Approximation of Ω—bandlimited Functions by Ω—bandlimited Trigonometric Polynomials

It is known that the space of Ω—bandlimited functions is dense in L2 norm on any finite interval [a, b]. In particular, for any Ω > 0 there exist so-called superoscillating Ω—bandlimited functions which can oscillate arbitrarily quickly on any finite interval of arbitrary length. This raises the question, is any Ω—bandlimited function in some sense the limit of a sequence of Ω—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 Ω—bandlimited function can indeed be seen both as the uniform pointwise limit on any compact set, and the L2 limit on any line parallel to ℝ of a sequence of spatially-truncated Ω—bandlimited trigonometric polynomials whose periods become infinite in length. That these results are indeed consistent with and supported by known results about superoscillations and Ω—bandlimited functions is explained.