Théorèmes du type Ingham et application à la théorie du contrôle

Ingham [4] improved a previous result of Wiener [10] on nonharmonic Fourier series. Modifying his weight function we obtain optimal results improving several earlier theorems of Kahane [7], Castro and Zuazua [2] and of Jaffard, Tucsnak, and Zuazua [5]. Then we apply these results to simultaneous observability problems.     

Discrete Ingham inequalities and applications

In this Note we prove a discrete version of the classical Ingham inequality for nonharmonic Fourier series whose exponents satisfy a gap condition. Time integrals are replaced by discrete sums on a discrete mesh. We prove that, as the mesh becomes finer and finer the limit of the discrete Ingham inequality is the classical continuous one. This analysis is partially motivated by control-theoretical applications. As an application we analyze the control/observation properties of numerical approximation schemes of the 1-d wave equation. The discrete Ingham inequality provides observability and controllability results which are uniform with respect to the mesh size in suitable classes of numerical solutions in which the high frequency components have been filtered. To cite this article: M. Negreanu, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On a theorem of Ingham

We prove a trigonometric inequality of Ingham’s type for nonharmonic Fourier series when the gap condition between frequencies does not hold any more. Partially supported by grants PB93-1203 of the DGICYT (Spain) and CHRX-CT94-04771 of the UE.     

Zariski density in lie groups

In [7] Furstenberg gave a proof of Borel’s density theorem [1], which depended not on complete reducibility but rather on properties of the action of a minimally almost periodic group on projective space. In [9] and [10] the basic idea of this proof was extended in various ways to deal with other particular classes of Lie groupsG and closed subgroupsH of cofinite volume. In [5] Dani gives a more general form of the density theorem in whichH need only be non-wandering. In the present paper we define the condition ofk-minimal quasiboundedness, and prove that this condition is necessary and sufficient for the density theorem to hold ((2.4) and (2.6)). Here we replace the arguments of [9] and [10] simply by proofs that the groups considered there satisfy this condition (2.10). We extend the results of [9] and [10] by considering groups which are analytic rather than algebraic, and in the solvable case we completely characterize thek-minimally quasibounded groups (2.9). In the last section we give two applications of the density theorem.     

Stability theorems for Fourier frames and wavelet Riesz bases

In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis, our estimate reduces to Kadec’ optimal 1/4 result. The second application proves that a phenomenon discovered by Daubechies and Tchamitchian [4] for the orthonormal Meyer wavelet basis (stability of the Riesz basis property under small changes of the translation parameter) actually holds for a large class of wavelet Riesz bases.     

On some riesz bases

In this paper we consider three problems concerning systems of vector exponentials. In the first part we prove a conjecture of V. Komornik raised in [14] on the independence of the movement of a rectangular membrane in different points. It was independently proved by M. Horváth [9] and S. A. Avdonin (personal communication). The analogous problem for the circular membrane was partly solved in [3] — the complete solution is given in [10]. In the second part we fill in a gap in the theory of Blaschke-Potapov products developed in the paper [19] of Potapov. Namely we prove that the Blaschke-Potapov product is determined by its kernel sets up to a multiplicative constant matrix. In the third part of the present paper we give a multidimensional generalization of the notion of sine type function developed by Levin [16], [17] and by our generalization we prove the multidimensional variant of the Levin-Golovin basis theorem [16], [6].     

Transformations of linear connections,II

We elucidate [9] with two applications. In the first we view connections as differential systems. Specializing this to trivial bundles overS ¹ and applying the theory of Floquet, we obtain equivalent connections with constant Christoffel symbols. In the second application we prove that the canonical connections of parallelizable manifolds (in particular Lie groups) can be obtained from the canonical flat connection of appropriate trivial bundles. Thus, the formalisms of [1], [4], [5] and [6] fit in the general setting of [9].     

Remark on a lower diameter bound for compact shrinking Ricci solitons

In this paper, inspired by Fernández-López and García-Río [11], we shall give a new lower diameter bound for compact non-trivial shrinking Ricci solitons depending on the range of the potential function, as well as on the range of the scalar curvature. Moreover, by using a universal lower diameter bound for compact non-trivial shrinking Ricci solitons by Chu and Hu [7] and by Futaki, Li, and Li [13], we shall provide a new sufficient condition for four-dimensional compact non-trivial shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Furthermore, we shall give a new lower diameter bound for compact self–shrinkers of the mean curvature flow depending on the norm of the mean curvature. We shall also prove a new gap theorem for compact self–shrinkers by showing a necessary and sufficient condition to have constant norm of the mean curvature.     

An approximation theorem of wong-zakai type for nonlinear stochastic partial differential equations

We present an extension of the Wong-Zakai approximation theorem for nonlinear 984 given by the Wiener process and a martingale. By approximating these disturbances we obtain in the limit equation the Ito correction term for the infinite dimensional case. Such form of the correction term connected with the Wiener process was proved in the author's papers [21] and [22], where the approximation theorem for semilinear stochastic evolution equations in Hilbert spaces was studied. Our model here is similar as the one considered by Pardoux [17]     

Isomorphism on Hypergroups

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A Fixed Point Theorem and an Equilibrium Point of an Abstract Economy in <Emphasis Type="Italic">H</Emphasis>-spaces

In this note we first prove a fixed point theorem in H-spaces which unities and extends the corresponding results in [6] and [9]. Then, by applying the fixed point theorem, we prove an existence theorem of an equilibrium point of an abstract economy in H-spaces which improves and generalizes similar result in [4].     

