https://math.codidact.com/categories/41/tags/5410.rssNew Posts Tagged 'real-analysis' - MathematicsMathematics - Codidact2024-02-10T19:26:14Zhttps://math.codidact.com/posts/290728Cyclical or “loop” fractals?Julius H.https://math.codidact.com/users/573372024-02-10T03:58:19Z2024-02-10T19:26:14Z<p>I want to ask an intuitively conceived question about “fractal loops”. However, I don’t know the mathematical definition of a fractal perfectly, off the top of my head.</p>
<p>Because you can “...https://math.codidact.com/posts/290642Fourier transform of an $L^1$ function is uniformly continuousSnoopyhttps://math.codidact.com/users/568172024-01-26T00:06:28Z2024-01-26T13:49:02Z<p>$\def\Rbb{\mathbf{R}}$$\def\Cbb{\mathbf{C}}$$\def\intw{\int_{\Rbb^n}}$If $f\in L^1(\Rbb^n)$, denote the Fourier transform of $f$ as
$$
\hat{f}(x) = \int_{\Rbb^n}f(t)e^{-2\pi x\cdot t}\ dt
$$<...https://math.codidact.com/posts/290599For any real number $m$, $ \left|\sum_{n=1}^{\infty}\frac{m}{n^2+m^2}\right|<\frac{\pi}{2} $Snoopyhttps://math.codidact.com/users/568172024-01-20T13:25:32Z2024-01-20T14:57:43Z<blockquote>
<p><strong>Problem.</strong> Prove that for any real number $m$,
$$
\left|\sum_{n=1}^{\infty}\frac{m}{n^2+m^2}\right|<\frac{\pi}{2}
$$</p>
</blockquote>
<hr>
<p><strong>Notes...https://math.codidact.com/posts/290567If both $\lim_{x\to\infty}f(x)$ and $\lim_{x\to\infty}f'(x)$ exist, then $\lim_{x\to\infty}f'(x)=0$.Snoopyhttps://math.codidact.com/users/568172024-01-13T22:56:11Z2024-01-17T12:52:50Z<p><strong>Question.</strong> Let $f:\mathbf{R}\to\mathbf{R}$ be a differentiable function. If both the limits $\displaystyle \lim_{x\to\infty}f(x)$ and $\displaystyle \lim_{x\to\infty}f'(x)$ exist...https://math.codidact.com/posts/290576The derivatives of a function at a boundary pointtommihttps://math.codidact.com/users/534072024-01-15T08:11:00Z2024-01-17T01:12:44Z<p>I have a function $f \colon [0, L[ \, \to \mathbb{R}$ and I want to use the derivatives of arbitrary high orders of this function at zero. The function is defined on the half-open interval $[0, ...https://math.codidact.com/posts/290570Using convexity in the proof of Hölder’s inequalitySnoopyhttps://math.codidact.com/users/568172024-01-14T15:38:57Z2024-01-14T15:42:11Z<p>A key fact for the algebra properties of $L^p$ spaces is <a href="https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality">Hölder’s inequality</a>:</p>
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<p>Let $f \in L^p$ and $g ...https://math.codidact.com/posts/290537What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1])$? Snoopyhttps://math.codidact.com/users/568172024-01-10T03:13:08Z2024-01-10T03:20:14Z<blockquote>
<p><strong>Question.</strong> What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1])$?</p>
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<p><strong>Notes.</strong> This is an example of questi...https://math.codidact.com/posts/290527$\liminf (a_n+b_n) = \liminf(a_n)+\liminf(b_n)$ provided that $\lim a_n$ existsSnoopyhttps://math.codidact.com/users/568172024-01-09T01:33:42Z2024-01-09T13:27:05Z<blockquote>
<p><strong>Question.</strong> Suppose $(a_n)$ and $(b_n)$ are two sequences of real numbers such that $\displaystyle \lim_{n\to\infty}a_n=a.$ Show that
$$
\liminf_{n\to\infty}(a_n+b...https://math.codidact.com/posts/290512$\sup(A\cdot B) = (\sup A)(\sup B)$ where $A$ and $B$ bounded sets of positive real numbersSnoopyhttps://math.codidact.com/users/568172024-01-07T20:48:55Z2024-01-09T01:00:26Z<blockquote>
<p><strong>Problem.</strong> Suppose $A$ and $B$ are two subsets of positive real numbers. In addition, assume that $A$ and $B$ are both bounded. Show that
$$ (\sup A)(\sup B) = \sup...https://math.codidact.com/posts/290508Why is $ \int_0^{\infty}\left|\frac{\sin x}{x}\right|\ dx=\infty$?Snoopyhttps://math.codidact.com/users/568172024-01-07T15:51:57Z2024-01-07T16:05:42Z<blockquote>
<p><strong>Question</strong>: Why is
$$
\int_0^{\infty}\left|\frac{\sin x}{x}\right|\ dx=\infty\quad ?
