Functional Analysis

Functional Analysis

  • Jean-Louis Honeine added an answer:
    Is it appropriate (correct) to calculate the median instantaneous frequency (scale) of a signal in time?

    I would like to quantify the changes in frequencies (scales to be more precise since I am using continous wavelet transformation) coefficient in time of signals composed of low frequencies 0.1-4 Hz. The signal goes from one steady state to another. Is it correct to calculate the median frequencies at each bin and see how the median frequency evolves in time? I obviously would loose some resolution since I would have to calculate my median frequency in bins that have the same size, so I have to accept the size of the biggest bin (of the lowest frequencies).  It would be really helpful if somebody has a better idea or can send me some references to read. Thank you.

    Jean-Louis Honeine · University of Pavia

    Thank you. I am getting so far very good results taking into consideration the limitations.  I am actually calculating the moving median frequency of a lot of signals. Now I wish to average them which hopefully will give a better result.

  • David Blazquez-Sanz added an answer:
    Is it possible to prove the following idea?

    Let ||x||_1\leq ||y||_1. Is it possible to say ||x||_p\leq||y||_p  for 1<p? In another way, the inequality in l1 is protect in lp or not?

    David Blazquez-Sanz · National University of Colombia

    As previous answers suggest, you can prove that for given p, e < K, these is an sequence x with ||x||_1 = e and ||x||_p > K. You can take the sequence:

      e/n , e /n , ...., e/n, 0, 0, 0, 0, ...

      providing that n^(p/(p-1)) > K/e. Thus, there is no possible condition, at least using only the || ||_1 norm. 

  • Dr. Indrajit Mandal added an answer:
    Is it possible to apply the method from the paper below to other functional or probabilistic situations?

    It worked well for level-3 large deviations, and for functional laws of the iterated logarithm in infinite dimensions.

    Dr. Indrajit Mandal · Rajiv Gandhi Institute of Technology, Bangalore

    Hi friend


    I have gone through the paper.

    I feel you can implement the method for other probabilistic situations. 

    Best regards

    Dr.indrajit MAndal

  • Octav Olteanu added an answer:
    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.

    Octav Olteanu · Polytechnic University of Bucharest

    @Rogier Brussee. Of course, your assertion is true (it is a well-known result in functional analysis). An other well-known nontrivial surprising result in Banach space theory is the famous Theorem of James: "Let A be a bounded and weakly closed subset of the real Banach space X. If every continuous linear functional on X attains its supremum on A, then A is weakly compact". In a way, this is partially a (hard) converse of the following obvious assertion: "Every continuous linear functional on X attains its supremum on a weakly compact subset A". For the Theorem of James, see R. B. Holmes, "Geometric Functional Analysis and its Applications", Springer, 1975.

  • Xiangnan Guan added an answer:
    Can someone help me with in vitro functional analysis of exhausted CD8+ T cells?

    PD-1/PDL-1 blockage restores functions to exhausted CD8+ T cells. In the literature, I always see that people isolate CD8+ T cells with high-PD-1 expression, characteristic of exhaustion. These isolated cells are cultured in vitro, and have impaired functions, but have improved functions when blockade antibody (anti-PD-1 or anti-PDL-1) is added. My question is: Isolated PD-1 high CD8+ T cells, when cultured in vitro, how do they maintain the exhausted phenotypes given that no PDL-1 ligand is added to the medium? If there is no ligand, PD-1 is not activated, then why the anti-PD-1 or anti-PDL-1 even matter? Thanks a lot. 

    Xiangnan Guan · The Ohio State University


    Thanks for your help! I will read that review. Thanks.

  • George Stoica added an answer:
    Is there a classification of all subspaces of measurable functions for which weak-convergence implies convergence in measure?
    I ask because this seems to allow one to replace compactness with weak-sequential-compactness(and weak-compactness for Banach spaces of measurable functions) in some arguments. For example, if X is such a space of measurable functions then any weakly-sequentially-compact subset will have the property that convergence in measure is equivalent to convergence in X.
    George Stoica · University of New Brunswick

    Dear Geoff,

    Long shot, but one never knows: Riesz' theorem asserts that norm bounded sequences in L^p (p>1) contain a weakly convergent subsequence in the same L^p. It is also known that, norm boundedness in L^p (p>1) and a.e. convergence (or in measure) imply the weak convergence in L^p of the entire sequence. However, there are classes of random variables, such as independent & identically distributed, exchangeable, spreadable (introduced by Kallenberg), for which norm boundedness in L^p (p \geq 1) implies weak convergence in L^p of the entire sequence, without the requirement of a.e. convergence or in measure.



