A summability method for the arithmetic Fourier transform

The Arithmetic Fourier Transform (AFT) is an algorithm for the computation of Fourier coefficients, which is suitable for parallel processing and in which there are no multiplications by complex exponentials. This is accomplished by the use of the Möbius function and Möbius inversion. However, the algorithm does require the evaluation of the function at an array of irregularly spaced points. In the case that the function has been sampled at regularly spaced points, interpolation is used at the intermediate points of the array. Generally theAFT is most effective when used to calculate the Fourier cosine coefficients of an even function.In this paper a summability method is used to derive a modification of theAFT algorithm. The proof of the modification is quite independent of theAFT itself and involves a summation by primes. One advantage of the new algorithm is that with a suitable sampling scheme low order Fourier coefficients may be calculated without interpolation.     

Sampling theorems and the contour integral method

Various forms of the Whittaker-Kotelnikov -Shannon sampling theorem, which allow certain entire functions to be represented by interpolatory series, are derived by viewing them as sums of residues. Series in which the sample points are distributed in one and in two dimensions are considered.This contour integral method provides a powerful way of generating the correct form of the series, and of obtaining the uniform convergence.Particular emphasis is placed upon series representations involving derivative samples as well as samples of the function itself. The general derivative sampling series is treated, as well as derivative sampling at points which are slightly perturbed from uniform spacing. Finally, derivative sampling at lattice points in the complex plane is considered     

Sampling multipliers and the Poisson Summation Formula

Periodization and sampling operators are defined, and the Fourier transform of periodization is uniform sampling in a well-defined sense. Implementing this point of view, Poisson Summation Formulas are proved in several spaces including integrable functions of bounded variation (where the result is known) and elements of mixed norm spaces. These Poisson Summation Formulas can be used to prove corresponding sampling theorems. The sampling operators used to understand and prove the aforementioned Poisson Summation Formulas lead to the introduction of spaces of continuous linear operators which commute with integer translations. Operators L of this type are appropriately called sampling multipliers. For a given function f, they give rise to new sampling formulas, whose sampling coefficients are of the form Lf. In practice, Lf can be used to model noisy data or data where point values are not available. By representation theorems of the second named author, some of these operator spaces are proved to be mixed norm spaces. The approach and results of this paper were developed in the context of Duffin and Schaeffer’s theory of frames. In particular, sampling multipliers L are related to the Bessel map used by Duffin and Schaeffer in their definition of the frame operator. The first named author was supported in part by AFOSR contract F49620-96-1-0193. The second named author was supported by the Cusanuswerk.     

Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces

Under the appropriate definition of sampling density D_ϕ, a function f that belongs to a shift invariant space can be reconstructed in a stable way from its non-uniform samples only if D_ϕ≥1. This result is similar to Landau's result for the Paley-Wiener space B _1/2 . If the shift invariant space consists of polynomial splines, then we show that D_ϕ<1 is sufficient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B _1/2 .     

Fourier Series with Linear Splines on an Interval

We construct orthonormal bases of linear splines on a finite interval [a, b] and then we study the Fourier series associated to these orthonormal bases. For continuous functions defined on [a, b], we prove that the associated Fourier series converges pointwisely on (a, b) and also uniformly on [a, b], if it convergences pointwisely at a and b.     

Sampling with Bessel Functions

The paper deals with sampling of σ-bandlimited functions in R^m with Clifford-valued, where bandlimitedness means that the spectrum is contained in the ball B(0, σ) that is centered at the origin and has radius σ. By comparing with the general setting, what is new in the sampling is using the explicit Bochner-type relations involving spherical harmonics and monogenics in the Clifford algebra setting. Convergence of the sampling formulas in the L² sense and in the uniform and absolute pointwise sense are studied. The study was supported by Research Grant of the University of Macau No. RG059/05- 06S/QT/FST.     

On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function

Summary. This paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point values (or a highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct an uniform exponentially convergent approximation to the function in any sub-interval of analyticity. The proof covers the cases of Fourier, Chebyshev, Legendre, and more general Gegenbauer collocation methods. A numerical example is also provided. Received July 17, 1994 / Revised version received December 12, 1994     

The banach algebra A and its properties

Beurling’s algebra is considered. A* arises quite naturally in problems of summability of the Fourier series at Lebesgue points, whereas Wiener’s algebra A of functions with absolutely convergent Fourier series arises when studying the norm convergence of linear means. Certainly, both algebras are used in some other areas. A* has many properties similar to those of A, but there are certain essential distinctions. A* is a regular Banach algebra, its space of maximal ideals coincides with[−π, π], and its dual space is indicated. Analogs of Herz’s and Wiener-Ditkin’s theorems hold. Quantitative parameters in an analog of the Beurling-Pollard theorem differ from those for A. Several inclusion results comparing the algebra A* with certain Banach spaces of smooth functions are given. Some special properties of the analogous space for Fourier transforms on the real axis are presented. The paper ends with a summary of some open problems.     

