# Functional Analysis

4
I am working on "Extensions of Special functions and their applications ". Is anyone interested in collaboration then they are most well come?

Since last 15 years, I am working on Special Functions and their applications. Recently, number of applications of special functions found in many fields. Now, I am interested in extensions of special functions?

I am interested and I am working at applications of hypergeometric functions

6
Where and which are the applications of operator algebras?

von Neumann algebras, C*-algebras, etc

K-theory groups of operator algebras, the algebras of operator on Hilbert spaces, simply C*-algebras are play an important role in string theory and M-theory. So the K-theory of operator algebras classifications d-brans and Ramond - Ramond fields and many other charges contains as an elements in K_0 and K_1 groups. On the Other hand C*-algebras associated to the noncommutative torus and noncommutative spheres are view as a natural algebras which represent the noncommutative space time in various dimensionals , the sheet of words.

Also, Operator algebras has a useful applications in quantum information, including non commutative probability, operator spaces, quantum entropy , quantum computing and quantum cryptography.

5
Is there any theorem / lemma/ theory regarding closed form expressions which says that we can find out some nth derivative of a function?

Consider a function

x_dot= f (x),  its 1st derivative can be written as   x(1)=f(x),

And its 2nd derivative can be x(2)=f '(x). x_dot,

And recursively, we can find out x(n) nth derivative of the x_dot= f(x) in the case if f(x) is linear, which is a reason for the formation of matrix exponential (eAT) If A is a linear matrix in f (x).

Or

One can also say that if f (x) results in a closed form expression for its Taylor expansion. Then nth derivative can be written. My question is that expression can be written for nonlinear systems if they come to have a closed form expression in their Taylor expansion.

1
Does any logistic kernel (e.g. the sigmoid) reproduce a Hilbert space?

I would like to use the logistic function as distance operator between two functions from a set of them. These two functions live in a Hilbert space, Im not sure if the result of successive measures is a set living in a Hilbert space also. In the case it is not true, how can be mathematically proved the computational consistence of this measure? How this consistence can be theoretically ensured before empirically test it? Thank you

If the logistic kernel is a radial function, say g(r), it will reproduce a Hilbert space in a space variables of any dimensionality if and only if the univariate function f(t) = g(\sqrt{t}) is a completely monotonous function on (0,\infty) and is bounded at zero. Complete monotonity means that (-1)^m f^{(m)} is nonegative on (0,\nfty) for any m. In other words f should be nonnegative, f' should be nonpositive, f'' should be nonnegative, and so on. Excluding the case when f(t)=t^n for natural number n.

If the dimensionality of space of independent variables is limited (say 2-dimensional or 3-dimensional), it is enough that the function and its first derivatives satisfy the monotonity property.

See detail in Wedland, Holger, "Scattered Data Approximation", Cambridge Monographs on Applied and Computational Math., 2005, Chapters 7 and 8.

1
Where and which are the applications of Voiculescu's non-commutative probability?
Where and which are the applications of Voiculescu's non-commutative probability?​_Where and which are the applications of Voiculescu's non-commutative probability?​

Where and which are the applications of Voiculescu's non-commutative probability in functional analysis, physics, probability theory, etc?

George, I think all applications have revolved around random matrices and noncommutative random variables and most relate to Von Neumann algebras. Various open questions have been asked ... like the possibility of a theory analogous to that of independence in probability theory.

http://www.msri.org/publications/ln/msri/2002/rmt/voiculescu/1/meta/aux/voiculescu.pdf

15
Can anyone help me with topological fields?

I am looking for examples of topological fields. It seems that they are scarce in the literature. continuous real functions on a compact space are just topological ring. In fact, I am interested in knowing a topological field with sequences. Simple examples are R or C. But sequences of these spaces are not topological field.

Your question remains for me unclear. If you are looking for a field consisting of sequences on some algebra with coordinatewise operations, it cannot exist because any element admitting a nonzero coordinate must be invertible, but this is not true.

Now, if you are looking for a field with a linear topology (even with continuous multiplication), you can find them. Actually, every field with its strongest locally convex topology is a topological field (here multiplication need not be continuous in both variables but only separately continuous). You can also have a fields with a linear topology such that the multiplication is continuous in both variables, but the topology need not be locally convex. You can have a look at  somes (old)  papers of L. Waelbrock, and Williamson.

Anyway, because of Gelfand-Mazur theorem, if you ask a "topological" field to have some stronger properties you just get $\mathbb{C}$ in the complex case. You can have a look at some papers of Mati Abel.

