Functional Analysis And Wavelets

We consider the problem of determining the Fourier integral in the Hilbert space of square integrable func- tions. Fourier integral is the scalar product of two functions belonging to the Hilbert space of square integrable functions and the Hilbert space of almost periodic functions. Scalar product for different Hilbert spaces defined at the intersection of these spaces, which contains only one zero element. Therefore, the Fourier integral is not defined in the Hilbert space of square integrable functions with nonzero norm.

The Kac-Baker model describes a 1-dim. classical lattice spin system with exponentially fast decaying two body interaction. The model was introduced by M. Kac and G. Baker to investigate the phenomenon of phase transition in systems with weak long-range interactions like van der Waals gas. Ruelle's dynamical zeta function for this model can be expressed in terms of Fredholm determinants of two transfer operators and hence is a meromorphic function. One of the two operators, found by M. Kac, is an integral operator with symmetric kernel acting in the Hilbert space of square integrable functions on the line. The other one is Ruelle's transfer operator acting in some Banach space of holomorphic observables of the system. In this paper we show how the Kac operator can be explicitly related basically through the Segal-Bargmann transform to the Ruelle operator restricted to a certain Segal-Bargmann space of entire functions in the complex plane. This allows us to show that Ruelle's zeta function for the Kac-Baker model has infinitely many non-trivial zeros on the real axis. In a special case we can show the existence of also infinitely many trivial zeros on the line Re s = ln 2 in the complex s-plane. Hence some kind of Riemann hypothesis seems to hold for this dynamical zeta function.

Spectrum of the Kerzman–Stein operator for a family of smooth regions in the plane

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.

We introduce the method of Geodesic Principal Component Analysis (GPCA) on the space of probability measures on the line, with finite second moment, endowed with the Wasserstein metric. We discuss the advantages of this approach, over a standard functional PCA of probability densities in the Hilbert space of square-integrable functions. We establish the consistency of the method by showing that the empirical GPCA converges to its population counterpart, as the sample size tends to infinity. A key property in the study of GPCA is the isometry between the Wasserstein space and a closed convex subset of the space of square-integrable functions, with respect to an appropriate measure. Therefore, we consider the general problem of PCA in a closed convex subset of a separable Hilbert space, which serves as basis for the analysis of GPCA and also has interest in its own right. We provide illustrative examples on simple statistical models, to show the benefits of this approach for data analysis. The method is also applied to a real dataset of population pyramids.

Harmonic analysis on a locally compact group is the study of certain spaces and algebras of functions and measures defined on the group. The function spaces in question are defined by measurability and integrability conditions; manipulation of the relevant measures requires delicate techniques from measure and integration theory. Also, the fundamental theorems of the theory of representations of locally compact groups [see §22] depend wholly upon the study of a certain space of square integrable functions. Thus the theories of measure and integration play a vital role in the present book.

The functional linear model with scalar response is a regression model where the predictor is a random function defined on some compact set of ℝ and the response is scalar. The response is modelled as Y=Ψ(X)+ɛ, where Ψ is some linear continuous operator defined on the space of square integrable functions and valued in ℝ. The random input X is independent from the noise ɛ. In this paper, we are interested in testing the null hypothesis of no effect, that is, the nullity of Ψ restricted to the Hilbert space generated by the random variable X. We introduce two test statistics based on the norm of the empirical cross‐covariance operator of (X,Y). The first test statistic relies on a χ2 approximation and we show the asymptotic normality of the second one under appropriate conditions on the covariance operator of X. The test procedures can be applied to check a given relationship between X and Y. The method is illustrated through a simulation study.

We provide a new upper bound for sampling numbers (g_n)_{nin mathbb {N}} associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants C,c>0 (which are specified in the paper) such that gn2≤Clog(n)n∑k≥⌊cn⌋σk2,n≥2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} g^2_n \\le \\frac{C\\log (n)}{n}\\sum \\limits _{k\\ge \\lfloor cn \\rfloor } \\sigma _k^2,\\quad n\\ge 2, \\end{aligned}$$\\end{document}where (sigma _k)_{kin mathbb {N}} is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding mathrm {Id}:H(K) rightarrow L_2(D,varrho _D). The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of H^s_{text {mix}}(mathbb {T}^d) in L_2(mathbb {T}^d) with s>1/2. We obtain the asymptotic bound gn≤Cs,dn-slog(n)(d-1)s+1/2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} g_n \\le C_{s,d}n^{-s}\\log (n)^{(d-1)s+1/2}, \\end{aligned}$$\\end{document}which improves on very recent results by shortening the gap between upper and lower bound to sqrt{log (n)}. The result implies that for dimensions d>2 any sparse grid sampling recovery method does not perform asymptotically optimal.

ABSTRACTWe study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by homogenized solution, the Green’s function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential. These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations.

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

Diffusion multi-compartment models (MCM) allow for a fine and comprehensive study of the white matter microstructure. Non linear registration of MCM images may provide valuable information on the brain e.g. through population comparison. State-of-the-art MCM registration however relies on pairing-based similarity measures where the one-to-one mapping of MCM compartments is required. This approach leads to non differentiabilties or discontinuities, which may turn into poorer registration. Moreover, these measures are often specific to one MCM compartment model. We propose two new MCM similarity measures based on the space of square integrable functions, applied to MCM characteristic functions. These measures are pairing-free and agnostic to compartment types. We derive their analytic expressions for multi-tensor models and propose a spherical approximation for more complex models. Evaluation is performed on synthetic deformations and inter-subject registration, demonstrating the robustness of the proposed measures.

The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new structures and criteria. One of them is a sufficient condition that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper aims at identifying functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In order to tackle (ii) in the future, we give some expressions of the scalar products. New and remarkable structures arise for the Gram matrix, in particular moment matrices for a suitable weight that may be the squared $\Xi$-function for instance.

We construct a family of generalized coherent states attached to Landau levels of a charged particle moving in the Poincaré disc under a perpendicular uniform magnetic field. The corresponding coherent state transforms enable us to connect, by an integral transform, spaces of bound states of the particle with the space of square integrable functions on the real line. The established connection provides us with a new way to obtain hyperbolic Landau states.

The three-body Schrodinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length \({\kappa + 1, \kappa = 0, 1, \ldots}\) As a result, the solutions to the three-body Schrodinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve—in these subalgebras—the radial Schrodinger equation in three dimensions with the inverse power potential of the form \({r^{-{\kappa}-1}}\) . As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with \({\kappa = 0}\) , are reduced to the well-known eigenvalues of bound states at threshold.

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.