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Register a new userA wavelet theory for local fields and related groups
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Authors
John J. Benedetto
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Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G=Q_p, the field of p-adic rational numbers (as a group under addition), which has compact open subgroup H=Z_p, the ring of p-adic integers. Classical wavelet theories, which require a non-trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of a quotient of the dual group of G. Wavelet bases are constructed by means of an iterative method giving rise to so-called wavelet sets in the dual group.
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Using the wavelet theory introduced by the author and J. Benedetto, we present examples of wavelets on p-adic fields and other locally compact abelian groups with compact open subgroups. We observe that in this setting, the Haar and Shannon wavelets (which are at opposite extremes over the real numbers) coincide and are localized both in time and in frequency. We also study the behavior of the translation operators required in the theory.
An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported.
Through combining the work of Jean-Loup Waldspurger (cite{waldspurger10} and cite{waldspurgertemperedggp}) and Raphael Beuzart-Plessis (cite{beuzart2015local}), we give a proof for the tempered part of the local Gan-Gross-Prasad conjecture (cite{ggporiginal}) for special orthogonal groups over any local fields of characteristic zero, which was already proved by Waldspurger over $p$-adic fields.
We unite two themes in dyadic analysis and number theory by studying an analogue of the failure of the Hasse principle in harmonic analysis. Explicitly, we construct an explicit family of measures on the real line that are $p$-adic doubling for any finite set of primes, yet not doubling, and we apply these results to show analogous statements about the reverse Holder and Muckenhoupt $A_p$ classes of weights. The proofs involve a delicate interplay among several geometric and number theoretic properties.
We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in particular with the use of wavelet techniques. We are able to provide a simple proof of norm estimates for integral operators with kernel in $B^{frac{1}{p}-frac{1}{2}}_{p,p}(mathbb R,L_2(mathbb R))$. This recovers, extends and sheds new light on a theorem of Birman and Solomyak. We also use these techniques to provide a simple proof of Schur multiplier bounds for double operator integrals, with bounded symbol in $B^{frac{1}{p}-frac{1}{2}}_{frac{2p}{2-p},p}(mathbb R,L_infty(mathbb R))$, which extends Birman and Solomyaks result to symbols without compact domain.
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