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).

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67 views

Daniell approach to $\int \sigma \mbox{ } d\mu \geq 0$ clearification

Can someone help me clarify the following proof. Theorem: If $\sigma \in C_0$ and $\sigma \geq 0$ a.e.$(\mu)$, then $\int \sigma \mbox{ } d\mu \geq 0$ Proof: The inequality is equivalent to proving ...
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1answer
64 views

det function in concave

Let $f(A)=(\det(A))^{\frac{1}{n}}$. And assume domain of $f$ is space of positive semi definite symmetric $n\times n$ matrices with real entries. Show that $f$ is concave: $$f((1-t)A+tB)) \ge ...
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42 views

Why is the partial derivative of this fuction locally bounded?

We have a function for $x_i, t_i >0$ $$|f(x_1,t_1)-f(x_0,t_0)| \leq C (|t_1-t_0|^{1/2} + |x_1-x_0|)$$ Why does this mean $f_t$ is locally bounded? $f$ is non-increasing and convex in $x$. $f$ is ...
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1answer
337 views

How to integrate over an arbitrarily positioned spherical cap in spherical coordinates

If you want to integrate over the SURFACE of a spherical cap that is positioned in the way it is on wikipedia, this is rather simple. since it has azimuthal symmetry you get a factor $2\pi$ and for ...
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2answers
131 views

Question about a particular linear operator

Let A be a linear operator. $A: L^2(0,1) \rightarrow L^2(0,1)$ given by $Ag(a) = \int_0^a(a-x)g(x)dx$ where $a \in (0,1)$. This is the integral operator, and we know ||A|| < 1 which is easy to ...
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2answers
222 views

Is there any value of zeta that is an integer?

Is there any value which we can substitute for $s$ in $\zeta (s)$ such that $$\sum_{n=1}^{\infty }n^{-s}\in \mathbb{Z}$$
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1answer
346 views

Show that the Fourier transform of a radial function $ L^1 (\mathbb{R}) $ is also radial

How do I prove that the Fourier transform of a radial function $ f \in L^1 (\mathbb{R}) $ is also radial function? I tried by polar coordinates but I dont got.
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1answer
140 views

Calculus prequisites book

Everytime I try read a calculus textbook I find that my books (serge lang and gelfand's )didn't cover a subject well (like say minimum of a quadratic polynomial) ...I need a recommendation for a ...
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2answers
215 views

Do integral with Legendre Polynomials

Is it possible to integrate this analytically: $$ \int_{0}^{2\pi} P_l(\cos(\theta-\theta')) P_l(\cos(\theta)) \sin(\theta) d\theta$$ I mean the integral would be pretty easy by using the ...
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2answers
114 views

Is the function of two strictly concave functions also concave?

This may be a trivial question to most, but here we go: I have two strictly concave functions, say $f(x)$ and $g(x)$. From this can I say that a function of those two functions, $h[f(x), g(x)]$, is ...
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2answers
103 views

Is $H_0^1([a,b]) \subset C([a,b],\mathbb{R})$?

i have a small question : how to see that $H_0^1([a,b])\subset C([a,b],\mathbb{R})$? Please Thank you
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1answer
146 views

Why they use the word “abstract” for naming mathematical fields of study?

I've found some books with titles such as abstract analysis - but I don't understand why they choose such word. For me it seems quite vague and perhaps misleading - consider the examples: Real ...
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1answer
350 views

Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)

I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out. 1 . I am not clear, why in step (1)'s proof he says that from unique ...
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1answer
77 views

It Suffices to Check Mixing on an Algebra

Let $X, \mathcal{A}, \mu$ be a probability space and $T: X \rightarrow X$ a measure preserving measurable map (i.e. $\mu (T^{-1}(A)) = \mu (A)$ for all $A \in \mathcal{A}$). We say $T$ is mixing for ...
5
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1answer
320 views

An unbounded continuous function on $(0,1)$ that is in $L^p(0,1)$ for $1\le p <\infty$

So this is a question on an old qualifying exam I was going over. Give an example of a function $g$ such that $g$ is continuous and unbounded on $(0,1)$ and that $g \in L^p(0,1) $ for $1 \le p < ...
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1answer
63 views

Properties about the Generalized Legendre-type polynomial sequence $f_n(x)=\frac{d^n}{dx^n}(p(x)^n)$

Suppose we have the Generalized Legendre-type polynomial sequence $f_n(x)=\dfrac{d^n}{dx^n}(p(x)^n)$ , where $p(x)$ is any polynomials of degree at least $2$ and has at least two terms. $1.$ Do these ...
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1answer
139 views

On the absolute integrability of radially symmetric functions

Let $\phi:\mathbb R\to\mathbb R$ be an smooth, even function and $\int_\mathbb R|\phi(t)|^p\,\mathrm dt<\infty$, that is, $\phi$ is pth-power integrable in $\mathbb R$ iff $p\geq p_0$ for ...
2
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1answer
107 views

Defining a Chebyshev series expansion

I'm trying to implement the Clenshaw algorithm for a truncated Chebyshev series. I think I've grasped the algorithm itself, but I'm a bit confused by an additional term in the definition. I have ...
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1answer
97 views

Does $\lim_{n\to\infty}\left|\sin n\right|^\frac1n$ where $n\in\mathbb Z^+$ exist?

