Questions tagged [mean-value-theorem]
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11 questions
1 vote
1 answer
212 views
Riemann sum error bounds without using derivatives or the Mean Value Theorem?
I know that if a function $f$ is continuously differentiable on $[a,b]$, one can bound the difference between its integral and a Riemann sum using either the Mean Value Theorem or Taylor expansions. ...
- 37
5 votes
2 answers
517 views
Extending a convex function to a higher dimensional domain
Let $D\subset{\mathbb R}^2$ be the unit disk, and $I=(-1,1)$. Let $v\in C^2(\bar I)$ be a convex function. Does there exist a convex function $u\in C^2(\bar D)$, such that on the one hand $u(\cdot,0)=...
- 53.1k
0 votes
0 answers
156 views
Estimate value of harmonic function in the annulus
Let $D = B_{2r}(0)\backslash \overline{B}_r(0)$. Assume $Lu = 0$ in $D$ where $L$ is a uniform elliptic operator with constant coefficients $$ Lu = \sum_{i,j} a_{ij}u_{x_i}u_{x_j}, \qquad \lambda |\xi|...
user124759
14 votes
1 answer
627 views
Mean value theorem for Dirichlet series - optimize?
Let $a_n\in \mathbb{C}$. We can prove a mean value theorem, meaning an inequality $$\int_0^T \left|\sum_{n=1}^\infty a_n n^{-i t}\right|^2 dt \leq \sum_{n=1}^\infty (c_0 T + c_1 n + c_2) |a_n|^2.$$ ...
- 21.7k
2 votes
0 answers
159 views
Mean values of $\zeta(s)$ for $\Re(s)=1/2$ vs $\Re(s)\ne 1/2$
Say I have a good estimate for the $L^2$ mean of the Riemann zeta function $\zeta(s)$ for $\Re s = 1/2$, $|t|\leq T$: $$\int_0^T |\zeta(1/2+i t)|^2 = T \log T - T (1 + \log 2 \pi - 2\gamma) + O(T^\...
- 21.7k
1 vote
0 answers
140 views
Mean Value Inequality with Linear Term
I am having trouble proving this modified mean-value inequality. Suppose that $\Delta u+cu\ge 0$ for $u:\mathbb{R}^n\to [0,\infty).$ Prove that there exists constants $r_0,C>0$ depending only on $c$...
- 11
2 votes
0 answers
227 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 ...
- 5,599
2 votes
1 answer
382 views
Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?
For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowledge to solve it. Problem definition: Let $f(\xi) \in \...
- 129
4 votes
1 answer
1k views
How to understand the integral?
In order to understander the nonlinear elliptic equation with natural boundary condition, $$\sigma_2(D^2u)=0 \text{ in } \Omega$$ I wish to understand the following integral, $$E(u,\Sigma)=\int_\Sigma ...
- 697
2 votes
1 answer
429 views
Mean value theorem in terms of Wirtinger calculus?
The mean value theorem for vector-valued function in the real domain $f: \mathcal{R}^n \rightarrow \mathcal{R}^d$ can be expressed as \begin{equation} f(x)-f(y)=\int_{0}^{1}\nabla f(x(\tau))d \tau \...
- 515
16 votes
1 answer
532 views
Bull's-eye Riemann sum
Let $f:[a,b] \to \mathbb{R}^2$ be a continuous curve on the plane. Question: Are there numbers $a \leq x \leq c \leq y \le b$ such that $$(c-a)f(x)+(b-c)f(y) = \int_a^b f(t) \, dt \ ?$$ In other ...
- 2,509