real analysis – prove the sequence $ sum_ {n = 1} ^ infty (n (f ( frac {1} {n} – f (- frac {1} {n}) – 2f & # 39; (0)) $ converges

Prove the series $ sum_ {n = 1} ^ infty (n (f ( frac {1} {n} – f ( frac {1} {n}) – 2f & # 39; (0)) $ where converges $ f $ is defined on $ (-1.1) $ and $ f & # 39; & gt; (x) $ is continuous.

I already have a solution, but I do not quite follow it.

It uses the Taylor extension over $ 0 $ to get:

$ f (x) = f (x) + f ((0) frac {x} {1} + f & # 39; # (0) frac {x ^ 2} {2} + f & # 39; & # 39; & # 39; (t) frac {x ^ 3} {6} $

$ f (-x) = f (x) – f ((0) frac {x} {1} + f & # 39; # (0) frac {x ^ 2} {2} – f & # 39; & # 39; & # 39; (s) frac {x ^ 3} {6} $

For some $ s, t in (-1,1) $,

Here is my problem: I'm not clear on that $ s, t $ used in the third derivative. We know that the second derivative is continuous, perhaps the statement should be that the third derivative is continuous?

real analysis – Necessary and sufficient convergence condition according to Borel-Cantelli lemma

Given a sequence of random variables $ {X_n } _ {n = 1} ^ infty $The first Borel-Cantelli lemma tells us that there is a positive sequence $ {a_m } _ {m = 1} ^ infty $ for which:

$$ a_m overset {m rightarrow infty} { longrightarrow} 0 quad text {and} quad sum limits_ {n, m = 1} ^ infty mathbb {P} big ( vert X_n vert> a_m big) < infty tag {$ circledast $} $$
Then $ X_n $ almost certainly converges $ 0 $, My question is whether there is an inverse relationship, i. E. $ X_n rightarrow0 $ almost certainly implies that there is a positive sequence $ {a_m } $ so that $ circledast $ holds?

fa. function analysis – meaning and motivation for external functions

To let $ mathbb {D} $ and $ mathbb {T} $ denote the open unit disk and the unit circle in $ mathbb {C} $ respectively. We write $ Hol ( mathbb {D}) $ for the space of all holomorphic functions $ mathbb {D}. $ The Hardy rooms open $ mathbb {D} $ are defined as: $$ H ^ {p}: = left {f in Hol left ( mathbb {D} right): sup _ {r <1} int ^ {2 pi} _ {0} left | f left (again ^ {i theta} right) right | ^ {p} d theta < infty right } ; ; ; ; (0 <p < infty), $$
$$ H ^ { infty}: = left {f in Hol left ( mathbb {D} right): sup_ {z in D} left | f left (z right) right | < infty right }. $$
A function $ g in H ^ p ( mathbb {D}) $ is outward if there is a function $ G: mathbb {T} longrightarrow (0, infty) $ With $ G in L ^ 1 ( mathbb {T}) $ so that
$$ g left (z right) = alpha text {exp} left ( int ^ {2 pi} _ {0} dfrac {e ^ {i theta} + z} {e ^ { i theta} -z} G left (e ^ {i theta} right) dfrac {d theta} {2 pi} right) qquad (z in mathbb {D}) $$ and $ | alpha | = 1 $,

The definition of an outer function seems to be so involved. Can someone say what has led to define external functions as such? What would be the motivation to remember such a definition?

I know that external functions in a Hardy space are important, for example, when we consider the canonical factorization of an element of a Hardy space. Can someone mention another important benefit of external functions?

complex analysis – whole function such that $ text {Im} f = text {Re} ^ 4 f + 1 $ is constant

So I came across the following problem:

Prove that if $ f $ is whole, $ u = text {Re} f $. $ v = text {im} f $ and $ v (z) = u ^ 4 (z) + 1 $ $ forall z in mathbb {C} $, then $ f $ is constant.

In this special case $ v (z) gt 0 for all z in mathbb {C} $so apply Liouville's theorem on $ g = exp (if) $ do the job, but I was wondering if there are any solutions that exploit the equality between $ u $ and $ v $ instead of the positivity of $ v $,

fa.functional analysis – Norm of the convolution operator

By the inequality of Young for some $ f in L ^ p ( mathbf {R}) $ the map $ T_f: g mapsto f star $ is a permanent operator of $ L ^ q ( mathbf {R}) $ to $ L ^ r ( mathbf {R}) $ from where $ 1 leq p, q, r leq infty $ fulfill $ 1 + frac1r = frac1p + frac1q $ and we even have

begin {align *}
| T_f | _ {p rightarrow r} leq | f | _q.
end {align *}

If I am not wrong $ | T_f | _ {p rightarrow r} $ and $ | f | _ {q} $ are not equivalent:

  • When $ r = q = 2 $ and $ p = 1 $ we have Plancherel's formula (for a correctly normalized Fourier transformation) $ | T_f (g) | _2 = | has {f} has {g} | _2 $ from which we get $ | T_f | _ {2 rightarrow 2} = | has {f} | _ infty $, and $ | f | _1 lesssim | has {f} | _ infty $ is just not reasonable.
  • On a more sophisticated level, on the Torus, I know the partial Fourier series $ S_N (f) $ according to the Dirichlet kernel $ D_N $ converge $ L ^ p ( mathbf {T}) $ for non-extreme values ​​of $ p $, The Dirichlet kernel is unlimited $ L ^ 1 ( mathbf {T}) $. $ | D_N | _1 lesssim | T_ {D_N} | _ {p rightarrow p} $ is not possible due to the Banach-Steinhaus theorem.