  • George Stoica asked a question:
    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.

  • George Stoica asked a question:
    Are there any Baum-Katz-type results proved for non-commutative random sequences?

    E.g., for free random sequences, in non-commutative L^p or Orlicz spaces.

  • Andrew J King added an answer:
    How to design primers from AT rich region?

    Hi All. I would like to isolate a full length cds taken from NCBI for functional analysis. I am designing a set of primer pairs beginning from the start and stop codon. I face a problem in designing the reverse primer. Towards the end of the template till stop codon, it is basically AT-rich region. Thus, causing the %GC to be low (20% to 30%) in my reverse primer. May I ask what are some of the ways to increase the %GC? Is it possible to add in additional GC onto the 3'end of my primer? Sorry for the amateur questions as I'm still new to molecular biology. Thanks in advance.

  • Mihai Prunescu added an answer:
    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.

    Mihai Prunescu · Institute of Mathematics of the Romanian Academy

    You must make a quotient operation in order to get back to a field. Take for example the field R* of nonstandard analysis. This is RN factorised by a relation x # y on infinite tuples if and only if  {i | xi = yi}  belongs to a maximal nonprincipal ultrafilter over N. Such an ultrafilter can be generated by the cofinite subsets of N, but contains more subsets. The field R* contains R, infinite small elements (around 0) and infinite elements, like the class of (1, 2, 3, 4, ...., n, ....) . 

  • Yazid Meftah added an answer:
    Can anyone suggest how to find the transition state with different functional?

    I have a problem to find transition state with PBE functional. I have already used the scan process and use the opt=(ts,calcfc,noeigentest) and redundant option 4 5 D to optimize the transition state. But I can't find any imaginary frequency. Then I tried use qst3 calculation to find the transition state, but I found the transition state with BLYP functional. Is there any way to find out the transition state or some suggestion to do this calculation?

    Yazid Meftah · Université de Biskra

    thank you professor

    the problem, that I find TS with M06 PBE0 BLYP B3LYP, other TS with PBEPBE BP86.
    difference in forming bound

  • Pankaj Kumar Singh added an answer:
    The space of Riemann integrable functions
    Let R[a,b] denote the space of all real (or complex) valued functions that are Riemann integrable on [a,b]. Endowed with the sup-norm (not ess sup) R[a,b] is a Banach space.

    What is known on the Banach space properties of R[a,b] ? It is clearly non-separable. Does it possess the predual? Is there a space X such that the dual of X is isomorphic to R[a,b]? What is R[a,b]/C[a,b]?
    Pankaj Kumar Singh · Motilal Nehru National Institute of Technology

    could it be a metric space?

  • Vladimir Kadets added an answer:
    Is an explicit form of maximal ideals/homomorphism known of C-star algebra with almost periodic functions?

    Due to the Gelfand-Neumark Theorem algebra of almost periodic functions in the sense of Bohr is isometrically isomorphic to the algebra of complex continuous functions on the space of maximal ideals of the first algebra. This space is compact and is known as Bohr's compactification of the real line. I cannot find any explicite form of elements of this space; only abstract description, Hewitt, Ross monograph for example.

    Vladimir Kadets · V. N. Karazin Kharkiv National University

    You did not precise if you consider the algebra of CONTINUOUS almost periodic functions, as it is supposed usually. If so, I would suspect that a typical complex homomorphism, that is not an evaluation functional, reflects behavior of elements at infinity. Say, if one takes an arbitrary tending to infinity sequence tn  of reals, and an arbitrary ultrafilter U on N, then the functional F that maps every bounded almost periodic function f to the U-limit of f(tn) is a complex homomorphism. If the algebra that you consider contains also discontinuous functions, there will be also homomorphisms similar to "limit at a point": just take in the previous  example tn tending to some finite point.

