Sampling and interpolating sequences for multiband-limited functions and exponential bases on disconnected sets (original) (raw)

Construction of Sampling and Interpolating Sequences for Multi-Band Signals. The Two-Band Case

International Journal of Applied Mathematics and Computer Science, 2007

Recently several papers have related the production of sampling and interpolating sequences for multi-band signals to the solution of certain kinds of Wiener-Hopf equations. Our approach is based on connections between exponential Riesz bases and the controllability of distributed parameter systems. For the case of two-band signals we derive an operator whose invertibility is equivalent to the existence of a sampling and interpolating sequence, and prove the invertibility of this operator.

Sampling and recovery of multidimensional bandlimited functions via frames

Journal of Mathematical Analysis and Applications, 2010

In this paper, we investigate frames for L2[−π, π] d consisting of exponential functions in connection to oversampling and nonuniform sampling of bandlimited functions. We derive a multidimensional nonuniform oversampling formula for bandlimited functions with a fairly general frequency domain. The stability of said formula under various perturbations in the sampled data is investigated, and a computationally managable simplification of the main oversampling theorem is given. Also, a generalization of Kadec's 1/4 Theorem to higher dimensions is considered. Finally, the developed techniques are used to approximate biorthogonal functions of particular exponential Riesz bases for L2[−π, π], and a well known theorem of Levinson is recovered as a corollary.

Interpolation of subspaces and applications to exponential bases

2000

We give precise conditions under which the real interpolation space [Y 0 , X 1 ] θ,p coincides with a closed subspace of [X 0 , X 1 ] θ,p when Y 0 is a closed subspace of codimension one. We then apply this result to nonharmonic Fourier series in Sobolev spaces H s (−π, π) when 0 < s < 1. The main result: let E be a family of exponentials exp(iλ n t) and E forms an unconditional basis in L 2 (−π, π). Then there exist two number s 0 , s 1 such that E forms an unconditional basis in H s for s < s 0 , E forms an unconditional basis in its span with codimension 1 in H s for s 1 < s. For s 0 ≤ s ≤ s 1 the exponential family is not an unconditional basis in its span.

Multidimensional of gradually series band limited functions and frames

We show frames for L 2 [−π, π] d consisting of exponential functions in connection to over sampling and nonuniform sampling of gradually series bandlimited functions. We derive a multidimensional nonuniform over sampling formula for gradually series bandlimited functions with a fairly general frequency domain. The stability of this formula under various perturbations in the sampled data is investigated, and a computationally manageable simplification of the main over sampling theorem is given. Also, a generalization of Kadec's 1/4 theorem to higher dimensions is considered. Finally, the developed techniques are used to approximate biorthogonal functions of particular exponential Riesz bases for L 2 [−π, π],and a well known theorem of Levinson is recovered as a corollary.

Sampling, interpolation and Riesz bases in small Fock spaces

We give a complete description of Riesz bases of reproducing kernels in small Fock spaces. This characterization is in the spirit of the well known Kadets-Ingham 1/4 theorem for Paley-Wiener spaces. Contrarily to the situation in Paley-Wiener spaces, a link can be established between Riesz bases in the Hilbert case and corresponding complete interpolating sequences in small Fock spaces with associated uniform norm. These results allow to show that if a sequence has a density stricly different from the critical one then either it can be completed or reduced to a complete interpolating sequence. In particular, this allows to give necessary and sufficient conditions for interpolation or sampling in terms of densities.

Sampling and Interpolation Problems for Vector Valued Signals in the Paley–Wiener Spaces

Ieee Transactions on Signal Processing, 2008

An approach to solving sampling and interpolation problems in the case of non-separated sequences is developed. The approach is based on the duality between Riesz bases of exponential divided differences and group sampling and interpolating sequences. The complete description of group sampling and interpolating sequences for the Paley-Wiener spaces is obtained and stability properties of these sequences are discussed.

Sampling and Interpolation Problems for Vector Valued Signals in the Paley–Wiener Spaces

IEEE Transactions on Signal Processing, 2000

Frame spectral pairs and exponential bases

2020

Given a domain Ω⊂ R^d with positive and finite Lebesgue measure and a discrete set Λ⊂ R^d, we say that (Ω, Λ) is a frame spectral pair if the set of exponential functions ℰ(Λ):={e^2π i λ· x: λ∈Λ} is a frame for L^2(Ω). Special cases of frames include Riesz bases and orthogonal bases.In the finite setting Z_N^d, d, N≥ 1, a frame spectral pair can be defined in a similar way. We show how to construct and obtain a new frame spectral pair in R^d by "adding" frame spectral pairs in R^d and Z_N^d. Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. In this paper, we will also obtain a connection between frame spectral pairs and the Whittaker-Shannon interpolation formula when the frame is an orthogonal basis.

Discrete Sampling and Interpolation: Universal Sampling Sets for Discrete Bandlimited Spaces

IEEE Transactions on Information Theory, 2000

We study the problem of interpolating all values of a discrete signal f of length N when d < N values are known, especially in the case when the Fourier transform of the signal is zero outside some prescribed index set J ; these comprise the (generalized) bandlimited spaces B J. The sampling pattern for f is specified by an index set I, and is said to be a universal sampling set if samples in the locations I can be used to interpolate signals from B J for any J. When N is a prime power we give several characterizations of universal sampling sets, some structure theorems for such sets, an algorithm for their construction, and a formula that counts them. There are also natural applications to additive uncertainty principles.

Sampling and interpolating sequences for multiband-limited functions and exponential bases on disconnected sets (original) (raw)

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