GROWTH ESTIMATES FOR SINE-TYPE-FUNCTIONS AND APPLICATIONS TO RIESZ BASES OF EXPONENTIALS

We present explicit estimates for the growth of sine-type-functions as well as for the derivatives at their zero sets, thus obtaining explicit constants in a result of Levin. The estimates are then used to derive explicit lower bounds for exponential Riesz bases, as they arise in Avdonin's Theorem on 1/4 in the mean or in a Theorem, of Bogmer, Horvath, Job and Seip. An application is discussed, where knowledge of explicit lower bounds of exponential Riesz bases is desirable.     

POINTWISE ESTIMATES OF SOLUTIONS TO CAUCHY PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS＊＊＊

This paper considers the pointwise estimate of the solutions to Cauchy problem for quasilin-ear hyperbolic systems, which bases on the existence of the solutions by using the fundamental solutions. It gives a sharp pointwise estimates of the solutions on domam under consideration. Specially, the estimate is precise near each characteristic direction.     

A NOTE ON MEYER—KONIG AND ZHLLER OPERATORS FOR FUNCTIONS OF BOUNDED VARIATION

In the present note we give the correct and improved estimate on the rate of convergence of integrated Meyer-Konig and Zeller operators for function of bounded variation.     

Growth estimates for sine-type-functions and applications to Riesz bases of exponentials

We present explicit estimates for the growth of sine-type-functions as well as for the derivatives at theirzero sets, thus obtaining explicit constants in a result of Levin. The estimates are then used to derive explicitlower bounds for exponential Riesz bases, as they arise in Avdonin's Theorem on 1/4 in the mean or in alower bounds of exponential Riesz bases is desirable.     

Uniformly bounded families of Riesz bases of exponentials, sines, and cosines

We construct a wide class of families of exponentials, sines, and cosines generating Riesz bases of the corresponding Hilbert spaces with uniformly bounded upper bounds and uniformly positive lower bounds.     

Multi-tiling and Riesz bases

Let S   be a bounded, Riemann measurable set in R^d${\mathbb{R}}^d$, and Λ be a lattice. By a theorem of Fuglede, if S   tiles R^d${\mathbb{R}}^d$ with translation set Λ, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multi-tiles  R^d${\mathbb{R}}^d$ with translation set Λ, S has a Riesz basis of exponentials. The proof is based on Meyer?s quasicrystals.     

Uniform partitions of frames of exponentials into Riesz sequences

The Feichtinger conjecture, if true, would have as a corollary that for each set E⊂[0,1] and Λ⊂Z, there is a partition Λ₁,…,ΛN of Z such that for each 1?i?N, is a Riesz sequence. In this paper, sufficient conditions on sets E⊂[0,1] and Λ⊂R are given so that can be uniformly partitioned into Riesz sequences.     

Nearness of the Riesz bases to orthonormal bases in a Hilbert space

One formulates existence conditions and certain properties of a special class of unconditional bases in a Hilbert space. One formulates consequences of these conditions for the system of eigenfunctions of the restriction operator, conjugate to the shift operator, and for the system of the resonance states of a certain scattering problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 201–208, 1983.     

Sharp weak type estimates for Riesz transforms

Let $d$ be a given positive integer and let $\{R_j\}_{j=1}^d$ denote the collection of Riesz transforms on $\mathbb {R}^d$ . For $1<p<\infty $ , we determine the best constant $C_p$ such that the following holds. For any locally integrable function $f$ on $\mathbb {R}^d$ and any $j\in \{1,\,2,\,\ldots ,\,d\}$ , $$\begin{aligned} ||(R_jf)_+||_{L^{p,\infty }(\mathbb {R}^d)}\le C_p||f||_{L^{p,\infty }(\mathbb {R}^d)}. \end{aligned}$$ A related statement for Riesz transforms on spheres is also established. The proofs exploit Gundy–Varopoulos representation of Riesz transforms and appropriate inequality for orthogonal martingales.     

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.