34 research outputs found
A process very similar to multifractional Brownian motion
In Ayache and Taqqu (2005), the multifractional Brownian (mBm) motion is
obtained by replacing the constant parameter of the fractional Brownian
motion (fBm) by a smooth enough functional parameter depending on the
time . Here, we consider the process obtained by replacing in the
wavelet expansion of the fBm the index by a function depending on
the dyadic point . This process was introduced in Benassi et al (2000)
to model fBm with piece-wise constant Hurst index and continuous paths. In this
work, we investigate the case where the functional parameter satisfies an
uniform H\"older condition of order \beta>\sup_{t\in \rit} H(t) and ones
shows that, in this case, the process is very similar to the mBm in the
following senses: i) the difference between and a mBm satisfies an uniform
H\"older condition of order ; ii) as a by product, one
deduces that at each point the pointwise H\"older exponent of is
and that is tangent to a fBm with Hurst parameter .Comment: 18 page
Variable exponent Besov-Morrey spaces
In this paper we introduce Besov-Morrey spaces with all indices variable and study some fundamental properties. This includes a description in terms of Peetre maximal functions and atomic and molecular decompositions. This new scale of non-standard function spaces requires the introduction of variable exponent mixed Morrey-sequence spaces, which in turn are defined within the framework of semimodular spaces. In particular, we obtain a convolution inequality involving special radial kernels, which proves to be a key tool in this work.publishe
The Mathematical Theory of Wavelets
ABSTRACT. We present an overview of some aspects of the mathematical theory of wavelets. These notes are addressed to an audience of mathematicians familiar with only the most basic elements of Fourier Analysis. The material discussed is quite broad and covers several topics involving wavelets. Though most of the larger and more involved proofs are not included, complete references to them are provided. We do, however, present complete proofs for results that are new (in particular, this applies to a recently obtained characterization of “all ” wavelets in section 4). 1