## Multiresolution expansion, approximation order and quasiasymptotic behavior of tempered distributions.(English)Zbl 1122.46020

Let $$S_r'$$ denote the space of tempered distributions of order $$r$$ and $$\widetilde S_r$$ denote those elements $$f$$ of $$S_r$$ which satisfy the inequality $$\{f^{(q)}(x)\}\leq C_m(1+ x^2)^{-m/2}$$, $$x\in\mathbb{R}$$, $$0\leq q\leq r$$. Let $$\{V_j\}_{j\in\mathbb{Z}}$$ denote a multiresolution analysis of $$L^2$$ (MRA of $$L^2$$) and $$\phi$$ its scaling function. The main result of this article is Theorem 3 which reads as follows:
Let there be given an $$r$$-regular MRA of $$L^2$$ such that $$\phi\in\widetilde S_r$$. Let $$E_j(x, y)$$, $$j\in\mathbb{Z}$$, denote the function $$E_j(x, y)= 2^j E(2^j x, 2^j y)$$, where $$E(x, y)= \sum_{k\in\mathbb{Z}} \phi(x- k)\overline\phi(y- k)$$, $$x,y\in\mathbb{R}$$. If $$\sigma\in S_{r+1}$$, then the sequence $$\langle E_j(x, y),\sigma(x)\rangle$$ converges to $$\sigma(y)$$ in $$S_r$$ as $$j\to \infty$$.
The consequences of Theorem 3 are: a result on the multiresolution expansion of $$f\in S'$$ in $$S_r'$$, some approximation results of order $$k$$ in $$S_r'$$, and the quasi-asymptotic behaviour of an $$f\in S'$$ through its projections $$f_j$$.

### MSC:

 46F05 Topological linear spaces of test functions, distributions and ultradistributions 42C99 Nontrigonometric harmonic analysis
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### References:

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