User arc - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:15:06Z http://mathoverflow.net/feeds/user/19795 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96488/function-with-fourier-coefficient-of-order-on-m Function with Fourier coefficient of order $o(n^{-m})$ arc 2012-05-09T17:54:17Z 2012-05-09T18:37:51Z <p>Let $(a_n),(b_n)$ be Fourier coefficients of periodic locally integrable function $f: R \rightarrow \R$. Assume that $n^ma_n, n^m b_n \rightarrow 0$ when $n \rightarrow \infty$. By Weierstrass test $f$ is of class $C^{m-2}$. Is maybe $f^{m-2}$ absolutely continuous or differentiable almost everywhere, everywhere, etc.?</p> http://mathoverflow.net/questions/93088/atoms-of-regular-borel-measure Atoms of regular Borel measure arc 2012-04-04T08:52:58Z 2012-04-04T13:29:29Z <p>Let $X$ be locally compact Hausdorff space. Let $\mu$ be a Borel measure on it which is finite on compact and outer regular with respect to open sets and inner regular with respect to compact sets. Does an atom of such measure have to be a singleton (up to set of zero measure)?</p> http://mathoverflow.net/questions/92748/can-convolution-on-r-be-discontinuous-everywhere Can convolution on $R_+$ be discontinuous everywhere ? arc 2012-03-31T13:18:47Z 2012-04-01T09:19:05Z <p>Let $L^+$ be a set of all real valued functions defined on a real line which are Lebesgue integrable on each $[0,c]$, where $c>0$, and are zero for $x&lt;0$. Let for $a>0$, $f_a(t)=(t-a)^{-3/4}$ for $t>a$ and $0$ for $t \leq a$. Then, for $a,b>0$, convolution $(f_a*f_b)(x):=\int_0^x f_a(x-y)f_b(y)dy=\beta (x-a-b)^{-\frac{1}{2}}$ for $x> a+b$ and $0$ for $x \leq a+b$, where $\beta=B(\frac{1}{4}, \frac{1}{4})$. It shows that convolution of two functions from $L^+$ need not be continuous. Is it possible, maybe by condensation of singularities and above example to show existence of two functions from $L^+$ which convolution is discontinuous everywhere<br> on $[0, \infty)$?</p> http://mathoverflow.net/questions/91116/approximation-by-polynomials Approximation by polynomials arc 2012-03-13T20:53:41Z 2012-03-16T11:36:47Z <p>Let $f:[a,b] \rightarrow \mathbb{R}$ be of class $C^n$. Let $x_0, ..., x_m$ be different numbers from $[a,b]$. </p> <p>Does for each $\varepsilon >0$ there exist a polynom $P$ such that $P^{(k)}(x_i)=f^{(k)}(x_i)$ for $i=0,...,m$, $k=0,...,n$ and $sup_{x \in [a,b]} |f(x)-P(x)|&lt; \varepsilon$?</p> http://mathoverflow.net/questions/87983/completness-of-borel-measure Completness of Borel measure arc 2012-02-09T11:45:52Z 2012-02-09T11:45:52Z <p>Let $X$ be compact Hausdorff space and $\mu$ a finite Borel measure without atoms which is outer regular with respect to open sets and inner regular with respect to compact sets. Can such measure be complete?</p> http://mathoverflow.net/questions/84743/taylor-theorem-for-function-of-several-variables Taylor theorem for function of several variables arc 2012-01-02T13:18:32Z 2012-01-02T13:18:32Z <p>I would like to ask about references or sketch of a proof of the following version of Taylor theorem for functions of many variables.</p> <p>Assume that $f$ is a function of class $C^{n+k}$ defined in a neighbourhood $W$ of zero in $\mathbb{R}^m$ with values in $\mathbb{R}.$ Then there exist a functions $R_\alpha$, where $\alpha$ is multiindex with length $n$, defined in $W$ with values in $\mathbb{R}$ such that </p> <ol> <li><p>$R_\alpha$ is of clas $C^k$ in $W$, $R_\alpha(0)=0$, $D^i R_\alpha(0)=\left(\frac{i!}{(n+i)!}\right) D^{n+i}f(0)$ for every multiindex $i$ with $0 \leq |i| \leq k$.