  • Myroslav Sparavalo added an answer:
    Do you have questions on the discussion announced on 04/27/14 related to Lyapunov's Functions and Concept of Stability?
    To see the invitation, please click the link
    The questions can be divided into three groups. The first group consists of the questions, to which the author has the answers directly resulting from the paper. The second group is composed of the questions, for which he has only conjectures or guesstimates. The third group represents the questions, the answers to which the author has no ideas about. The questions that interest me particularly as the author are as follows:
    1. What is the mathematical nature (algebraic, geometrical, topological, etc.) of Lyapunov functions? What physical interpretations can be given to them?
    2. Are there any direct or indirect relations between Lyapunov functions, first integrals and the right-hand sides of systems of differential equations? If yes, then what kinds they are?
    3. How to approach a nonlinear non-autonomous system of the most general form by means of the second Lyapunov method? What and why do we need in the very beginning to know and how to get on with the system from this initial point further using the general procedure of utilization of Lyapunov functions? Is the procedure workable enough to crack the concrete practical problems of stability despite the presence of general nonlinearity, non-autonomousness, structural and coefficient uncertainties?
    4. What are advantages and disadvantages of the utilization of Lyapunov functions at the investigation of the stability of nonlinear non-autonomous systems in the light of the results of the paper?
    5. What role if any does the Lyapunov concept of stability play for quantum-mechanical and biochemical physical processes? Can it be considered one of the fundamental principles of the creation, formation and existence of living and nonliving matter?
    Myroslav Sparavalo · NYC Transit Authority

    David, I am awfully sorry for the very belated reply. Only today I've found your comment. First of all, the main objects of my research are NOT equilibrium or singular points. They have been studied quite well. But what badly lacks the attention of researchers are non-singular orbits describing various transient and unsteady processes, for example, hypersonic airplane maneuvering in dense atmospheric layers. In geometric terms its flight trajectory is an arbitrary curve. Second, the goal of the research is NOT to invent a general method of constructing analytical formulae for V. The prime objective is to show what is the intrinsic mathematical nature of Lyapunov functions in the structure of a given system of ODE and how they are related to its first integrals, what are their weak points and what advantages give us Poincare's approach over Lyapunov functions in stability problems. As to "global", the concept of foliations is used in the paper. This means that by the definition of an one-codimensional foliation, which leaves are the elementary “bricks” of phase spaces, that is by default we tackle the problems of stability globally. If you have any further questions, please don't hesitate to ask. Once again, please accept my apologies for the very belated reply.

  • Muhammad Zubair Ahmad added an answer:
    How is wavelet transform method used for an edge image in order to find the discontinuous point in a line?

    need matlab/opencv 2.1 coding examples

    Muhammad Zubair Ahmad · National University of Science and Technology

    An other good source for you would be the book:

    sparse image and signal processing by starck

    you will find the CH 2 help full

  • Oleg Reinov added an answer:
    How can I calculate the determinant of a linear operator defined on Hilbert spaces?

    I want to calculate the determinant of a linear operator L from H to H, where H is a Hilbert space: y=Lx. In my particular problem L is bounded, differentiable end invertible.

    Thank you

    Oleg Reinov · 1. St. Petersburg State University & 2. ASSMS of CGU in Lahore, Pakistan

    All about the determinants - see

    "I. Gohberg, S. Goldberg, N. Krupnik: Traces and Determinants of Linear Operators, Birkhauser Verlag, Basel-Boston-Berlin (2000)."

  • Zeev Sobol added an answer:
    Are there any classes of norm bounded functions in L^p that converge weakly in L^p?

    Riesz' theorem asserts that norm bounded sequences in L^p (p>1) contain a weakly convergent subsequence in the same L^p. It is also known that, norm boundedness in L^p (p>1) and a.e. convergence (or in measure) imply the weak convergence of the entire sequence in L^p. However, there are classes of random variables, such as i.i.d., exchangeable, spreadable (Kallenberg) for which norm boundedness in L^p (p \geq 1) implies weak convergence of the entire sequence in L^p, without the requirement of a.e. convergence or in measure. Are there other such classes of functions? The usual connection with the Banach-Saks property and the strong law of large numbers does not seem to provide an answer.

