All Questions
Tagged with real-analysis reference-request
419 questions
16
votes
6
answers
2k
views
Alternative proofs sought after for a certain identity
Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so
QUESTION. can you provide another verification for the problem below?
Problem. Prove ...
- 43.2k
0
votes
0
answers
48
views
definition of functions that "weakly vanishes as $y\to\infty$" and find a proof of Theorem 9 in the article
I'm reading "Extension problem and fractional operators: semigroups and wave equations" by P.R. Stinga, i have two questions:
in Theorem 7 the author use the state "weakly vanishes as $...
- 339
0
votes
1
answer
106
views
Existence of uniform approximator that also approximates derivative
Let $S$ be a subset of $C^1([0, 1], \mathbb{R})$. It is a well-known fact that given a function $f\in C^1([0, 1], \mathbb{R})$ and a sequence $\{f_n\}\subset C^1([0,1], \mathbb{R})$ such that $f_n\to ...
- 119
2
votes
0
answers
86
views
Eigenvalues of the operator $A = -v'' + B(x) v$
How can I prove that for the eigenvalues of the operator $$A := -v'' + B(x) v$$ on $(0,L)$ with zero Dirichlet boundary condition it holds that
$$
\left| \lambda_n - \frac{\pi^2n^2}{L^2}\right| \le ||...
- 217
6
votes
2
answers
775
views
Upper bounds for Bessel functions
Cosider the K-Bessel function $$K_\nu(x):= \frac\pi 2 \frac{I_{-\nu}(x)-I_\nu(x)}{\sin(\nu\pi)}.$$
See also Watson, G. N., A treatise on the theory of Bessel functions., Cambridge: University Press, ...
- 63
2
votes
0
answers
141
views
For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc
Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying
$$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$
where $g_0$, $...
- 969
1
vote
0
answers
42
views
Energy estimate for $\theta_t + H(\theta)_x = 0$ in $t>0, x >0$?
Consider the IBVP for $$\theta_t + H(\theta)_x = 0, \qquad t>0, \ x>0$$ with $$H(\theta) = \frac{1}{\pi} \text{pv}\int_{0}^\infty \frac{\theta(y)}{y-x} dy$$
with Dirichlet boundary conditions. ...
- 161
4
votes
2
answers
197
views
Symmetry of one-sided partial derivatives
Consider some $f: [0,1)\times [0,1)\to \mathbb{R}$. I'm interested in conditions that guarantee that the following one-sided second partial derivatives at $(x,y)=(0,0)$ are symmetric:
$$
\partial_x^+ ...
- 695
3
votes
1
answer
156
views
Inequality for generalized Laguerre polynomials
Please. Does anybody know a proof of this inequality
$$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and
$L^{\alpha}_n$ ...
- 81
1
vote
1
answer
277
views
Checking the uniform denseness of a set in $C([0, 1], \mathbb{R}^2)$
Let $\lambda:[0, 1]\to \mathbb{R}$, and $b_{1j}, b_{2j}:[0, 1] \to \mathbb{R}$, $j = 1, \ldots, m$ be smooth functions. Consider the following two sets
$$\begin{align*}
S_1 &= \left\{ \begin{...
- 119
2
votes
1
answer
246
views
Eigenfunctions of the fractional Laplacian are smooth?
Let $\Omega\subset\mathbb{R}^n$ open, bounded with smooth boundary, let $s\in(0,1)$. I know that the fractional Laplacian has a sequence of eigenfunctions $\{e_k\}_{k\in\mathbb{N}}\subset H^s(\mathbb{...
- 339
1
vote
0
answers
55
views
Projection of a real analytic manifold onto subspace is union of real analytic submanifolds
Let $M$ be a compact connected real analytic submanifold of the Euclidean space $\mathbb{R}^{n} \times \mathbb{R}$ and denote by $\pi : \mathbb{R}^{n} \times \mathbb{R} \rightarrow \mathbb{R}^{n}$ the ...
- 19
0
votes
1
answer
86
views
Kolmogorov entropy of a subset of $L^1$
How can we estimate the Kolmogorov $\epsilon$-entropy
$$H_\epsilon (A,L^1(\mathbb R))$$
where
$
A = \{f:\mathbb R \to [0,K] \text{ s.t. $f \in L^1$ and has total variation $TV(f) \le M$}\}
$?
- 839
3
votes
1
answer
246
views
Stone-Weierstrass theorem for modules of non-self-adjoint subalgebras
In "Weierstrass-Stone, the Theorem" by Joao Prolla, there is a Stone-Weierstrass theorem for modules, stated as the following:
Let $\mathcal{A}$ be a subalegebra of $C(X, \mathbb{R})$ and $...