$$</p>
</blockquote>
<p>There are several other ways to state the fact in t...https://math.codidact.com/posts/289216Classification for involutory real infinite seriesnsajkohttps://math.codidact.com/users/577582023-08-03T17:09:38Z2023-08-18T10:07:47Z<p>Out of curiosity I'm playing around with the concept of involutory functions. An involutory function (<em>involution</em>) is a function whose composition with itself is the identity function (i...https://math.codidact.com/posts/288129Which other Real Analysis textbooks unusually recommend ending delta-epsilon proofs with a cluttered, bedecked $\epsilon$? TextKithttps://math.codidact.com/users/536962023-05-17T00:21:42Z2023-05-17T03:48:27Z<ol>
<li>
<p>Most textbooks conclude $\delta-\epsilon$ proofs tidily with $\epsilon > 0$ alone, as in red beneath. But what's the official term for this alternative $\delta-\epsilon$ proof, as...https://math.codidact.com/posts/286961Is $f(x)=\sin(x)$ the unique function satisfying $f'(0)=1$ and $f^{(n)}(\Bbb R)\subset [-1,1]$ for all $n=0,1,\ldots$?Grovehttps://math.codidact.com/users/534522022-09-06T15:22:23Z2023-01-25T08:14:28Z<blockquote>
<p><strong>Question.</strong> Is there a function $f:\Bbb R \to \Bbb R$ with $f'(0)=1$ and $f^{(n)}(x)\in [-1,1]$ for all $n=0,1,\ldots$ and $x\in \Bbb R$, other than $f(x)=\sin(x)$?<...https://math.codidact.com/posts/287762Is there a "regular" quasi-convex function $f:\Bbb R^2 \to \Bbb R$ that is not a monotone transformation of any convex function?Pavel Kocourekhttps://math.codidact.com/users/588182023-01-23T14:58:24Z2023-01-23T14:58:24Z<h3>Question</h3>
<blockquote>
<p>Can you find an example of a differentiable quasi-convex function $f:\Bbb R^2 \to \Bbb R$ that is <em>non-degenerate</em>, but there does not exist any strictly...https://math.codidact.com/posts/287756Is there a two variable quartic polynomial with two strict local minima and no other critical point?Pavel Kocourekhttps://math.codidact.com/users/588182023-01-21T15:22:38Z2023-01-21T15:22:38Z<blockquote>
<p>Does there exist a degree 4 polynomial $p:\Bbb R^2 \to \Bbb R$ that has two strict local minima and no other critical point?</p>
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<p>This is the same as <a href="htt...https://math.codidact.com/posts/287492Example of $f:[0,1]\to\mathbf{R}$ with $\lim_{a\to 0^+}\int_a^1f(x)dx=L $ for some real number $L$ but $\int_0^1|f(x)|dx=\infty $Snoopyhttps://math.codidact.com/users/568172022-11-27T19:08:16Z2022-11-27T23:35:05Z<p>In the Wikipedia article on <a href="https://en.wikipedia.org/wiki/Improper_integral#Types_of_integrals">improper integrals</a>, the function $f(x)=\frac{\sin x}{x}$ gives an example that is imp...https://math.codidact.com/posts/287484Finding the limit $ \lim_{x\to 0^+}e^{1/x}\sum_{n=\lfloor 1/x\rfloor}^\infty\frac{x^n}{n} $Snoopyhttps://math.codidact.com/users/568172022-11-27T01:26:33Z2022-11-27T19:19:42Z<blockquote>
