Tag: Analysis

functional analysis – $ L ^ 2 $ bound and interpolation of the Hölder norm

Consider the function

$$ F (x): = int _ { mathbb R} f (t + x) f (t-x) dt. $$

Clearly we have by Cauchy-Schwarz

$$ Green F (x) Green Green Green ^ 2 {L ^ 2} $$
$$ green F (green) green 2 f n Green_ {L ^ 2} Green f Green_ {L ^ 2} the 2 Green f Green_ {H ^ 1} ^ 2 $$

or $ H ^ 1 $ is the $ L ^ 2 $-Sobolev order space 1.

This shows that bind the higher standard of $ F $ we require that $ f in L ^ 2 $ and to link the $ C ^ 1 $ standard of $ F $ it is enough to have $ f in H ^ 1. $

I wonder now if it is true that the standard Hölder $ C ^ { gamma} $ with $ gamma in (0,1) $ can be delimited by the $ H ^ { gamma} $ standard of $ f $ and if that's not the case, I'd be curious to know what is the right interpolation space for $ f $ is bound to Hölder's standard of $ F $.

Fa.functional analysis – Reference request on the Min-Max theorem

Consider the following min-max problem

$$ inf_ {x in M} sup_ {y in N} F (x, y), $$

or $ F: M times N to mathbb R $ is Lipschitz and $ y mapsto F (x, y) $ is concave for all $ x in $ M. Could we drift $ inf_ {x in M} sup_ {y in N} F (x, y) = sup_ {y in N} inf_ {x in M} F (x, y) $ if $ M subset mathbb R ^ m $ and $ N subset mathbb R ^ n $ are the two compact?

PS: To the best of my knowledge, the reference to the Min-Max theorem comes from Mr. Sion: https://msp.org/pjm/1958/8-1/pjm-v8-n1-p14-p.pdf , the convexity of $ x mapsto F (x, y) $ is missing in my case. All comments or references are very appreciated!

complex analysis – If $ int_ {| z | = 1} z ^ nf (z) dz = 0 forall n = 0, 1, 2, … $, then $ f $ has a removable singularity at $ z = 0 $

True or false: $ f $ holomorph in $ A = {z in mathbb {C}: 0 lt | z | lt 2 } $ and $ int_ {| z | = 1} z ^ n f (z) dz = 0 forall n = 0, 1, 2, … $ then $ f $ has a removable singularity to $ z = $ 0.

I do not know if this is true or false, because I can only prove that $ f $ is either analytic at zero or has a removable singularity.

$ f $ holomorph in $ A $ means we can express $ f $ like a Laurent series of form

$$ f (z) = sum_ {n = 1} ^ { infty} frac {b_n} {z ^ n} + sum_ {n = 0} ^ { infty} {a_n} {z ^ n} $$

with
$$ a_n = frac {1} {2 pi i} int _ { gamma} frac {f (z)} {z ^ {n + 1}} dz $$

$$ b_n = frac {1} {2 pi i} int _ { gamma} {f (z)} {z ^ {n-1}} dz $$

$ int_ {| z | = 1} z ^ n f (z) dz = 0 forall n = 0, 1, 2, … $ means that $ b_n = 2 pi i cdot 0 = 0 forall n $, so $ f $ has as worse a removable singularity to $ z = $ 0.

Now to prove that $ f $ actually has a removable singularity, one has to prove that $ f $ is not analytic to $ z = $ 0. But $ f (z) = sum_ {n = 0} ^ { infty} {a_n} {z ^ n} $ at $ A $, so for $ f $ do not be analytical at $ z = $ 0 you have to have that $ f (0) neq a_0 = frac {1} {2 pi i} int _ { gamma} frac {f (z)} {z} dz $, which I could not prove.

complex analysis – Since Riemann maps $ f: Omega to mathbb D $, prove that $ inf_ {z in partial Omega} | za | frac1 {f) (a)} z in partial Omega} | za | $.