On the other hand, you can prove that $ | T_f | _ {1 rightarrow 1} $ and $ | T_f | _ { infty rightarrow infty} $ are both equivalent to $ | f | _1 $,

My questions :

  1. Are they other exponents for which this equivalence holds?
  2. If the equivalence does not hold, there is a description of $ | T_f | _ {p rightarrow r} $ (focusing on the case $ p = r $)?
  3. Is there an elementary (= not as above) proof for this? $ | T_f | _ {p rightarrow p} $ is not synonymous with $ | f | _1 $ when $ p notin {1,2, infty } $ ?

I have found several results in the literature related to this question, but they either treat the optimality of Young's inequality as the continuity of the operator $ (f, g) mapsto f star $ (not interested) or specify the equivalence if $ f $ is not negative.

Calculus and Analysis – Unexpected difference between integral and summation?

I try to integrate something like: Integrate(Exp(-i*(k*x+k*z))*Exp(-(x^2+z^2)),{x,-largenumber,largenumber},{z,-largenumber,largenumber})

My problem is that it takes too much time to compute the whole integral, so I try to approximate it. I have tried two approaches: 1) Using the analytic form of the integral and replacing the integration limits. 2) approximation of the integral as sum and summation over the x and z index.

However, both methods result in a result that is significantly different from the full integral. The strangest thing is that both (1) and (2) provide exactly the same (wrong) solution. I think that may have something to do with the complex values ​​of the integrand. The absolute value of the solution is what I ultimately need. I wonder if I have to be careful to consider the real and imaginary part of the integrand when approaching it as a sum.

I have previously made similar integrals and had no problems. What are the potential problems that I could face? How does Integrate behave other than the analytic form of the integral and the insertion of values ​​or the summation of the integrand over the relevant indexes?

real analysis – $ C ^ 2 $ functions is an open subset of $ C ^ 0 $?

To let $ C ^ k ( mathbb {R} ^ d; mathbb {R} ^ D) $ denote the amount of $ k $-fold differentiable functions $ mathbb {R} ^ d $ to $ mathbb {R} ^ D $; from where $ d, D $ are positive integers. is $ C ^ k ( mathbb {R} ^ d; mathbb {R} ^ D) $ an open subset of $ C ( mathbb {R} ^ d; mathbb {R} ^ D) $; If the latter is equipped with the compact open topology (topology of uniform convergence on compact)?

Does it generally have a non-empty interior?

real analysis – is $ lim_ {n to infty} sum_ {k = 0} ^ {m_n-1} o ( delta x_i ^ n) = 0 $ or not always?

To let $ {x_i ^ n } _ {i = 0} ^ {m_n-1} $ a subdivision of $ (0.1) $ s.t. $ max_ {i = 1, …, m_n-1} delta x_i ^ n to 0 $, does $$ lim_ {n to infty} sum_ {i = 0} ^ {m_n-1} o ( delta x_i ^ n) = 0 ? $$

That's always the case for me, but with my question here I can have doubts. The thing is that I can not get a counterexample (all the examples I have work). So can someone confirm whether it is always true or provide a counterexample?

Algorithms – Doubt on some inequalities in asymptotic analysis

I worked through this recent paper and had some doubts about the final analysis of the complexity of the schema (you do not have to go through the entire paper). I included three questions here because I thought they were simple questions.

  1. In p. 17 (last four lines, see equation 7) this inequality is used (here $ k (n) = sqrt { frac {n delta (n)} { log n}} $ and $ Delta (n) = o (n / log n) $):

$ frac { binom {n} {a}} { binom {^ n / _k} {^ a / _k} ^ k} leq (^ n / _k) ^ k $

Can I find proof of this?

  1. Similarly, at the beginning of page 18, how is this possible? (the above inequality is used to come here, $ (n / k) ^ {k / 2} $ and do not worry about the meaning of $ mathtt {ss} $)

$ mathtt {ss} leq sqrt { binom {n} { frac {n} {2} – delta (n)}} cdot (n / k) ^ {k / 2} cdot binom {k} {2 delta (n)} cdot 2 ^ {(2 delta (n) +1) n / k} leq 2 ^ {n / 2 + o (n)} $

  1. Again this might be a bit trivial, in Pg 17 an inequality about Equation 7, the $ O (n) $ Term is not relevant, is that not relevant?