On strong Nörlund summability of Fourier series

In this note, a sufficient condition for summability of Fourier series has been obtained which in conjunction with the author's Tauberian theorem [M.L. Mittal, A Tauberian theorem on strong Nörlund summability, J. Indian Math. Soc. 44 (1980) 369-377] on strong Nörlund summability gives a sufficient condition for summability [C,1,2] of a Fourier series. This generalizes results due to Prasad [G. Prasad, On strong Nörlund summability of Fourier series, Univ. Roorkee Res. J. 9 (1966-1967) 1-10] and Varshney [O.P. Varshney, Note on H₂ summability of Fourier series, Boll. Un. Mat. Ital. 16 (1961) 383-385].     

On generalizations of a theorem of Ferenc Lukács

A theorem of Ferenc Lukács determines the jumps of a periodic Lebesgue integrable function f at each point of discontinuity of the first kind in terms of the partial sums of the conjugate Fourier series of f. The aim of this note is to prove analogous theorems for functions and series, introduced by Taberski ([10], [11]).     

Analytic density in Lie groups

A subgroupH of an analytic groupG is said to beanalytically dense if the only analytic subgroup ofG containingH isG itself. The main purpose of this paper is to give sufficient conditions onG (analogous to those of [8], [9], and [7] in the case of Zariski density) which guarantee the analytic density of cofinite volume subgroupsH. First we consider the case of arbitrary cofinite volume subgroups (Theorem 5 and its corollaries). Then we specialize to lattices, and prove the following result (Theorem 8):Let G be an analytic group whose radical is simply connected and whose Levi factor has no compact part and a finite center. Then any lattice in G is analytically dense. In proving this use is made of a result of Montgomery which also implies that for any simply connected solvable group, cocompactness of a closed subgroup implies analytic density. In the case of a solvable group with real roots this means analytic density and cocompactness are equivalent and thus completes a circle of ideas raised in Saito [13]. In Corollary 9 we deal with a related local condition. Finally in Theorem 10 and its corollaries we apply these considerations to prove a homomorphism extension theorem and an isomorphism theorem for 1-dimensional cohomology.     

Non-analytic functional calculi in several variables

Generalizing a result of [1] and [2], we show that every-scalar system (see Def. 3) is decomposable (in the sense of [6], [7]). By means of this fact and by some results on decomposable n-tuples (contained in [6] and [7]) we prove the theorem of support and (in the case of an inverse closed admissible algebra) the spectral mapping theorem for-functional calculi in several variables (see Def. 3). In the case of a single operator we obtain a simplification of the definition of an-spectral function (in the sense of [5], Def. 3.1.3).     

On determination of jumps in terms of the Abel-Poisson mean of Fourier series. II

In this paper, we generalize two important results of Bagota and Móricz [1], and generalize our earlier results in [6] from one-variable to two-variable case. As special applications, we prove that the generalized jump of f(x, y) at some point (x ₀, y ₀) can be determined by the higher order mixed partial derivatives of the Abel-Poisson mean of double Fourier series and the higher order mixed partial derivatives of the Abel-Poisson means of the three conjugate double Fourier series.     

Some results for a finite family of uniformly L-Lipschitzian mappings in Banach spaces

The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly L-Lipschitzian mappings in Banach spaces. The results presented in the paper improve and extend some recent results in Chang [1], Cho et al. [2] Ofoedu [5], Schu [7] and Zeng [8, 9]. The present studies were supported by “the Natural Science Foundation of China (No. 10771141),” the Natural Science Foundation of Zhejiang Province (Y605191), the Natural Science Foundation of Heilongjiang Province (A0211), the Scientific Research Foundation from Zhejiang Province Education Committee (20051897).     

A theorem on weighted means in non-archimedean fields

K denotes a complete, non-trivially valued, non-archimedean field. Infinite matrices, sequences and series have entries in K. In this paper, we prove an interesting result, which gives an equivalent formulation of summability by weighted mean methods. Incidentally this result includes the non-archimedean analogue of a theorem proved by Móricz and Rhoades (see [2], Theorem MR, p.188).     

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.     

Semilattice decompositions of semigroups

The purpose of this paper is to develop a general theory of semilattice decompositions of semigroups from the point of view of obtaining theorems of the type: A semigroup S has propertyD if and only if S is a semilattice of semigroups having property β. As such we are able to extend the theories of Clifford [3], Andersen [1], Croisot [5], Tamura and Kimura [14], Petrich [9], Chrislock [2], Tamura and Shafer [15], Iyengar [7] and Weissglass and the author [10]. The root of our whole theory is Tamura's semilattice decomposition theorem [12, 13]. Of this, we give a new proof. The results of this paper were obtained by the author between January and July of 1971, while an undergraduate at the University of California, Santa Barbara.