- 119
1
vote
0
answers
151
views
Fourier transforms exhibiting symmetries about their critical points
Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
- 113
4
votes
0
answers
84
views
Elementary functions (growing faster than exponential ones) with elementary Legendre–Fenchel transforms
Let $F$ be the set of all convex functions $f\colon[0,\infty)\to[0,\infty)$ with $f(0)=0=f'_+(0)$ and $f_+(\infty-)=\infty$, where $f'_+$ is the right derivative of $f$. For any function $f\in F$, its ...
- 128k
1
vote
0
answers
218
views
Asymptotic inverses and de Bruijn conjugates (etc.) for complex-valued functions
I recently got my hands on a copy of Regular Variation by Bingham, Goldie, and Teugels ("BGT"), and it's been an absolute revelation for my research. The thing is, my current work centers ...
- 1,284
6
votes
3
answers
555
views
Uniformly approximating a function and its derivative using polynomials
I'm struggling either proving or disproving the following statement:
Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If ...
- 263
0
votes
2
answers
241
views
Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$
Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions).
Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$.
Link to the problem (paper "...
- 698
1
vote
1
answer
71
views
Terminology: Co-completion of Met?
In main-stream mathematical literature, the term metric space is reserved for $(X,d)$ where $X$ is a set and $d:X\times X\rightarrow [0,\infty)$ satisfies the usual properties of a metric. However, ...
- 5,405
1
vote
0
answers
56
views
Moduli of continuity and Wasserstein differentiability of functions between measures
Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
- 11
2
votes
0
answers
65
views
Request for resources or techniques for bounding the infinity norm of an infinite product convolved with a simple function
I'm attempting to bound an expression of the form.
$$
\lVert(\prod_{i=1}^{\infty} \phi_i) * s \rVert_{\infty}
$$
Where $\phi_i$ are bounded periodic step functions which can be replaced by smoothed ...
- 41
2
votes
1
answer
186
views
Set of null-sequences is not $\sigma$-compact
I am interested in a reference for the following fact (or a similar result).
PROPOSITION. Let $X$ denote the set of real null sequences; i.e., the set of $(a_n)_{n=0}^{\infty}$ with $a_n\to 0$, with ...
- 6,548
1
vote
1
answer
99
views
Mean deviation in $p$-norm for $1 < p < 2$
Let $(X, \mu)$ be a probability space, and let $p \in (1, 2)$ be arbitrary. It is known from Corollary 2.4 of this paper by G. Sinnamon that for any measurable $f : X \to [0, +\infty],$ we have
$$0 \...
- 255
4
votes
1
answer
1k
views
The cotangent sum $\sum_{k=0}^{n-1}(-1)^k\cot\Big(\frac{\pi}{4n}(2k+1)\Big)=n$
On the Wolfram Research Reference page for the cotangent function (https://functions.wolfram.com/ElementaryFunctions/Cot/23/01/), I saw the following partial sum formula
$$\sum_{k=0}^{n-1}(-1)^k\cot\...
- 195
1
vote
1
answer
387
views
$L^p$ compactness for a sequence of functions from compactness of product with cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
-3
votes
1
answer
361
views
Basis for space of continuous, surjective monotone functions on $\mathbb{R}$ [closed]
$\DeclareMathOperator\CM{CM}$
I recently came across Okhezin - Study of families of monotone continuous functions on Tychonoff spaces describing monotone functions on general topological spaces and I ...
- 5,405
1
vote
1
answer
426
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 161
7
votes
5
answers
667
views
Reference for graduate-level text or monograph with focus on "the continuum"
I always had the dream to design a course for my graduate students like "mathematical models of the continuum". This course should cover history of real numbers, the Measure Problem, the ...
- 125
1
vote
1
answer
484
views
Convexity at a point and Jensen inequality
I am looking for a reference for the following claim:
Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed.
Suppose that "$\phi$ is convex at $c$". ...
- 6,741
1
vote
0
answers
71
views
Reference request : Convergence of radial basis function interpolation or spline interpolation as points become dense, for a continuous function
Is there any proof for this. Kindly request a reference in case available or any related documents towards this.
PS : I am specifically interested in the case of scattered data (irregularly placed), ...
- 698
4
votes
1
answer
116
views
$AC^p$ curves and pointwise metric speed in abstract metric spaces?
For a fixed "reasonable" metric space $(X,d)$ (say complete, separable, whatever is needed...), a curve $\gamma:[0,1]\to X$ is said to be $AC^p(0,1)$ (absolutely continuous) if
$$
d(\gamma(s)...
- 5,380
4
votes
2
answers
592
views
From Zurab's integral representation for the Apéry's constant to almost impossible integrals
I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...