6
Can we say that a linear space X is not reflexive?
Suppose X be a linear space. My question is: How can we say that X is not reflexive in any norm. Reflexive means X is linearly isometric to its second dual X''

The following geometric characterization for the reflexivity of a Banach space X holds: X is reflexive if and only if every closed convex subset of X is proximinal (see "Geometric functional analysis and its applications" by Richard B. Holmes)..

16
Does the collection of all self adjoint operators have any property?
There are unitary, self adjoint and normal operators in operator theory. Do each collection posses nice property?

Unitary operators U (U*=U^(-1)), have important geometric properties. Namely, U and U* preserve the scalar products (hence the norms and the angles), so both of them are isometries.. The spectrum of a unitary operator is contained in the unit circle. Self-adjoint operators have real spectrum. They admit  an integral representation associated to a spectral measure. Normal operators T have the form T = A + i B, with A, B commutting self - adjoint operators. Unitary operators have a similar representation, where A^(2)+B^(2)=I, (AB=BA). Normal and unitary operators admit spectral measures and associated integral representations too.

41
Is there a book in English where one can find characterizations of zero-derivative (stationary) points ?

In non-English literature two  such characterizations for C2 functions of the single variable can be found in the text Neralic, Sego: Matematika (second edition), Element, Zagreb, 2013 (ISBN 978-953-197-644-2) but they do not seem to be widely known. They appear to be important in analysis, calculus, optimization and other areas.  Where can one find such results in functional analysis ?

Characterizations of zero-derivative points have some interesting applications. Illustration: Consider a C1 Lipschitz function f and an interior point x* in a compact interval K of its domain. Denote by I(x*,x) the integral of f from x* to x in K and F(x*,x) =  I(x*,x) - f(x*) (x-x*). Then F'(x*,x*) = f(x*) - f(x*) = 0. The quadratic envelope characterization says that abs F(x*,x) is overestimated by 1/2 max abs f'(x) on K times square of (x - x*) for every x in K. This "third part" of the fundamental theorem of calculus compares integration with differentiation. (The first two parts of the theorem say that these processes are "inverse". )Thanks.

• Ioana Ghenciu asked a question:
Open
Is there an example of a Banach space which has property (BD) but does not have property wGP?

A Banach space has property BD if every limited subset of it is relatively weakly compact.

A subset $A$ of $X$ is called a Grothendieck set if every operator $T:X\to c_0$ maps $A$ onto a relatively weakly compact set.

A Banach space $X$ has the weak Gelfand-Phillips  (wGP) property if every Grothendieck set in $X$ is relatively weakly compact.

Every limited set is a Grothendieck set. If X has the wGP property , then X has the BD property.

It is known that if X does not contain $\ell_1$, then X has property BD. Moreover, it has property wGP.

16
A power series summation a_n z^n such that a_n tends 0 as n goes infinity. How can we show it does not have pole on unit circle?

A power series summation a_n zn such that a_n tends 0 as n goes infinity. How can we show it does not have pole on unit circle?

It is easy to see that the problem reduces to the case when the radius of convergence equals 1. In this case, if one additionally assumes that the power series is absolutely and uniformly convergent in the closed unit disc, we have |s(z)|<=Sum |a^n| < infinity, since the power series is absolutely convergent at z_0 = 1. Hence this is a sufficient condition.  Example: |a_n| = M/(n^p), p>1 constant, M>0 constant.

27
How can I calculate the Lyapunov exponent?

In Mathematics the Lyapunov exponent of a dynamical systems is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Is there some new algorithms  for calculate the Lyapunov exponent?

Thank you Julien Clinton Sprott for your answer.

Yes, this what I find in some publications. They declare that the system is chaotic from the scrambled graph.

• Helmut Ziegfeld Baumert added an answer:
4
Reynolds Stress realizability constraints: positive determinant?

Giving the construction of Reynolds stress tensor there some basic constraints which limits how the values for components may assume in a given realisable flow. I currently have no problem understanding that

- the Reynolds stresses are positive semi-definite;

- and have both first and second invariants greater than zero, due to results in functional analysis.

However, I am failing to see why should also the determinant of R be also a positive (or zero) quantity. At the appendix in Schumann's paper there is such a proof, but I didn't quite well follow all his steps in the demonstration.

Has anyone ever demonstrated somewhat differently than Schumann's version? I mean, in a more geometrically way?

Best regards to all and thanks in advance for any advice,

Fernando Soares.

Fernando, I missed the Schumann paper so I am not aware whether I can help or not, but attached you may find a paper which might help indirectly. It is a closed theory based on Kolmogorov 1942 with remarks by Landau, same year. It gives for the first time not only stresses but also the fundamental constants of turbulence like von-Karman as
1/SQRT(2 pi) =0.399. Princeton superpipe gives 0.40 +/- 0.02.