I look at the following problem from an old qualifying exam that I found:

Let $ Omega $ to be an open subset just connected $ Bbb C $, let $ a in Omega $ and suppose that we are given an analytical bijection $ f: Omega to mathbb D: = {z: | z | <1 } $ satisfactory $ f (a) = $ 0 and $ f (a)> $ 0. Prove $$ inf_ {z in partial Omega} | z-a | frac1 {f) (a)} z in partial Omega} | z-a |. $$

To be honest, I do not know where to start. my first thought was that $ 1 / f (a) = (f ^ {- 1}) & # 39; (0) $, and hoped to use this to try to apply the maximum module principle to a function involving $ f $ or $ (f ^ {- 1}) & # 39; $ and $ z-a $but the first problem that I got even before trying to write a function was to realize that $ f $ and $ (f ^ {- 1}) & # 39; $ may not even be defined on the respective limits, so it does not appear that I can use the maximum module to determine anything about the limit points in relation to these functions.

I have also assumed, seeing this, that it was simply something proven and used in the demonstration of Riemann's mapping theorem, but that does not really seem to be to be found the.

real analysis – Determine if $ f (x, y) = frac {1} {(1-xy) ^ 2} $ is $ embeddable[0,1]^ 2 $

Find out if $ f (x, y) = frac {1} {(1-xy) ^ 2} $ is embeddable on $ (0,1) ^ 2 $.

If I could use Fubini:

$$ int_0 ^ 1 int_0 ^ 1 frac {1} {(1-xy) ^ 2} dxdy = int_0 ^ 1 frac {1} {1-y} dy = infty $$

But can not use Fubini, because this function is not embeddable (in fact, I do not know it yet). How can I approach it without this theorem?

real analysis – Convergent subsequence of $ sin (n ^ 2) $

What can we say about convergent subsequences of $ sin (n ^ 2) $ whose existence is guaranteed by the Bolzano-Weierstrass Theorem?

Can we, as a corollary, affirm that for all $ epsilon gt $ 0 it exists $ m, n in mathrm {N} $ such as $ | sin (n) – sin (m) | lt epsilon $ ?

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functional analysis – meaning and motivation of external functions

Let $ mathbb {D} $ and $ mathbb {T} $ denotes the disk of the open unit and the circle of the unit $ mathbb {C} $ respectively. We write $ Hol ( mathbb {D}) $ for the space of all holomorphic functions on $ mathbb {D}. $ Hardy's spaces on $ mathbb {D} $ are defined as: $$ H ^ {p}: = left {f in left ( mathbb {D} right): n {r <1} int ^ {2 pi} _ {0} left | f left (re ^ {i theta} right) right | ^ {p} d theta < infty right } ; ; ; ; (0 <p < infty), $$
$$ H ^ { infty}: = left {f in Hol left ( mathbb {D} right): sup_ {z in D} left | f left (z right) right | < infty right }. $$
A function $ g in H ^ p ( mathbb {D}) $ is external if there is a function $ G: mathbb {T} longrightarrow (0, infty) $ with $ G in L ^ 1 ( mathbb {T}) $ such as
$$ g left (z right) = alpha text {exp} left ( int ^ {2 pi} _ {0} dfrac {e ^ {i theta} + z} {e ^ { i theta} -z} G left (e ^ {i theta} right) dfrac {d theta} {2 pi} right) qquad (z in mathbb {D}) $$ and $ | alpha | = $ 1.

The definition of an external function seems so involved. Can any one say what led to defining the external functions as such? What would be the motivation behind retaining such a definition?

I know that external functions in a Hardy space are important, for example if we consider the canonical factorization of an element of a Hardy space. Can any one mention another major utility of external functions?

computation – $ int _ {- infty} ^ infty frac { sin (x- frac 1x)} {x + frac 1x} dx $ via complex analysis

I am a little new in complex analysis, so be patient with me:

First, I have defined a function $ f (x) = frac { sin (x- frac 1x)} {x + frac 1x} $, so that I can define a new function $ f (z) = frac {e ^ {i (z- frac 1z)}} {z + frac 1z} $, which I then simplified to be $ f (z) = frac {ze ^ {i (z- frac 1z)}} {z ^ 2 + 1} $

Second, I defined an outline $ C $ which contains the interval $ (- R, – epsilon) $, contour $ gamma $ who has a half circle of radius $ epsilon $ centered at the origin, the interval $ ( epsilon, R) $and the semicircular outline $ Gamma $ with radius $ R $ centered at the origin. I want to take the limit as $ R rightarrow infty $ and $ epsilon rightarrow0 $