4
votes
1
answer
539
views
Derivatives of Riemann $\xi$ and traces of zeros
Looking for references essentially corroborating (to authoritatively satisfy some editors) the sketch below of the relationship between even power (2,4,...) sums (traces) of the imaginary part of the ...
- 10.5k
1
vote
2
answers
859
views
Linear independence of exponential functions: a reference
Is there a publication containing this obvious fact: For any real $T>0$, any natural $n$, any complex $c_1,\dots,c_n$, and any distinct complex $z_1,\dots,z_n$ such that $\sum_1^n c_k e^{tz_k}=0$ ...
- 128k
12
votes
2
answers
698
views
Is the square root of a monotonic function whose all derivatives vanish smooth?
Let $g:[0,\infty] \to [0,\infty]$ be a smooth strictly increasing function satisfying $g(0)=0$ and $g^{(k)}(0)=0$ for every natural $k$.
Is $\sqrt g$ is infinitely (right) differentiable at $x=0$?
...
- 6,741
5
votes
0
answers
163
views
Minimizing total variation
Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by
$$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over ...
- 20.2k
1
vote
1
answer
195
views
Is the minimum of a constraint optimization problem differentiable in the constraint parameter?
Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$.
Let $s&...
- 6,741
2
votes
1
answer
71
views
Estimates on divergence-type operator for the matrix
Is there any result (Schauder-like estimates, $L^2$ estimates or similar) to equations of the form
$$
{\rm div}(Av)=f
$$
where $A$ is the "unknown" (i.e. I would like estimates on $A$ depending on $f$,...
4
votes
0
answers
125
views
Is there a name for this slightly stronger version of Cesàro convergence which "more quickly ignores earlier terms"?
Let $V$ be a normed vector space, let $l \in V$, and let $(a_n)$ be a sequence in $V$. We say that $a_n$ is Cesàro-convergent to $l$ if $\frac{1}{n}\sum_{i=1}^n a_i \to l$ as $n\to\infty$.
Now I will ...
- 708
18
votes
4
answers
4k
views
Problems in advanced calculus
I have been teaching Advanced Calculus at the University of Pittsburgh for many years. The course is intended both for advanced undergraduate students and the first year graduate students who have to ...
-2
votes
1
answer
147
views
Asymptotics for certain integrals
I stumbled on the following problem, if you can see a way through it.
Let $x$ be a real variable and fix a real value $\frac14\leq\nu\leq\frac34$.
QUESTION. For $x\rightarrow0$, does there exist a ...
- 43.2k
4
votes
1
answer
208
views
Stationary phase method for $\varphi''(x_0)= 0$
Stationary phase method (in the usual setup) gives asymptotic for
$$
I(\lambda)=\int_{a}^{b} f(t) e^{i \lambda \varphi(t)} d t,
$$
when at any stationary point $x_0$ ($\varphi'(x_0)=0$) second ...
- 12.3k
4
votes
3
answers
499
views
Are $\pm f\sqrt{1+g^2}$ and $\pm fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?
This is a follow-up on the previous question.
Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb ...
- 128k
8
votes
1
answer
384
views
Are $f\sqrt{1+g^2}$ and $fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?
Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb R)$. Does it then necessarily follow that the
...
- 128k
0
votes
0
answers
84
views
Relation between two matrices associated with a positive definite function
Let $f:\mathbb{R}^N \to \mathbb{R}$ be a positive definite function. Let $$g(h) = \int_{\mathbb{R}^N}f(x)f(h-x)\mathrm{d}x$$ Due to Bochner's and Parseval's theorems, $g$ is also a positive definite ...
- 698
2
votes
0
answers
200
views
The collection of mean value abscissas in the Mean value theorem
The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have
$$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$
for at least one $c\in [a,b]$. We ...
- 5,474
5
votes
1
answer
329
views
Reference for the rectifiablity of the boundary hypersurface of convex open set
The boundary of any convex open set $X$ is $\mathbb R^n$ is a rectifiable hypersurface.
To see this, intuitively, simply take a sphere $S_d$ with diameter $d\in(0,+\infty]$ that contains $X$. The ...
- 263
1
vote
1
answer
644
views
Most general form of Jensen's inequality
What is the most general form of Jensen's inequality?
Wikipedia gives for example this more general form, which holds in every topological vector space.
Are there even more general forms, for ...
- 2,129
1
vote
1
answer
167
views
Convex-like properties of the polar parametrization of the boundary a convex body on the plane
Let $B$ be a compact convex set on the complex plane, containing zero in its interior. The boundary $\partial B$ of $B$ has the polar parametrization
$\mathbf p:\mathbb R\to \partial B$ assigning to ...
- 41.9k