Does $\lim_{n\to\infty}\left|\sin n\right|^\frac1n$ where $n\in\mathbb Z^+$ exist? How can I determine this using freshman calculus?
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64 views

Properties about the Generalized Hermite-type polynomial sequence $f_n(x)=e^{-p(x)}\frac{d^n}{dx^n}e^{p(x)}$

Suppose we have the Generalized Hermite-type polynomial sequences $f_n(x)=e^{-p(x)}\dfrac{d^n}{dx^n}e^{p(x)}$ , where $p(x)$ is any polynomials of degree at least $2$ . $1.$ Do these polynomial ...
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1answer
105 views

Example of the equality of an inequality

This question is related to Daniel Fischer's answer here. Suppose $f$ is a real $C^{1}$ function on $[0, 1]$ such that $f(0) = 0$ and $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$. Then (essentially by ...
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1answer
133 views

Result of a corollary of Bolzano-Weierstrass theorem.

This is a problem of the book Flemming-Functions of several variables Let A be a closed, convex, nonempty set, and $ \vec{x}_{0} \notin A. $ Show that there is exactly one point $ \vec{x}_{1} \in A ...
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1answer
89 views

Limits and continuity.

I need help with these exercises of Analysis about limits and continuity. Construct a set $ A \subset [0,1] \times [0,1]$ such that $A$ has at most one point en each horizontal line and one in each ...
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2answers
100 views

Does this hold for three numbers [duplicate]

If $a\ge b\ge c\ge0$, does it hold that $\sqrt[3]{\left(a-b+c\right)^{2}}\ge\sqrt[3]{a^{2}}-\sqrt[3]{b^{2}}+\sqrt[3]{c^{2}}$? Thanks for any help.
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3answers
332 views

Mathematical Analysis advice

Claim: Let $\delta>0, n\in N. $ Then $\lim_{n\rightarrow\infty} I_{n} $exists, where $ I_{n}=\int_{0}^{\delta} \frac{\sin\ nx}{x} dx $ Proof: $f(x) =\frac{\sin\ nx}{x}$ has a removable ...
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1answer
23 views

Approximation of Delatadistribution

I'm trying to understand a computation in my physics script. To describe the Deltadistribution $\delta(x) $ correctly we would need the formalism of distributions, but one can also much less ...
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1answer
396 views

about addition and multiplications of real number

We all know that for every two real number $x,y$, the operation of addition $x+y$ satisfies the following conditions: \begin{gather} x+y=y+x;\\ x+0=x;\\ (x+y)+z=x+(y+z);\\ x+(-x)=0. \end{gather} ...
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1answer
170 views

Find the limit sup and limit inf of a given sequence of sets

Suppose we have a set $X_b = \{\frac{a}{b}:a \in \mathbb{Z^{+}}\} $ where $b \in \mathbb{Z^{+}}$. We want to find $\lim_{b \to +\infty} \inf{X_b}$ and also find find $\lim_{b \to +\infty} \sup{X_b}$. ...
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1answer
299 views

Proof involving gamma function, infinite product and Gauss

How can I rigorously and directly prove that $$\Gamma (z)=\lim_{n\rightarrow \infty }\frac{n!n^{z}}{z(z+1)\cdots(z+n)}$$
4
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1answer
87 views

If $a\ge b\ge-c\ge0$, is $\sqrt[3]{a-b-c}\ge\sqrt[3]{a}-\sqrt[3]{b}-\sqrt[3]{c}$?

Let $a\ge b\ge-c\ge0$. Is it true that $\sqrt[3]{a-b-c}\ge\sqrt[3]{a}-\sqrt[3]{b}-\sqrt[3]{c}$?
3
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1answer
160 views

Convergence of the Zeta and Phi functions

I want to show that the following functions (in the picture) are absolutely and locally uniformly convergent if real part of complex number $s$ is bigger than 1. Absolute part for zeta function is ...
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2answers
36 views

Why can't a $C^1$-class mapping with nonzero derivative fill a square? [duplicate]

Let $f: [0,1] \rightarrow \mathbb R^2$ be of class $C^1$ with $f'(t)\neq (0,0)$. Why can't $f$ be a Peano-type curve, i.e. $f(I) \neq I\times I$?
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2answers
146 views

Showing a series is not uniformly convergence

Suppose you want to show a series does not converge uniformly on some interval. If you know the point wise limit is $f$, and you can show the $\sup |f_{n} - f|$ does not go to zero on your interval, ...
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4answers
174 views

Another version of the variable change

Let $f:[a,b] \to \mathbb{R}$ integrable and $g:[c,d] \to \mathbb{R}$ a monotonic function with $g'$ integrable. If $g([[c,d]) \subset [a,b]$, show that $\displaystyle \int_{g(c)}^{g(d)} f(x)dx = ...
5
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2answers
234 views

Prove that g(0)=0?

