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Approximation of [OMEGA]-bandlimited functions by [OMEGA]-bandlimited trigonometric polynomials.
Abstract
It is known that the space of [OMEGA]-bandlimited functions is dense in [L.sup.2] norm on any finite interval [a, b]. In particular, for any [OMEGA] > 0 there exist so-called superoscillating [OMEGA]-bandlimited functions which can oscillate arbitrarily quickly on any finite interval of arbitrary length. This raises the question, is any [OMEGA]-bandlimited function in some sense the limit of a sequence of [OMEGA]-bandlimited trigonometric polynomials whose periods become infinite in length? Although the existence of superoscillating bandlimited functions may appear to suggest that the answer is negative, it is shown in this paper that any [OMEGA]-bandlimited function can indeed be seen both as the uniform pointwise limit on any compact set. and the [L.sup.2] limit on any line parallel to R of a sequence of spatially-truncated [OMEGA]-bandlimited trigonometric polynomials whose periods become infinite in length. That these results are indeed consistent with and supported by known results about superoscillations and [OMEGA]-bandlimited functions is explained. Key words and phrases : trigonometric polynomials, bandlimited, Paley-Wiener space, reconstruction, sampling. 2000 AMS Mathematics Subject Classification--42A10,42A15,42A65. 1 Introduction For any fixed strictly [OMEGA]-bandlimited function f this paper provides a method of constructing a sequence of [OMEGA]-bandlimited trigonometric polynomials {[f.sub.N}N[member of]N] which converge to the bandlimited function f uniformly on any compact set in C. It will be further shown that the members of this sequence of trigonometric polynomials {[f.sub.N]} can be multiplied by characteristic functions [chi]N of certain vertical strips of increasing width to yield a new sequence {[[phi].sub.N] := [f.sub.N[chi]N]} which converges to f in [L.sup.2] norm on rely line parallel to the real axis in the complex plane C. For any fixed [OMEGA]-bandlimited function, sequences of [OMEGA]-bandlimited trigonometric polynomials which converge to it have been constructed in the past [5, 15]. The sequences considered in this paper, however, can be seen as a more natural approximation of the original [OMEGA]-bandlimited function since they are directly the image of the original function under a sequence of spectral projections of self-adjoint Laplacians on spatial intervals of increasing size. The above convergence results may appear to be in conflict with the fact that the space of [OMEGA]-bandlimited functions with an arbitrarily small but finite bandlimit [OMEGA] > 0 is dense in [L.sup.2] norm on any finite interval of arbitrarily large size [16]. For example, given any finite interval [a, b], and any positive value [OMEGA] > 0 one can construct a sequence of 'spheroidal prolate wave functions' which are [OM...See the full content of this document
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