Density theorems for sampling and interpolation in the Bargmann-Fock space I.
Beurling type density theorems in the unit disk.
Curves of maximum modulus in coherent state representations
Density theorems for sampling and interpolation in the Bargmann-Fock space II.
Density theorems for sampling and interpolation in the Bargmann-Fock space III.
Note on some greatest common divisor matrices
Some quadratic forms related to "greatest common divisor matrices" are represented in terms of L²-norms of rather simple functions. Our formula is especially useful when the size of the matrix grows, and we will study the asymptotic behaviour of the smallest and largest eigenvalues. Indeed, a sharp bound in terms of the zeta function is obtained. Our leading example is a hybrid between Hilbert's matrix and Smith's matrix.
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Convergence and Summability of Gabor Expansions at the Nyquist Density.
Sampling and Interpolating Sequences for Multiband-Limited Functions and Exponential Bases on Disconnected Sets.
Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's (A) condition.
We describe the complete interpolating sequences for the Paley-Wiener spaces L (1 < p < ∞) in terms of Muckenhoupt's (A) condition. For p = 2, this description coincides with those given by Pavlov [9], Nikol'skii [8] and Minkin [7] of the unconditional bases of complex exponentials in L(-π,π). While the techniques of these authors are linked to the Hilbert space geometry of L , our method of proof is based in turning the problem into one about boundedness...
Weak conditions for interpolation in holomorphic spaces.
An analogue of the notion of uniformly separated sequences, expressed in terms of extremal functions, yields a necessary and sufficient condition for interpolation in L spaces of holomorphic functions of Paley-Wiener-type when 0 < p ≤ 1, of Fock-type when 0 < p ≤ 2, and of Bergman-type when 0 < p < ∞. Moreover, if a uniformly discrete sequence has a certain uniform non-uniqueness property with respect to any such L space (0 < p < ∞), then it is an interpolation...
GCD sums from Poisson integrals and systems of dilated functions
Upper bounds for GCD sums of the form are established, where is any sequence of distinct positive integers and ; the estimate for solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for . The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish...
Developments from nonharmonic Fourier series.
Convergence of series of dilated functions and spectral norms of GCD matrices
We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.
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