(taken from Spivak's calculus page 281) Suppose that $f$ and $g$ are differentiable functions satisfying $$ \int_{0}^{f(x)} (fg)(t) \, \mathrm{d}t=g(f(x))$$ Prove that $g(0)=0$. Now if ...
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2answers
447 views

Convergent Sequence and Cauchy Criterion- Counter Example

Consider the sequence $\left \{ x_{n} \right \}$ that satisfies the condition: $$\left | x_{n+1}-x_{n} \right |< \frac{1}{2^{n}} \ \ \ for\ all\ n=1,2,3,...$$ Part (1): Prove that the sequence ...
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1answer
472 views

Problem on using the definition of Riemann Integral

Consider the function: $f:[a,b]\rightarrow \mathbb{R}$ be a bounded function on $[a,b]$. Suppose that $f$ is Riemann integrable on $[a,c]$ for any $c\in (a,b)$. The question is to prove that $f$ is ...
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1answer
95 views

Problem involving Properties of continuous functions

I am given two real functions $f$ and $g$ that are continuous on the interval $[a,b]$ such that for all $x$ in $[a,b]$, we have: $f(x)< g(x)$. The question is to prove that there exists a number ...
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1answer
96 views

How can every $p$-adic integer be the limit of a sequence of non-negative integers?

See Andrew Baker's P-adic Notes. Every element of $\mathbb{Z}_p = \{a \in \mathbb{Q}_p : |a|_p \leq 1 \}$ is a limit of a sequence of non-negative integers, with respect to the $|\cdot|_p$ norm. How ...
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62 views

Nonnegative series converges implies terms decay exponentially?

Let $\{a_n\}$ be a sequence with nonnegative terms ($a_n\geq 0$). If $\sum_{n=0}^\infty a_n < \infty$, does this imply that there exists $\alpha<1$ such that $a_n \leq \alpha^n$ for all but ...
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1answer
38 views

Properties of some family of subsets

Let $X$ be a linear space (without topology), over field of real or complex numbers. Let $B$ be a family of subsets of $X$ satisfying conditions: sets from $B$ are balansed and absorbing (in ...
3
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1answer
85 views

Derivative at $0$ of $\int_0^x \sin \frac{1}{t} dt$

Let $f(x)=\int_0^x \sin \frac{1}{t} dt \textrm{ for } x \in \mathbb R$. Is $f$ differentiable at $0$ ?
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1answer
351 views

Recognizing uppersemicontinuous function as a pointwise decreasing limit.

Let $X$ be a compact metric space and $f:X\rightarrow \mathbb{R}$ be upper semicontinuous. Then why is it that $f$ is the pointwise decreasing limit of continuous functions? My attempt has been to ...
2
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1answer
68 views

Change of coordinates and derivatives in $\mathbb{R}^n$ on a boundary integral

I'm am slightly confused while trying to keep everything straight between looking at integration as on a manifold vs. the diffeomorphism change of variables. Consider a smooth domain $B\subset ...
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1answer
181 views

Question about Titchmarsh's proof of the Vitali Convergence Theorem

Consider the following version of the Vitali Convergence Theorem presented in Titchmarsh's Theory of Functions: Let $f_{n}(z)$ be a sequence of functions, each regular in a region $D$; let ...
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2answers
38 views

Show that $f:I\to\mathbb{R}^{n^2}$ defined by $f(t)=X(t)^k$ is differentiable

Let $I$ be a interval, $\mathbb{R}^{n^2}$ be the set of all $n\times n$ matrices and $X:I \to\mathbb{R}^{n^2}$ be a differentiable function. Given $k\in\mathbb{N}$, define $f:I\to\mathbb{R}^{n^2}$ by ...
3
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1answer
415 views

Essential Supremum vs. Uniform norm

I just went to check something about the $||\cdot||_\infty$ norm and realized that it can perhaps refer to two quite different things. I'm coming at this from an Analysis class so I am use to having ...
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1answer
97 views

Double Integral: $f(x,y)=x$ if $x=y$, and $f(x,y)=0$ otherwise

Let $\Omega = [0,1] \times [0,1]$. Let $f \colon \Omega \to \mathbb{R}$ be $f(x,y)=x$ if $x=y$, and $f(x,y)=0$ otherwise. I would like to show the integral exists or not using the criterion of Riemann ...
0
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1answer
51 views

In a compact subset of a metric space, given an open cover. (test prep)

Let $(X,d)$ be a compact metric space and suppose that $\Omega$ is a compact subset of $X$ and $U$ is an open cover of $\Omega$. Prove that there exists a $\delta$ such that, for every $\omega \in ...
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0answers
489 views

Derivative Riccati-Bessel function

I have found two derivatives of the so-called Riccati-Bessel functions in a textbook $$ (x j_n(x))'=xj_{n-1}(x)-nj_{n}(x)$$ and $$ (x h_n^{(1)}(x))'=x h_{n-1}^{(1)}(x)-n h_n^{(1)}(x)$$ so $j_n$ is ...