<p>Let $\lfloor x \rfloor$ be the maximum integer $n\le x$. Find the limit
$$
\lim_{x\to 0^+}e^{1/x}\sum_{n=\lfloor 1/x\rfloor}^\infty\frac{x^n}{n}
$$</p>
</blockquote>
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<p>...https://math.codidact.com/posts/285819Is it impossible to prove Jensen's inequality by way of step functions?whybecausehttps://math.codidact.com/users/550222022-02-04T01:03:31Z2022-02-11T21:32:00Z<p>Jensen's Inequality: Let $\varphi:\Bbb R\to \Bbb R$ be convex, and $f:[0,1]\to\Bbb R$ be integrable, and suppose $\varphi\circ f$ is integrable over [0,1]. Then
$$ \varphi\left(\int_{[0,1]} f\...https://math.codidact.com/posts/285925"Pointwise equicontinuity implies uniform" implies compactwhybecausehttps://math.codidact.com/users/550222022-02-11T05:02:37Z2022-02-11T20:05:08Z<p>Suppose $(M,d)$ is a metric space such that every sequence $f_n:M\to \Bbb R$ which is pointwise equicontinuous is also uniformly equicontinuous. Does this imply that $M$ is a compact metric spa...https://math.codidact.com/posts/285756$\{f_n\}\to f$ in $X$ implies $||f||\le\liminf||f_n||$ whybecausehttps://math.codidact.com/users/550222022-01-31T02:27:52Z2022-01-31T16:28:15Z<p>Let $X$ be a normed linear space and suppose that, for each $f\in X$ there exists a bounded linear functional $ T\in X^* $ such that $T(f)=||f||$ and $||T||_*=1$. Prove that if $\{f_n\}\to f$ in...https://math.codidact.com/posts/285424$\int_{E_n} |g|^q = \left| \int_E \chi_{E_n}\cdot \text{sgn}(g)\cdot g \cdot |g|^{q-1}\cdot |g| \right|$ whybecausehttps://math.codidact.com/users/550222021-12-31T01:18:40Z2021-12-31T02:13:44Z<p>I am trying to understand why the following equation is true. Here $E$ is a measurable set and all functions are defined and measurable on it. $1<p,q,<\infty$ such that $\frac 1 p+\frac ...https://math.codidact.com/posts/282681Find the intervals on which it is increasing and those on which it is decreasing of the following function.deleted user#2021-07-14T09:36:43Z2021-07-14T10:44:36Z<blockquote>
<p>Find the intervals on which it is increasing and those on which it is decreasing of the following function. $$f(x)=x^3-9x^2+24x-12,0\leq x\leq 6$$</p>
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<p>After diff...https://math.codidact.com/posts/282631Rules of checking differentiability for Rolle's theoremdeleted user#2021-07-10T05:48:27Z2021-07-10T06:06:59Z<p>I was doing some exercises of Rolle's theorem. But, they didn't check the differentiability the way we checked differentiability normally. I am giving some examples.</p>
<p>When I was checking ...https://math.codidact.com/posts/282623Second mean value theorem proof (differentiation)deleted user#2021-07-09T10:37:02Z2021-07-09T15:45:39Z<blockquote>
<p>Since $F(x)$ is continuous in the closed interval [a,a+h] and differentiable in the open interval (a,a+h). Also f(a) = f(b) so by Rolle's theorem we get</p>
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<p>$$F'...