    Zeev Sobol · Swansea University

    I apologize for a tautology, but since every weakly converegent sequence is necessary norm-bounded, you will not have a very much better necessary and sufficient condition than the definition itself. However, you may generate a lot of such conditions using the following idea: a sequence f_n norm-bounded in L^p and such that f_n(g) converges for every g from a set spanning a weakly dense set in L^{p'}, converges weakly. My favorite is, g being indicators of sets generating the underlying field (convergence of integrals). Alternatively, you may take g being smooth compactly supported functions (convergence in distributions). 

  • Elena Lebedeva added an answer:
    Can anyone prove how Legendre wavelet forms an orthonormal basis for L^2(R)?

    Please explain......

    Thanks in advance.............

    Elena Lebedeva · St. Petersburg State Polytechnical University

    It can be done using multiresolution analysis starting with a scaling mask (a low-pass filter). It is a standard procedure, and it can be found in any classical textbook on wavelet theory. 

  • Oleg Reinov added an answer:
    I am looking for a brief, well understanding and new book in real analysis and measure theory. Could you please guide me?

    I want to teach real analysis course this semester. I want to teach more about functional analysis and less about real analysis. I need a brief and new book. Could you please guide me?

    Oleg Reinov · 1. St. Petersburg State University & 2. ASSMS of CGU in Lahore, Pakistan

    Boris Makarov, Anatolii Podkorytov (auth.) Real Analysis- Measures, Integrals and Applications 2013 - Springer, Universitext.

  • Dona Strauss added an answer:
    Is every neighborhood of identity in an abelian topological group absorbing?

    Let G be an abelian topological group and U be a neighborhood at identity. Do we have the identity that the union of nU in which n is a positive integer equals to whole of G? We have the same situation in a topological vector space. I am interested to know if it is true for topological groups, too.

    Dona Strauss · University of Leeds

    There appears to be some confusion about the notation nU and U^n. I am assuming

    that, in an abelian group, nU denotes the set of elements nu with u in U; i.e. the set of elements of the form u+u+...+u, where u is in U and there are n terms in the sum. In a general group, U^n denotes the set of products of the form u_1u_2...u_n, with each u_i in U. If one is using additive notation, this is the set of sums of the form u_1+u_2+...+u_n, with each u_i in U.

  • Christopher Jason Larsen added an answer:
    Is it possible to show explicitly the function $\varphi_{x}\in C^{0,1}(R^{n-1})$ that is related with each $x\in\partial B_{r}(0)$?

    We know that an open ball $B_{r}(0)\subseteq R^{n}$ is a smooth domain. It follows that this is a Lipschitz domain. Is it possible to show explicitly the function $\varphi_{x}\in C^{0,1}(R^{n-1})$ that is related with each $x\in\partial B_{r}(0)$?

    Christopher Jason Larsen · Worcester Polytechnic Institute

    You could use identical functions, oriented correctly, so all Lipschitz constants are the same. You could also make the constant arbitrarily small by taking small arcs. 

  • Hong-Bin Chen added an answer:
    Is there any criterion for k-positivity of an operator acting on a C*-algebra?

    Suppose that H is an n-dimensional Hilbert space. B(H) is the set of bounded operators on H forming a C*-algebra. Given a linear mapping L : B(H) -> B(H).

    Choi's theorem helps us determine whether L is a completely positive mapping. Is there any similar criterion on the k-positivity of L?

    Hong-Bin Chen · National Cheng Kung University

    Thanks a lot!!

  • Elena Martín Peinador added an answer:
    Can we have Heine-Borel property in topological groups?

    We know that Heine-Borel property and related theorems for topological vector spaces. Do we have similar notions for abelian topological groups? For example when a topological group has this property or not? Can you introduce me a reference about that?

    Elena Martín Peinador · Complutense University of Madrid

    To James Peters

    Is available  this Thesis:  X. Shi, Graev Metrics and Isometry Groups of Polish Ultrametric Spaces, Ph.D. thesis, University of North Texas, 2013:?.

  • Wang he added an answer:
    Can someone help me with DNA polymerase activity analysis?
    I've expressed Taq polymerase in E.coli. I have found a few reports for its purification also using sephadex columns. Next, I need to check its efficacy preferably by methods other than PCR. Can anybody suggest such functional tests?
    Wang he · Beijing Genomics Institute

    Hi!   Have you solved this problem? I have met the same problem , and I tried a  method according to a paper which used Picogreen to maesure the dsDNA. Unfortunately, the results could not be stable, so I want to know much about the method measuring radioactivity, such as the amount of DNA ,the reaction system. How many is the least amount  of radioactivity that can be detected. Waiting for your reply . Thank you ! 

