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Recent developments in the theory of function spaces with dominating mixed smoothness

Schmeisser, Hans-Jürgen (2007)

Nonlinear Analysis, Function Spaces and Applications

The aim of these lectures is to present a survey of some results on spaces of functions with dominating mixed smoothness. These results concern joint work with Winfried Sickel and Miroslav Krbec as well as the work which has been done by Jan Vybíral within his thesis. The first goal is to discuss the Fourier-analytical approach, equivalent characterizations with the help of derivatives and differences, local means, atomic and wavelet decompositions. Secondly, on this basis we study approximation...

Refinement type equations: sources and results

Rafał Kapica, Janusz Morawiec (2013)

Banach Center Publications

It has been proved recently that the two-direction refinement equation of the form f ( x ) = n c n , 1 f ( k x - n ) + n c n , - 1 f ( - k x - n ) can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation f ( x ) = n c f ( k x - n ) , which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation f ( x ) = c ( y ) f ( k x - y ) d y has also various interesting applications....

Régularité des bases d'ondelettes et mesures ergodiques.

Albert Cohen, Jean-Pierre Conze (1992)

Revista Matemática Iberoamericana

Nous reprenons la construction des bases orthonormées d'ondelettes à partir des filtres miroirs en quadrature tel qu'elle apparaît dans [4]. Nous montrons que leur régularité est liée à une mesure invariante pour la transformation ω → 2ω mod-2π. Cette méthode permet d'obtenir le facteur exact qui relie asymptotiquement la régularité des ondelettes constriutes dans [4] à la taille de leur support.

Ridgelet transform on tempered distributions

R. Roopkumar (2010)

Commentationes Mathematicae Universitatis Carolinae

We prove that ridgelet transform R : 𝒮 ( 2 ) 𝒮 ( 𝕐 ) and adjoint ridgelet transform R * : 𝒮 ( 𝕐 ) 𝒮 ( 2 ) are continuous, where 𝕐 = + × × [ 0 , 2 π ] . We also define the ridgelet transform on the space 𝒮 ' ( 2 ) of tempered distributions on 2 , adjoint ridgelet transform * on 𝒮 ' ( 𝕐 ) and establish that they are linear, continuous with respect to the weak * -topology, consistent with R , R * respectively, and they satisfy the identity ( * ) ( u ) = u , u 𝒮 ' ( 2 ) .

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