By Karlheinz Gröchenig, Yurii Lyubarskii, Kristian Seip

This booklet collects the court cases of the 2012 Abel Symposium, held on the Norwegian Academy of technological know-how and Letters, Oslo. The Symposium, and this publication, are excited by vital fields of recent mathematical research: operator-related functionality conception and time-frequency research; and the profound interaction among them.

Among the unique contributions and evaluate lectures collected listed here are a paper offering multifractal research as a bridge among geometric degree conception and sign processing; neighborhood and worldwide geometry of Prony platforms and Fourier reconstruction of piecewise-smooth features; Bernstein's challenge on weighted polynomial approximation; singular distributions and symmetry of the spectrum; and plenty of others.

Offering a range of the most recent and most enjoyable effects acquired by way of world-leading researchers, the publication will gain scientists operating in Harmonic and intricate research, Mathematical Physics and sign Processing.

**Read Online or Download Operator-Related Function Theory and Time-Frequency Analysis: The Abel Symposium 2012 (Abel Symposia) PDF**

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**Example text**

1 such that 2. x0 / Ä 2. H C"/jn I (53) let Dj D f W 2. H "/j Ä d Ä 2. C1 H C"/j g A Bridge Between Geometric Measure Theory and Signal Processing:. . 41 Let now ı > 0 be fixed. H / C 2ı; so that "; H C "/ Ä 2. H /C2ı/j : Therefore Dj is covered by 2. H / C 2ı: Since this estimate holds for any ı > 0, the first upper bound is proved. We now turn to the second bound in Theorem 5. We first assume that the support of f is a closed interval of the form ŒHmin ; Hmax . H // Ä 2. H // covers ŒHmin ; Hmax .

Proposition 3. e. A/ D 1). A/ 1=C . Proof. Let fBi g be an arbitrary "-covering of A. Bi / Ä C jBi jı : Bi Ä The result follows by passing to the limit when " ! 0. As an example of application of Proposition 3, let us derive a lower bound for the dimension of K. l1 /ı ; (26) so that, by (25), m0 C m1 D 1. lik /ı D jI jı : 18 P. Abry et al. I / Ä jI jı . Let J be the void interval of smallest generation included in I . J 0 /. 1 l0 l1 /ı jJ jı Ä C 0 jI jı : The mass distribution principle therefore implies that dim K obtained the following result.

It can also be used for exponents of different nature, see for instance [3, 44] where the exponent considered is the size of ergodic averages, or [4, 20] where it is the rate of divergence of Fourier series. In Sect. 1 we will give an easy example in such an alternative setting: We will consider the rate of divergence of the wavelet series of a function in a given Sobolev or Besov space. It follows that, for a given function, several notions of pointwise exponents can be considered, leading to different notions of multifractal spectra.