</p></li> <li><p>$R_\alpha$ is of class $C^{n+k}$ in $W\setminus {0}$.</p></li> <li><p>$\lim_{x\rightarrow 0} \|x\|^{|i|} D^{i+k}R_\alpha(x)=0$ for each multiindex $i$ with $1\leq |i|\leq n$ and for each $\alpha$ (with $| \alpha|=n$).</p></li> <li><p>$f(x)=\sum_{|\alpha|\leq n}\left(\frac{D^\alpha f(0)}{\alpha !}\right)x^\alpha+\sum_{|\alpha| =n} R_\alpha(x) x^\alpha$ for $x \in W$.</p></li> </ol> http://mathoverflow.net/questions/83734/a-smoothness-of-f-sqrtp-x A smoothness of $f(\sqrt[p] x)$ arc 2011-12-17T18:21:02Z 2011-12-18T20:20:38Z <p>Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function let $p \in \mathbb{N}$, $p \geq 2$. Assume that $f^{(k)}(0)=0$ for all $k \notin p \mathbb{N}$. Is it true that then $g(x)=f(\sqrt[p] x)$ for $x\geq 0$ is smooth?</p> <p>Thanks.</p> http://mathoverflow.net/questions/83213/embeddings-l-p-in-l-q-for-pq Embeddings $L_p$ in $L_q$ for $p<q$. arc 2011-12-11T23:04:47Z 2011-12-12T00:54:06Z <p>Let (X.S,m) be measure space and $p &lt; q$.</p> <p>I would like the references or idea of the proof of the followig result:</p> <p>Lp(X, m) is contained in Lq(X, m) iff S does not contain sets of arbitrarily small non-zero measure.</p> <p>Thanks.</p> http://mathoverflow.net/questions/82915/restriction-of-haar-measure-to-borel-sigma-algebra Restriction of Haar measure to Borel $\sigma$ -algebra arc 2011-12-07T22:55:28Z 2011-12-08T18:10:48Z <p>Let $(G,M,\mu)$ be a measure space, where $\mu$ is the Haar measure on topological group $G:=\mathbb R\times\mathbb R_d$, ($\mathbb R$ is the group of reals with the natural topology whereas $\mathbb R_d$ is the group of reals with the discrete topology) and $M$ be the $\sigma$-algebra of all Haar measurable subsets of $G$. </p> <p>Let $\mu_0 :=\mu|_B$, where $B$ is the $\sigma$-algebra of all Borel subsets of $G$, and let $(G,M_1,\mu_1)$ be the smallest completion of the measure space $(G,B,\mu_0)$.</p> <p>Is it true that $M_1=M$ and consequently $\mu_1=\mu$ ?</p> http://mathoverflow.net/questions/92748/can-convolution-on-r-be-discontinuous-everywhere Comment by arc arc 2012-03-31T18:21:41Z 2012-03-31T18:21:41Z Sorry, I did mistake. I just have edited. http://mathoverflow.net/questions/91116/approximation-by-polynomials/91133#91133 Comment by arc arc 2012-03-14T13:59:04Z 2012-03-14T13:59:04Z Very thanks. I have one question. You used the following fact: if a continuous function $g:[a,b]\rightarrow R$ vanishes at points $x_0,...,x_m$ then for every $\delta &gt;0$ there exists a polynomial $Q$ which vanishes at $x_0,...,x_m$ such that $\|g(x)-Q(x)\|_{sup} &lt;\delta$. It is easy for $m=0$, because by Weierstrass threorem we find polynomial $P$ such that $\|f-P\| &lt;\frac{\delta}{2}$ and polynomial $Q:=P-P(x_0)$ will be good. How to prove its for $m\geq 1$? http://mathoverflow.net/questions/83734/a-smoothness-of-f-sqrtp-x Comment by arc arc 2011-12-18T08:11:48Z 2011-12-18T08:11:48Z Thanks, but Iwould like to have at least a references. http://mathoverflow.net/questions/82915/restriction-of-haar-measure-to-borel-sigma-algebra/82982#82982 Comment by arc arc 2011-12-09T11:19:25Z 2011-12-09T11:19:25Z Very thanks for answer. http://mathoverflow.net/questions/82915/restriction-of-haar-measure-to-borel-sigma-algebra Comment by arc arc 2011-12-07T23:28:00Z 2011-12-07T23:28:00Z Not necessary. Haar measure is complete, because it is introduced by using Caratheodory theorem, but Borel measure generally is not complete. http://mathoverflow.net/questions/82915/restriction-of-haar-measure-to-borel-sigma-algebra Comment by arc arc 2011-12-07T23:14:44Z 2011-12-07T23:14:44Z Thanks, I have corrected.