So,
$$ int_C f (z) dz = int _ {- R} ^ {- epsilon} f (z) dz + int_ gamma f (z) dz + int_ epsilon ^ R f (z) dz + int_ Gamma f (z) dz $$

For the leftmost integral, I can simply use the residue theorem, as $ C $ is a closed outline with a single pole to $ i $. So,
$$ int_C f (z) dz = 2 pi i Res (f, i) $$
$$ = 2 pi i frac {1} {2e ^ 2} = frac { pi i} {e ^ 2} $$
Then, looking at the first and third integrals on the right,
replace $ z = -u $ and $ dz = -du $ in the first integral, then
$$ int _ {- R} ^ {- epsilon} f (z) dz + int_ epsilon ^ R f (z) dz $$
$$ int _ { epsilon} ^ {R} frac {-ue ^ {i ( frac 1u-u)}} {u ^ 2 + 1} of + int # epsilon ^ R frac {ze ^ {i (z- frac 1z)}} {z ^ 2 + 1} dz $$
which can then be combined to get
$$ int _ { epsilon} ^ {R} frac {z (e ^ {i (z- frac 1z)} – e ^ {- i (z- frac 1z)})} {z ^ 2 + 1} dz $$
Then, multiplying by $ frac {2i} {2i} $, we have
$$ 2i int _ { epsilon} ^ {R} enac {z sin (z- frac 1z)} {z ^ 2 + 1} dz $$
And then, leaving $ R rightarrow infty $ and $ epsilon rightarrow0 $ we have
$$ 2i int_ {0} ^ { infty} enac {z sin (z- frac 1z)} {z ^ 2 + 1} dz $$
Moreover, since this integral is only on the real line, we can exchange the $ z $ for a $ x $, and also simplify for it to look like the original
$$ 2i int_ {0} ^ { infty} enac { sin (x- frac 1x)} {x + frac 1x} dx $$
who then leaves us with
$$ frac { pi i} {e ^ 2} = 2i int_ {0} ^ { infty} frac { sin (x- frac 1x)} {x + frac 1x} dx + int_ gamma f (z) dz + int_ Gamma f (z) dz $$
The problem is that I do not really know how to deal with the other two integral parts of the equation. I'm sure the integral on $ Gamma $ tends to 0 simply using the M-L inequality, but I'm not so sure of how to evaluate the integral on $ gamma $

actual analysis – Find the increasing function $ f: mathbb {R} to mathbb {R} $ such that $ int limits_ {0} ^ {x} f (t) dt = x ^ 2, forall x in mathbb {R} $

Find the increasing function $ f: mathbb {R} to mathbb {R} $ such as $$ int limits_ {0} ^ {x} f (t) dt = x ^ 2, forall x in mathbb {R} $$
My solution: Let $ F: mathbb {R} to mathbb {R} $,$ F (x) = int limits_ {0} ^ {x} f (t) dt $.
Then it follows that $ F $ is differentiable and that $ F (x) = 2x, forall x in mathbb {R} $.
For $ a in mathbb {R} $ we have that $$ 2a = F & # 39; (a) = lim_ {x searrow a} frac {F (x) -F (a)} {xa} = lim_ {x searrow a} frac { int_ {a} ^ {x} f (t) dt} {xa} ge lim_ {x searrow a} frac { int_ {a} ^ {x} f (a) dt} {xa} = frac {(xa) f (a)} {xa} = f (a) $$
and $$ 2a = F & # 39; (a) = lim_ {x nearrow a} frac {F (x) -F (a)} {xa} = lim_ {x nearrow a} frac { int_ {a} ^ {x} f (t) dt} {xa} the lim_ {x nearrow a} frac { int_ {a} ^ {x} f (a) dt} {xa} = Frac {(xa) f (a)} {xa} = f (a). $$
It follows from these two relations that $ f (x) = 2x, forall x in mathbb {R} $.
I think my solution works, but what I would like to know is why we can not take $ f $ to be down. Assuming that it is decreasing, reversing the inequalities from above, we get that $ f (x) = 2x, forall x in mathbb {R} $, which is a contradiction. What I want to know is if there is another way to come to a contradiction in this case. Specifically, I would like to know what motivated the authors of this problem to consider $ f $ to be increasing instead of decreasing (I doubt they just tried both cases and chose the one that actually worked).