  • James F Peters added an answer:
    Anyone familiar with the Arzela-Ascoli Theorem?

    n the proof of the Arzela-Ascoli Theorem, it seems that only pointwise boundedness rather than uniform boundedness is needed. Is it right?

    James F Peters · University of Manitoba

    This is a good question.

    A general form the Arzela-Ascoli theorem is given in

    J. Conradie, J. Swart, A general duality result for precompact sets, Indag. Mathem, N.S., 1 (1990), no. 4, 409-416:

    See Theorem 3.2 (with proof), page 413.

    A basic introduction to the Arzela-Ascoli theorem is given in

    F. Botelho, Topics of Functional Analysis, Calculus of Variations and Dualilty, Acad. Pubs., 1991:

    See Section 1.7, starting on page 20 (Theorem 1.7.2).   The proof of Theorem 1.7.2 is starts on pare 21, extending to page 22.

  • Vittorino Pata added an answer:
    Is there any metric in which the set of real numbers R is compact?
    I want to check those metrics in which the set $R$ of real numbers is compact! Note that if that metric is induced from a norm, then R with that metric is not compact.
    Vittorino Pata · Politecnico di Milano

    Let C be a compact set of the same cardinality of R (say, C=[0,1]), with the metric d. Let f:R->C a bijection. Then define on R the metric D(x,y):=d(f(x,f(y)). This D should be compact.

  • Carsten Trunk added an answer:
    If A is a subspace of a normed space X and \{x_n\} is a sequence in A such that \{x_n\} converges to z, does z \in A?
    Please note that A is a subspace.
    Carsten Trunk · Technische Universität Ilmenau

    I have to support Geoff. Quite often (especially in Russian books or in papers from authors with an Russian background) a subspace is always understood to be CLOSED. And this is just by definition. Unfortunately, often this is not stated explicitely...

    On the opposite, in the western literature (like in the US) a subspace generally is just a subset of a vector space which itself is again a vector space but which need not to be closed.

    And yes, you are right, this is annoying.

  • Wojciech Kalas added an answer:
    Which is the best way to characterize an oncogene?

    Hi, now I´m working with a protein wich behaives as a oncogene ( after expression microarrays, functional analysis showed that promotes proliferation, growth and avoid apoptosis). I have some ideas about in vitro studies but, as we know,  two heads are better than one. Thanks 

    Wojciech Kalas · Polish Academy of Sciences

    Stable overexpression and trying to grow tumour in mice is definitely the best method, but is costly and laborious.

    Before, you can get some answer by introducing it to non-tumorigenic cell line and performing colony formation assay or trying to grow them to confluence, to find out if the cells losed the contact inhibition - common feature of cancer cells. You can try also compare expressing and non-expressing cell lines.It will be only circumstantial, but it can give you a hint if it behaves as oncogene and will ease decision to invest more time and money.

  • Parasuraman Basker added an answer:
    Is it appropriate to give the partial answer to an open problem in the literature?

    Is solving an open problem in mathematics with some extra conditions, that do not make it trivial, considered appropriate?

    Parasuraman Basker · California Department of Public Health

    Dear Sir

     How could I express my joy as you have given extra energies to explore further in mathematics. I have been initiated to learn mathematics from its alphabet onwards as I am the pure Biologist. But one thing I firmly confident that mathematics alone give ultimate solution like E= mc squared. In this regard please kindly suggest to read basic mathematics books.

    sincerely Yours


  • Reza Mehrnia added an answer:
    Hi all, how can I convert scale axis to frequency in wavelet transform?

    I wanna to produce time frequency representation after using wavelet  transform.

    Reza Mehrnia · Payame Noor University

    Hi, you should transform all values into frequency responses by spatial based software (such as Geo-soft , Surfer and Arc-Soft Families) according to Wavelet algorithm facilities. therefore it is automatically caused changing your scale bar into frequency measurements..

About Functional Analysis

Geometry of Banach space, Infinitely dimensional functional theory

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