8
For a subset X of RxR with the property that every continuous function f:A-->R attains its maximum in R. Is X compact?

For a subset X of RxR with the property that every continuous function f:A-->R attains its maximum in R. Is X compact? What if f is bounded but does not attain maximum in X.

Topological spaces on which every continuous function is bounded are called pseudo-compact.

By Theorem 30 in

1. E. Hewitt, Rings of real-valued continuous functions. I.
Trans. Am. Math. Soc. 64, 45-99 (1948). (Theorem 30)

a normal pseudo-compact space is compact. Since metric spaces are normal it follows  that a pseudo-compact metric space is compact.

The paper  [1] contains also other caharcaterizations of pseudo-compact spaces.

2. R. M. Stephenson, Pseudocompact spaces, Trans. Am. Math. Soc. 134, 437-448 (1968).

and the book

3. L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, New York, 1960.

8
A converse of the implicit function theorem?

I have a function in terms of a vector x, and a parameter a, say F(x,a). I also know that there exists a smooth unique parametrized curve x(a) such that F(x(a),a) = 0 for all a. I want to know if the Jacobian matrix of F differentiated with respect to x is nonsingular when evaluated on the curve x(a).

In other words, if we have a dynamical system can we say that local solvability of an equilibrium guarantees that stability cannot change assuming a Hopf bifurcation does not take place?

Consider simple form  of the implicit function theorem:

Suppose  that  (i):  F maps a  nbg  V in   R^2 into R , F(0,0)=0 and that   F’_x(x,a)  different from 0  in  V .

Then   (ii)  there is b>0    such that  the set  of points  F(x,a)=  0      in  Q= [-b,b]^2,   is graph   of a  function x=x(a),   -b <=    a <=b.

For example   if  F(x,a)= x-  a^3, then the set of points F(x,a)= 0    is graph  G1 of function x(a)=a^3.

The same graph is also  solution   of equation  F1(x,a)= (x- a^3)5=0.

In this example, the Jacobian matrix of F with respect to x is non singular on G1,    but  the Jacobian matrix of F1  with respect to x is   singular  on G1.

Thus the implicit function theorem  states : (i) implies (ii) .

What is   a converse of the implicit function theorem?

As I understand    (ii) implies (i) ?

Suppose (ii):

In other words  we suppose  that  we have smooth graph  G  given by  a function  x=x(a)   which maps [-b,b]  into itself.

Can we find a function F(x,a)  such that  a form of (i) holds:

the Jacobian matrix of F with respect to x is non singular on   Q= [-b,b]^2  and

F( x(a),a )= 0  for  in [-b,b].

Take  F(x,a)=  x-x(a).  Then   F’_x (x,a)=1.

Is this related to the question.

To Jason Bramburger  and Giovanni Dore:

Do you have something different in mind?

1
How to prove that > 0 ?

Let H1, H2 be Hilbert spaces and

A : H1 → H2 be a compact operator with the singular

system (λn, vn, un).

How to prove that <x, A*Ax> > 0  where <...,...> denotes the inner product ?

By the definition of the adjoint operator, <x, A*Ax> = <Ax, Ax> = ||Ax||2. Thus, <x, A*Ax> > 0 provided x does not belong to the kernel of A and it is zero otherwise.

• Peter I. Kogut added an answer:
2
Is it correct to say that $div(A^t \nabla y)\in L^2(\Omega)$ provided $div(A \nabla y)\in L^2(\Omega)$,
Here
$A$ is bounded nonsymmetric matrix and $y$ belongs to the Sobolev space $H^1_0(\Omega)$?

Thanks a lot, Gianni. Your arguments sound perfect!

• Yuriy Borisovich Zelins’kyi asked a question:
Open
Anyone familiar with m-convex compacts in R n?

Definition. A compact K ⊂ R n is called to be m-convex (m < n) if for each x ∈ Rn \ K there exists an m-plane T(x) such that x ∈ T(x) and T(x) ∩ K = ∅.

Problem. Let K ⊂ R n be a compact. Suppose that for each hyperplane T ⊂ R n the intersection T ∩K is (m−1)-convex. Find a condition on K which together with the above one would be sufficient for K to be m-convex.

http://www.usc.es/dmle/pdf/EXTRACTAMATHEMATICAE_2005_20_01_05.pdf

• Giuseppe Di Fazio added an answer:
3
How to use the Moser iteration technique to improve the regularity of very weak solution?

