Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets

In this paper we study the general reconstruction of a compactly supported function from its Fourier coefficients using compactly supported shearlet systems. We assume that only finitely many Fourier samples of the function are accessible and based on this finite collection of measurements an approximation is sought in a finite dimensional shearlet reconstruction space. We analyze this sampling and reconstruction process by a recently introduced method called generalized sampling. In particular by studying the stable sampling rate of generalized sampling we then show stable recovery of the signal is possible using an almost linear rate. Furthermore, we compare the result to the previously obtained rates for wavelets.     

Estimates for dimensions of spaces of siegel modular cusp forms

Every Siegel modular form has a Fourier-Jacobi expansion. This paper provides various sets of Fourier coefficients whose vanishing implies that the associated cusp form is identically zero. We call such setsestimates because in the Fourier series case, an upper bound for the dimension of the vector space of cusp forms is provided by the cardinality of the set. Our general estimates have, among others, those estimates of Siegel and Eichler as corollaries. In particular, one new corollary of our general estimates appears to be superior for computational purposes to all other known estimates. To illustrate the use of this corollary, we prove the known result that the theta series of the latticesD ₁₆ ⁺ andE ₈ ⊕E ₈ are the same in degreen = 3 by computing just one Fourier coefficient. 1991 Mathematics Subject Classification. 11F46 (11F30, 11F27).     

Convergence of fourier series with respect to multiplicative systems and the p-fluctuation continuity modulus

We obtain some sufficient conditions for convergence of series of the Fourier coefficients with respect to multiplicative systems for functions of bounded p-fluctuation. In some cases we establish the unimprovability of these conditions.     

The polyharmonic local sine transform: A new tool for local image analysis and synthesis without edge effect

We introduce a new local sine transform that can completely localize image information both in the space domain and in the spatial frequency domain. This transform, which we shall call the polyharmonic local sine transform (PHLST), first segments an image into local pieces using the characteristic functions, then decomposes each piece into two components: the polyharmonic component and the residual. The polyharmonic component is obtained by solving the elliptic boundary value problem associated with the so-called polyharmonic equation (e.g., Laplace's equation, biharmonic equation, etc.) given the boundary values (the pixel values along the boundary created by the characteristic function). Subsequently this component is subtracted from the original local piece to obtain the residual. Since the boundary values of the residual vanish, its Fourier sine series expansion has quickly decaying coefficients. Consequently, PHLST can distinguish intrinsic singularities in the data from the artificial discontinuities created by the local windowing. Combining this ability with the quickly decaying coefficients of the residuals, PHLST is also effective for image approximation, which we demonstrate using both synthetic and real images. In addition, we introduce the polyharmonic local Fourier transform (PHLFT) by replacing the Fourier sine series above by the complex Fourier series. With a slight sacrifice of the decay rate of the expansion coefficients, PHLFT allows one to compute local Fourier magnitudes and phases without revealing the edge effect (or Gibbs phenomenon), yet is invertible and useful for various filtering, analysis, and approximation purposes.     

Multidimensional pseudo-spectral methods on lattice grids

When multidimensional functions are approximated by a truncated Fourier series, the number of terms typically increases exponentially with the dimension s. However, for functions with more structure than just being L²-integrable, the contributions from many of the Ns terms in the truncated Fourier series may be insignificant. In this paper we suggest a way to reduce the number of terms by omitting the insignificant ones. We then show how lattice rules can be used for approximating the associated Fourier coefficients, allowing a similar reduction in grid points as in expansion terms. We also show that using a lattice grid permits the efficient computation of the Fourier coefficients by the FFT algorithm. Finally we assemble these ideas into a pseudo-spectral algorithm and demonstrate its efficiency on the Poisson equation.     

Maximal regularity of evolution equations on discrete time scales

We consider the question of Lp-maximal regularity for inhomogeneous Cauchy problems in Banach spaces using operator-valued Fourier multipliers. This follows results by L. Weis in the continuous time setting and by S. Blunck for discrete time evolution equations. We generalize the later result to the case of some discrete time scales (discrete problems with nonconstant step size). First we introduce an adequate evolution family of operators to consider the general problem. Then we consider the case where the step size is a periodic sequence by rewriting the problem on a product space and using operator matrix valued Fourier multipliers. Finally we give a perturbation result allowing to consider a wider class of step sizes.     

K. Saito's Conjecture for nonnegative eta products and analogous results for other infinite products

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z=1 case is an identity for the generating function for p-cores due to Klyachko [A.A. Klyachko, Modular forms and representations of symmetric groups, J. Soviet Math. 26 (1984) 1879-1887] and Garvan, Kim and Stanton [F. Garvan, D. Kim, D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990) 1-17]. A number of other infinite products are shown to have nonnegative coefficients. In the process a new generalization of the quintuple product identity is derived.     

NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE AND UNIQUENESS OF PERIODIC SOLUTIONS OF NEUTRAL LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS：A METHOD BY FOURIER SERIESES

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Improved sparse fourier approximation results: faster implementations and stronger guarantees

We study the problem of quickly estimating the best k-term Fourier representation for a given periodic function f: [0, 2π] → ?. Solving this problem requires the identification of k of the largest magnitude Fourier series coefficients of f in worst case k ² · log ^O(1) N time. Randomized sublinear-time Monte Carlo algorithms, which have a small probability of failing to output accurate answers for each input signal, have been developed for solving this problem (Gilbert et al. 2002, 2005). These methods were implemented as the Ann Arbor Fast Fourier Transform (AAFFT) and empirically evaluated in Iwen et al. (Commun Math Sci 5(4):981–998, 2007). In this paper we present a new implementation, called the Gopher Fast Fourier Transform (GFFT), of more recently developed sparse Fourier transform techniques (Iwen, Found Comput Math 10(3):303–338, 2010, Appl Comput Harmon Anal, 2012). Experiments indicate that GFFT is faster than AAFFT. In addition to our empirical evaluation, we also consider the existence of sublinear-time Fourier approximation methods with deterministic approximation guarantees for functions whose sequences of Fourier series coefficents are compressible. In particular, we prove the existence of sublinear-time Las Vegas Fourier Transforms which improve on the recent deterministic Fourier approximation results of Iwen (Found Comput Math 10(3):303–338, 2010, Appl Comput Harmon Anal, 2012) for Fourier compressible functions by guaranteeing accurate answers while using an asymptotically near-optimal number of function evaluations.     

On Padé-type model order reduction of J-Hermitian linear dynamical systems

A simple, yet powerful approach to model order reduction of large-scale linear dynamical systems is to employ projection onto block Krylov subspaces. The transfer functions of the resulting reduced-order models of such projection methods can be characterized as Padé-type approximants of the transfer function of the original large-scale system. If the original system exhibits certain symmetries, then the reduced-order models are considerably more accurate than the theory for general systems predicts. In this paper, the framework of J-Hermitian linear dynamical systems is used to establish a general result about this higher accuracy. In particular, it is shown that in the case of J-Hermitian linear dynamical systems, the reduced-order transfer functions match twice as many Taylor coefficients of the original transfer function as in the general case. An application to the SPRIM algorithm for order reduction of general RCL electrical networks is discussed.     

Functional conditions for the convergence of Fourier series with respect to general orthonormal systems

In this paper we establish some functional conditions for the convergence of the Fourier series for functions from the class C(0, 1) with respect to general orthonormal systems.     

A Consistent and Stable Approach to Generalized Sampling

We consider the problem of generalized sampling, in which one seeks to obtain reconstructions in arbitrary finite dimensional spaces from a finite number of samples taken with respect to an arbitrary orthonormal basis. Typical approaches to this problem consider solutions obtained via the consistent reconstruction technique or as solutions of an overcomplete linear systems. However, the consistent reconstruction technique is known to be non-convergent and ill-conditioned in important cases, such as the recovery of wavelet coefficients from Fourier samples, and whilst the latter approach presents solutions which are convergent and well-conditioned when the system is sufficiently overcomplete, the solution becomes inconsistent with the original measurements. In this paper, we consider generalized sampling via a non-linear minimization problem and prove that the minimizers present solutions which are convergent, stable and consistent with the original measurements. We also provide analysis in the case of recovering wavelets coefficients from Fourier samples. We show that for compactly supported wavelets of sufficient smoothness, there is a linear relationship between the number of wavelet coefficients which can be accurately recovered and the number of Fourier samples available.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

Maharam-type kernel representation for operators with a trigonometric domination

Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in \(L^{2}[0,1]\) by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.     

Dirichlet characters,Gauss sums and arithmetic Fourier transforms

In this paper, a general algorithm for the computation of the Fourier coefficients of 2π-periodic(continuous) functions is developed based on Dirichlet characters, Gauss sums and the generalized M¨obius transform. It permits the direct extraction of the Fourier cosine and sine coefficients. Three special cases of our algorithm are presented. A VLSI architecture is presented and the error estimates are given.     

Some classes of functions and Fourier coefficients with respect to general orthonormal systems

The a.e. convergence of an orthogonal series on [0, 1] depends strongly on the coefficients of this series. It is well known that a sufficient condition for the a.e. convergence of such a series is given by the Men’shov-Rademacher theorem. On the other hand, S. Banach proved that good differential properties of a function do not guarantee the a.e. convergence on [0, 1] of the Fourier series of this function with respect to general orthonormal systems (ONSs). In the present study, we find conditions on the functions of an ONS under which the Fourier coefficients of functions of some differential classes satisfy the hypothesis of the Men’shov-Rademacher theorem.     

La classification des systèmes aux q--différences singuliers réguliers par la matrice de connexion de Birkhoff

G.D. Birkhoff extended the Riemann-Hilbert problem for fuchsian linear q-difference systems with rational coefficients. He solved it in the generic (semi-simple) case: the classifying object is made up of the connection matrix P, together with the exponents at 0 and ∞. We follow his method in the case of a general regular singular system, but treat symetrically 0 and ∞ and use neither wildly growing nor multivalued functions; the matrix P then has elliptic coefficients.     

Jeux à champ moyen. II – Horizon fini et contrôle optimal

We continue in this Note our study of the notion of mean field games that we introduced in a previous Note. We consider here the case of Nash equilibria for stochastic control type problems in finite horizon. We present general existence and uniqueness results for the partial differential equations systems that we introduce. We also give a possible interpretation of these systems in term of optimal control. To cite this article: J.-M. Lasry, P.-L. Lions, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Equidistribution of signs for modular eigenforms of half integral weight

Let f be a cusp form of weight k + 1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato–Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp ²)} _p , where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn ²)} _n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato–Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind–Dirichlet density.