## Difference spaces and invariant linear formsDifference spaces arise by taking sums of finite or fractional differences. Linear forms which vanish identically on such a space are invariant in a corresponding sense. The difference spaces of L2 (Rn) are Hilbert spaces whose functions are characterized by the behaviour of their Fourier transforms near, e.g., the origin. One aim is to establish connections between these spaces and differential operators, singular integral operators and wavelets. Another aim is to discuss aspects of these ideas which emphasise invariant linear forms on locally compact groups. The work primarily presents new results, but does so from a clear, accessible and unified viewpoint, which emphasises connections with related work. |

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### Contents

Difference spaces | 1 |

Differentiation differences and the behaviour of the Fourier transform near the origin | 2 |

Multiplication spaces | 4 |

Copyright | |

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27r-periodic abelian group absolutely convergent Fourier abstract distributions Banach space bijection Borel measurable bounded linear operator Chapter characterization Co(G codimension compact abelian group compact subsets consider convergent Fourier series convolution Corollary corresponding defines a bounded definition denoted described difference spaces differential operators disjoint equivalent following hold Fourier series Fourier transform fp(Rn given Haar measure Hamel basis Hamel basis argument Hausdorff locally compact Hence invariant linear forms isometry kernel Lemma Let f Let G let h locally compact abelian LP(A Lp(Rn measure on G Meisters multiplication set multiplication space non-compact non-trivial linear combination non-zero norm numbers oo and let proof of Theorem Proposition 3.3 Proposition 5.2(a proves Riesz potential S-invariant S(Rn shows Sobolev space space LP(G subintervals subsets of G subspace of Rn tempered distributions Theorem 3.6 topologically invariant Tp(G Tp(Rn Tp{G translation invariant linear vector space vector subspace