The very weak solution like

-\Delta u=f in \Omega, u=0 on \partial\Omega,

when f \in L^p with 1\le p <2

\int_\Omega u(-\Delta ) \xi dx=\int_\Omega f \xi dx,   \xi\in C^{1.1}_0(\Omega),

so in this sense, how can choose the test functions to improve the regularity?

Give a look at this paper

http://www.ams.org/mathscinet/search/publdoc.html?arg3=1992&co4=AND&co5=AND&co6=AND&co7=AND&dr=pubyear&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=di%20fazio&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq

and let me know if you get trouble.

Best,

Giuseppe

4
Is there such a norm on any totally disconnected local field?

how can i prove the following statement? In other words, how can i prove existence of the following norm?

Let K be any totally disconnected local field. Then there is an integer q=pr, where p is a fixed prime element of K and r is a positive integer, and a norm ∣⋅∣ on K such that for all x∈K we have ∣x∣≥0 and for each x∈K other than 0, we get ∣x∣=qk for some integer k.

I second the reference suggested by Thong Nguyen Quang Do, it is better than the one I initially suggested.

• José Augusto Molina Garay added an answer:
10
Can a sequence of local diffeomorphisms with attracting periodic points have a limit with only expanding periodic points?

We are in a compact C1 manifold. Consider a sequence (gn) of local C1 diffeomorphisms with a periodic point qn which is attracting, each gcan have only attracting or expanding periodic points and gn converges in Ctopology to f, where f is also a local C1 diffeomorphism but all its periodic points are expanding. Can the existence of that sequence lead a contradiction?

Remark:

A periodic point p of h is attracting (expanding) if all eigenvalues of Dhtau (p) (p) have moduli less (greater) than 1.

In my question there exists a neighbourhood  C1 U of f such that all g in U can't be approximated by local diffeomorphisms  with a saddle type periodic point.

• James F Peters added an answer:
12
How can one characterize the boundary of a convex set?

I am working on a part of a paper related to topological properties of boundary points. It is important for me to realize the topological and algebraic behavior the boundary points of the convex sets. I would be grateful if someone could help me around this issues by giving some ideas or references related to it.

The general question provided in following;

Let B be a closed set in n−dimensional Euclidean space. What other properties B should have in order to be guaranteed that there exist the closed convex set A such that ∂A=B. How about infinite dimensional spaces?

This is a good question.

A good place to start in answering this question is

S. Alexander, M. Ghomi, The convex hull property of non compact hyper surfaces with positive curvature:

http://www.math.uiuc.edu/~sba/nchp.pdf

Theorem 1.1,  page 1, is helpful.   Let M be a metrically complete, positively curved immersed hyper surface with compact boundary $\partial M$ and let

\begin{allgn*}

f: M &\longrightarrow \mathbb{R}^{n+1}\\

C  &= conv( f (\partial M ) ) \mbox{convex hull of image of boundary of M}\\

f(int M) \cap C &= \emptyset

\end{align*}

which says that the image of the interior of M lies completely outside the convex hull of the image of the boundary.    A neat result.     The proof of Theorem is given in Section 3, starting on page 3.   An example of an immersed surface is given in Fig. 1 (see the attached image).

More to the point, consider

U. Eckhardt, W. Scherl, Z. Yu, Representations of plane curves by means of descriptors in Hough space:

http://www.math.uni-hamburg.de/home/eckhardt/HOUGH.pdf

This thesis has a crisp way to denote the boundary of any set M in the plane ( bd M ).  See Chapter 3, starting on page 6, on the classification of descriptors in terms of a continuous, piecewise smooth curve K in the plane.   Then see Chapter 8, starting on page 17, on convex boundary segments.    In ch. 8, starting on page 20, a curve K is reconstructed up to boundary tangents.   This is an important chapter, since it considers the boundery of a convex set and boundary of the complement of a convex set.

2
Which empirical measures, associated to infinite dimensional stochastic processes, satisfy the moderate deviation principle?

In 1998 we proved that the Antoniadis-Carmona processes satisfy the above requirements, in connection with the tunneling effect. Am interested in the current situation.

Alternatively, the infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion coefficient also satisfy the requirements, see: http://arxiv.org/pdf/1503.05322.pdf

3
Are there characterization of harmonic gradient mapping from the unit ball onto itself in 3-space?

In communication between V. Zorich and the author, the
question was asked to find examples  of harmonic gradient mapping from the unit ball onto itself in 3-space. For example, if $u=x^2 +y^2 - 2 z^2$, then $f= \nabla u=(2x,2y,-4z)$ is injective harmonic gradient mapping from
$\mathbb{B}^3$ onto the ellipsoid.