Reconstruction of Band-Limited Functions from Local Averages

In this paper, we show that every band-limited function can be reconstructed by its local averages near certain points. We give the optimal upper bounds for the support length of averaging functions with respect to both regular and irregular sampling points. Our results improve an earlier result by Gröchenig.     

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

We study an extension of the classical Paley–Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier–Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer’s expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier–Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel.     

Transplanting maximal inequalities between Laguerre and Hankel multipliers

We supplement our previous paper [9] by adding a theorem that transplantsL ^p -norm maximal inequalities for Laguerre multipliers. As an immediate consequence we obtain negative results concerningL ^p -estimates of partial sum maximal operators for Laguerre expansions.Research supported in part by KBN grant No. 2 PO3A 030 09.     

Hermite interpolation with trigonometric polynomials

Interpolation methods of Hermite type in translation invariant spaces of trigonometric polynomials for any position of interpolation points and any number of derivatives are constructed. For the case of an odd number of interpolation conditions-periodic trigonometric polynomials of minimum order are chosen as interpolation functions while for the case of an even number of interpolation conditions-antiperiodic trigonometric polynomials of minimum order are appropriate.     

Regularity of refinable function vectors

We study the existence and regularity of compactly supported solutions φ = (φ_v) _v=0 ^/r−1 of vector refinement equations. The space spanned by the translates of φ_v can only provide approximation order if the refinement maskP has certain particular factorization properties. We show, how the factorization ofP can lead to decay of |̸_v(u)| as |u| → ∞. The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm.     

Converse Sturm–Hurwitz–Kellogg theorem and related results

We prove that if Vⁿ is a Chebyshev system on the circle and f is a continuous real-valued function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S¹ that takes f to a function L²-orthogonal to V. We also prove that if f is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of that takes f to the Schwarzian derivative of a function on . We show that the space of piecewise constant functions on an interval with values ± 1 and at most n + 1 intervals of constant sign is homeomorphic to n-dimensional sphere. To V. I. Arnold for his 70th birthday     

On the structure of spaces of uniformly convergent Fourier series

We first give some new examples of translation invariant subspaces of C or U without local unconditional structure. In the second part, we prove that U and U ⁺ do not have the Gordon–Lewis property. In the third part, we show that absolutely summing operators from U to a K-convex space are compact. As a consequence, U and U ⁺ are not isomorphic. At last, we prove that U and U ⁺ do not have the Daugavet property.     

A characterization of the Clifford torus

We provide a characterization of the Clifford torus via a Ricci type condition among minimal surfaces in S⁴. More precisely, we prove that a compact minimal surface in S⁴, with induced metric ds² and Gaussian curvature K, for which the metric is flat away from points where K = 1, is the Clifford torus, provided that m is an integer with m > 2.Received: 8 September 2004     

Locally constant dyadic derivatives

We show that in order for a Walsh series to be locally constant it is necessary for certain blocks of that series to sum to zero. As a consequence, we show that a functionf with a somewhat sparse Walsh—Fourier series is necessarily a Walsh polynomial if its strong dyadic derivative is constant on an interval. In particular, if a Rademacher seriesR is strongly dyadically differentiable and if that derivative is constant on any open subset of [0, 1], thenR is a Rademacher polynomial.     

Divergent fourier series: Numerical experiments

An open problem in the theory of Fourier series is whether there are functions f L ₁ such that the partial sums S _n(f, x) diverge faster than log log n, almost everywhere in x. For a class of particularly bad functions Kahane proved that the rate of divergence is faster than o(log log n). We give here a probabilistic interpretation of the Kahane result, which shows that the record values of the sums S _n(f, x) should behave essentially as the record values of a sequence of independent identically distributed random variables, for which we deduce the divergence rate log log n. Numerical computation is in good agreement with the prediction. One can argue that the Kahane examples are in some sense optimal, and conclude that, under this assumption, ...(log log n) is the highest possible rate for divergence almost everywhere of the Fourier partial sums for L ₁ functions.     

From continuous to discrete Weyl-Heisenberg frames through sampling

In this article we consider the question when one can generate a Weyl- Heisenberg frame for l ² (ℤ) with shift parameters N, M ⁻¹ (integer N, M) by sampling a Weyl-Heisenberg frame for L ² (ℝ) with the same shift parameters at the integers. It is shown that this is possible when the window g ε L ² (ℝ) generating the Weyl-Heisenberg frame satisfies an appropriate regularity condition at the integers. When, in addition, the Tolimieri-Orr condition A is satisfied, the minimum energy dual window ^o γ ε L ² (ℝ) can be sampled as well, and the two sampled windows continue to be related by duality and minimality. The results of this article also provide a rigorous basis for the engineering practice of computing dual functions by writing the Wexler-Raz biorthogonality condition in the time-domain as a collection of decoupled linear systems involving samples of g and ^o γ as knowns and unknowns, respectively. We briefly indicate when and how one can generate a Weyl-Heisenberg frame for the space of K-periodic sequences, where K=LCM (N, M), by periodization of a Weyl-Heisenberg frame for ℓ ² ℤ with shift parameters N, M ⁻¹ .