These are some easy functions you might want to check out first, in order to understand your own questions.
to your first question: sin(x)+sin(pi*x) 's period=? . by the way, f is period iff f(x)=f(x+T) for some T.
to your second: sin(x)/cos(x)=tan(x)'s period=?
to your third: the...
Homework Statement
Let f\in C^{1} on [0,1], and f(0)=f(1)=0, prove that
\int_{0}^{1}(f(x))^{2}dx \leq \frac{1}{4} \int_{0}^{1} (f'(x))^{2}dx
Homework Equations
The Attempt at a Solution
what is the trick to produce a 1/4 there? and how to make use of f(0)=f(1)=0? well I know that...
Oh, this is what I know about measure: a set A has measure zero means there is a countable collection of open/or closed rectangle that covers A that the sum of the volumn of each rectangle is less than an arbitrary given epsilon.
my original goal is to construct the subset S of Q, then \chi_{S}...
thank you. I don't find an example in textbooks yet but your idea sounds workable, in which f seems to be unbounded, and I'll try to figure it out... well, I forgot to tell that f is supposed to be bounded in my original question. Anyway it's helpful and thanks a lot.
Homework Statement
Let Q=I\times I (I=[0,1]) be a rectangle in R^2. Find a real function f:Q\to R such that the iterated integrals
\int_{x\in I} \int_{y\in I} f(x,y) \; and \int_{y\in I} \int_{x\in I} f(x,y)
exists, but f is not integrable over Q.
Edit: f is bounded
Homework...
Homework Statement
Suppose that f is an integrable function (and suppose it's real valued) on the circle with c_n=0 for all n, where c_n stands for the coefficient of Fourier series. Then f(p)=0 whenever f is continuous at the point p.
Homework Equations
The Attempt at a Solution...
use the definition. note that |(a1+a2+...+an)/n-L| = |((a1-L)+...+(an-L))/n|, and I guess you know something about the behavior of |a_n-L| when n goes to infinity :)
hope this helps u
hi, could you please explain a little about how to write it as an Riemann-Stieljes integral? I learned something about Riemann-Stieljes integral in principle of mathematics but havn't met any concrete examples. Thanks
...I think it is necessary to know the graph of cos(x), which may help a lot. so, find one.
edit (:shy: trying not to be ambiguous)
...I think it is necessary for one to know the graph of cos(x), which may also help a lot. (regardless of this particular problem)...
"periodic" is really the...
Many things beyond my knowledge:( I'll come back to this problem when I am ready for it. Anyway, thanks very much! I'll keep an eye on Jordan's curve theorem and Brouwer's theorem.