If $u$ is real-valued function such that $f= \nabla u=(x,y,z)$, then $u= x^2/2 +y^2/2 +z^2/2 +c$.

In particular, $Id$ is not harmonic gradient mapping.

In complex plane, if $u$ is real-valued harmonic function, then $u_z=\frac{1}{2}(u'_x -u'_y)$ is analytic function and therefore
$\nabla u=\overline{F}$, where $F= 2 u_z$ is analytic function.

Znam da je pretpostavka jaka.

• Qingping Zeng asked a question:
Open
Any advice on isolated points of the approximate point spectrum of a bounded operator?

Let $X$ be a complex Banach space and $T$ be a bounded operator acting on $X$. Let $\sigma(T)$ and $\sigma_{ap}(T)$ denote the spectrum and approximate point spectrum of $T$, respectively. Let $\lambda \in \sigma(T)$. It is classical that, with the aid of the spectral projection, $\lambda$ is isolated in $\sigma(T)$ if and only if $T - \lambda$ can be decomposed as a direct sum of a quasinilpotent operator and a invertible one. Now my nature question is as follows: Let $\lambda \in \sigma_{ap}(T)$. Is it true that $\lambda$ is isolated in $\sigma_{ap}(T)$ if and only if $T - \lambda$ can be decomposed as a direct sum of a quasinilpotent operator and a bounded belowness one? Here we say that an operator is bounded below if it is injective and its range is closed. It is also nature to find the answer for other spectra (eg. left spectrum, surjective spectrum, right spectrum, essential spectrum,...). Thank you!

• Gerardo Maximiliano Mendez added an answer:
6
Can anyone help with a membership function question?

Can I use different membership functions in Fuzzification layer for ANFIS type model? e.g. U1 = Bell-shape and U2 = Triangular, so that the output of this layer will be wi = U1 x U2. In most of the papers they used the same type of membership function.

Yes you can do that, but there is not a good reason to do that. On the other side, you can do anything using C or C++.

• Daoud Bshouty Bshouty added an answer:
1
Is there any appropriate version of the Rad\'{o}-Kneser-Choquet theorem (RKC-Theorem) in space?

Let $\gamma$ be a closed Jordan curve and $f_0 : S^1 \overset{\text{onto}}{\longrightarrow} \gamma$. The basic question that they address in this paper is
under which conditions on $f_0$ we have that $F=P[f_0]$ is a
homeomorphism of $\mathbb{B}$ onto $D$, where $D$ denotes the
bounded open, simply connected set for which $\partial D = \gamma$. The fundamental benchmark for this issue is a classical
theorem, first conjectured by T. Rad\'{o} in 1926 , which was
proved immediately after by H. Kneser [12], and subsequently
rediscovered, with a different proof, by G. Choquet. Let us recall the result.

Theorem 1.1 (T. Rad\'{o} H. Kneser G. Choquet ) If $D$ is convex, then $F$ is a
homeomorphism of $\mathbb{B}$
onto $D$.

Laugesen (see Duren's book}, p. 54-56) constructed a homeomorphisam
of the the unit sphere in $\mathbb{R}^3$ onto itself, whose
Poisson extension to a vector-valued harmonic function fails to
be univalent in the ball.

None that I am aware of.

• Gul E Hina Aslam added an answer:
3
Minkowski type inequality in Banach algebras
Under which circumstances it is true that ||(A+B)ⁿ||¹/ⁿ ≤ ||Aⁿ||¹/ⁿ + ||Bⁿ||¹/ⁿ for elements A and B in a Banach algebra and a natural number n?

How can you even talk about the relation $\leq$ in a Banach algebra? Unless you mention it as an ordered Banach algebra or Banach lattice algebra.

4
Does a Hölder continuous function have a bounded fractional derivative?
Some nowhere differentiable functions are fractional differentiable and comply with the Hölder condition.

By example, boundedness of the Caputo derivative of f at every interval, implies Hölder continuity of f in the same order

1
Can someone help with the interpolation between the Bloch and the Moebius invariant H^1 space?

The space $H'$ consists of functions $f$ analytic in the unit disc such that $f'\in H^1$. The Bloch $B$ space is defined by the requirement

$$\sup (1-|z|^2)|f'(z)| <\infty,$$

Endowed with the obvious semi-norms, both $H'$ and $B$ are M-invariant in the strong sense: $\|f\circ \m\|=\|f\|,$ for every Moebius transformation $m$ of the disc.

The question is: What is the real or complex interpolation space $(H',B)_...$ ?

#### About Functional Analysis

Geometry of Banach space, Infinitely dimensional functional theory