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Register a new user$p$-Adic multiresolution analysis and wavelet frames
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We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating MRAs (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions. We also suggest a method for the construction of wavelet functions and prove that any wavelet function generates a $p$-adic wavelet frame.
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We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating a MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate Haar MRA. We also suggest a method of constructing sets of wavelet functions and prove that any set of wavelet functions generates a $p$-adic wavelet frame.
In the paper we obtain sufficient conditions for a trigonometric polynomial to be a refinement mask corresponding to a tight wavelet frame. The condition is formulated in terms of the roots of a mask. In particular, it is proved that any trigonometric polynomial can serve as a mask if its associated algebraic polynomial has only negative roots (at least one of them, of course, equals $-1$).
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.
Let $Pin Bbb Q_p[x,y]$, $sin Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{P,s}f)(y):=frac{1}{1-p^{-1}}int_{Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(Bbb Z_p)$. We show that if $P$ is homogeneous then for each character $chi$ of $Bbb Z_p^times$ the characteristic function $det(1-uA_{P,s,chi})$ of the restriction $A_{P,s,chi}$ of $A_{P,s}$ to the eigenspace $L^2(Bbb Z_p)_chi$ is the $q$-Wronskian of a set of solutions of a (possibly confluent) $q$-hypergeometric equation. In particular, the nonzero eigenvalues of $A_{P,s,chi}$ are the reciprocals of the zeros of such $q$-Wronskian.
The general construction of frames of p-adic wavelets is described. We consider the orbit of a mean zero generic locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss relation of this result to the multiresolution wavelet analysis.
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