{Z}_p$
I feel really dumb asking this question since it feels really elementary, but I am having problems showing that the following $\mathbb{Z}_p$-submodules of $\prod_{\mathbb{N}} \mathbb{Z}_p$ are $p$-...
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Strassman's Theorem: why is this series equal to the first term? In module
I'm studying $p$-adic numbers. More precisely, the proof of Strassman's Theorem: Let $$f(X) = \sum_{n=0}^\infty a_n X^n \in \mathbb{Q}_p[[X]]$$
a nonzero power series (not all coefficients are zero). ...
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lower bound for a messy rational function
Suppose that $\tau>0$ and $r>0$ are real parameters, and define $\alpha:\mathbb{R}^{+}\mapsto\mathbb{R}$ to be
$$
\alpha(\omega):=\frac
{[(\omega^2+r)\cos^2(\omega\tau)+\frac{\omega}{2}(1-r)\sin(...
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What exactly is a infinitesimal generator?
I am reading a book on symmetry methods for differential equations and trying to understand what exactly the infinitesimal generator is?
(And what it describes geometrically, if it describes ...
- 1,209
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A sub-collection of a sub-basis to a topology generates the same topology - proof assistance
Background and context
This is once again another follow-up question to this question I asked yesterday. Attempting to understand the concept of sub-basis to topologies and the weak topology induced ...
- 113
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Why do we care about invariant solutions of differential equations (under a one parameter Lie group)?
I am reading about symmetry methods on differential equations.
It starts by considering invariant points, then orbits of points and invariant curves. It ends in a method to determine all (w.r.t Lie ...
- 1,209
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How to evaluate the integral $ \int_{0}^{1}\int_{0}^{1}\frac{\ln(xy)}{x+y}\,\mathrm{d}x\,\mathrm{d}y $?
How can one evaluate the integral
$$
\int_{0}^{1}\int_{0}^{1}\frac{\ln(xy)}{x+y}\,\mathrm{d}x\,\mathrm{d}y\,?
$$
I used WolframAlpha to compute
$$
-\int_{0}^{1}\int_{0}^{1}\frac{\ln(xy)}{x+y}\,\mathrm{...
- 373
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Proving span of Hilbert space subset is dense iff orthogonal complement is trivial with sequences [duplicate]
Let $M$ be a nonempty subset of a Hilbert space $H$. The span of $M$ is dense in $H$ if and only if $M^\perp=\{0\}$.
I know how the proof goes for both directions. For the forward direction, the usual ...
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Ergodicity in the Wiener-Wintner Ergodic Theorem
I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the ...
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Proving every open cover having finite subcover entails sequential compactness
I am trying to prove that if it is true for a set $K$ that every open cover of it has some finite subcovering, then $K$ is sequentially compact and am looking to verify steps that I'm unsure about in ...
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Borel Functional Calculus for non-unital commutative $C^*$ Algebra
Suppose $A$ be a commutative unital $C^*$ algebra acting on a Hilbert space $H$. We know that $A$ is isometrically isomorphic to $C(X)$ for a compact Hausdorff space $X$, to be more precise $X$ is the ...
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Test functions are zero on boundary of bounded set
Test functions in $C_0^\infty(\Omega) = \{ \varphi:\Omega \to \mathbb{R}\ |\ \varphi \in C^\infty(\Omega) \text{ and } supp(\varphi) \text{ is a compact subset of } \Omega\}$, where $\Omega \subseteq ...
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A Problem on the Mean Value Theorem for Improper Integrals
Assume that $ \int_{1}^{+\infty} f(x) dx $ converges (as improper Riemann integral). Does there necessarily exist some $ \xi \in (1, +\infty) $ such that
$$
\int_{1}^{+\infty} \frac{f(x)}{x} dx = \...
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Condition on Measure for Continuity of Integral
Consider the integral $I(x) = \int_{\mathbb{R}^b}f(x,y)d\mu(y)$ with $x \in \mathbb{R}^a$ and $\mu$ a Borel measure on $\mathbb{R}^b$. Assume all of the following :
$f: \mathbb{R}^a \times \mathbb{R}^...
- 3,684
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Constructing a multidimensional partition of unity from a 1D partition of unity on a uniform grid
Let $\phi: \mathbb{R} \rightarrow \mathbb{R}$ be:
A smooth non-negative function compactly supported on $(-1, 1)$
Identically equal to 1 on $(-\frac12, \frac12)$
$\sum_{n \in \mathbb{Z}} \phi^2(x - n)...
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Hahn Banach extension in finite dimension: explicit construction
Suppose $u$ is a nonzero vector in a finite-dimensional vector space. Let
$x': \langle u\rangle\to \mathbb{R}$ a linear function from the span of $u$ to the real numbers. I want to explicitly ...
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