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A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points.
      
An affine pseudo-plane X is a smooth affine surface defined over ${\Bbb C}$ which is endowed with an ${\Bbb A}^1$-fibration such that every fiber is irreducible and only one fiber is a multiple fiber.
      
Moreover, we prove that if C is not smooth, then C has exactly one singular point and theMakar-Limanov invariant of S is trivial.
      
It follows that if such a surface has only one singular point, then it is isomorphic to a quotient C2/G, where G is a finite group acting linearly on C2.
      
When one expands a Schur function in terms of the irreducible characters of the symplectic (or orthogonal) group, the coefficient of the trivial character is 0 unless the indexing partition has an appropriate form.
      
Aliasing error bounds are derived for one- and two-channel sampling series analogous to the Whittaker-Kotel'nikov-Shannon series, and for the multi-band sampling series, and a "derivative" extension of it, due to Dodson, Beaty, et al.
      
Aliasing in the one-channel case is shown to arise from a transformation with similarities to a projection.
      
The golden thread connecting the various extensions and generalizations is the concept of logarithmic convexity, one that goes back to the work of J.
      
A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one.
      
One outcome is a simple proof that for $g_{m \alpha , n \beta}$ to span $L^2,$ the lattice $(m \alpha , n \beta )$ must have at least unit density.
      
There is an intrinsic definition of solution that is equivalent to the extrinsic one.
      
It is shown that the one-dimensional sampling sets correspond to Bessel sequences of complex exponentials that are not Riesz bases for $L^2[-R,R].$ A signal processing application in which such sampling sets arise naturally is described in detail.
      
The proof of the results uses an extension of the Littlewood-Paley-Stein theory of square functions to the vector-valued case and inequalities previously proved by one of the authors in the context of the Riesz transforms of order one.
      
Extending this theory, we show how to use correlations between two processes to predict one from the other.
      
Extending this theory, we show how to use correlations between two processes to predict one from the other.
      
Of special interest are the Mellin operators of differentiation and integration, more correctly of anti-differentiation, enabling one to establish the fundamental theorem of the differential and integral calculus in the Mellin frame.
      
This wavelet basis is obtained from three wavelet generators by scaling, translation and rotation, and the wavelets are supported either by one corner triangle or a pair of adjacent triangles in the triangulation of level k - 1.
      
It is known [7] that dualizing a form of the Poisson summation formula yields a pair of linear transformations which map a function ? of one variable into a function and its cosine transform in a generalized sense.
      
In this article we consider the question when one can generate a Weyl- Heisenberg frame for l2(?) with shift parameters N, M-1 (integer N, M) by sampling a Weyl-Heisenberg frame for L2(?) with the same shift parameters at the integers.
      
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 ?2? with shift parameters N, M-1.
      
 

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