Aliasing and oblique dual pair designs for consistent sampling

In this paper we study some aspects of oblique duality between finite sequences of vectors \cF and \cG lying in finite dimensional subspaces \cW and \cV, respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to \cF lying in \cV; we then compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for \cF under some restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces \cV and \cW has in oblique duality. We apply this analysis to compute those rigid rotations $U$ for \cW such that the canonical oblique dual of U\cdot \cF minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations $U$ for \cW such that the canonical oblique dual pair associated to U\cdot \cF minimize the aliasing. We point out that these two last problems are intrinsic to the theory of oblique duality.Comment: 23 page

Frames of translates with prescribed fine structure in shift invariant spaces

For a given finitely generated shift invariant (FSI) subspace \cW\subset L^2(\R^k) we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences E(\cF) induced by finite sequences of vectors \cF\in \cW^n that have a prescribed fine structure i.e., such that the norms of the vectors in \cF and the spectra of S_{E(\cF)} is prescribed in each fiber of \text{Spec}(\cW)\subset \T^k. We complement this result by developing an analogue of the so-called sequences of eigensteps from finite frame theory in the context of SG Bessel sequences, that allows for a detailed description of all sequences with prescribed fine structure. Then, given $0<\alpha_1\leq \ldots\leq \alpha_n$ we characterize the finite sequences \cF\in\cW^n such that $\|f_i\|^2=\alpha_i$, for $1\leq i\leq n$, and such that the fine spectral structure of the shift generated Bessel sequences E(\cF) have minimal spread (i.e. we show the existence of optimal SG Bessel sequences with prescribed norms); in this context the spread of the spectra is measured in terms of the convex potential P^\cW_\varphi induced by \cW and an arbitrary convex function $\varphi:\R_+\rightarrow \R_+$.Comment: 31 pages. Accepted in the JFA. This revised version has several changes in the notation and the organization of the text. There exists text overlap with arXiv:1508.01739 in the preliminary section

Geomorfološki indikatori promjene morske razine tijekom gornjeg pleistocena (virma) i holocena u području Kvarnera

This paper presents the data on the influence oft he sea level change during the last 150000 years on the morphogenetic events int he Kvarner region (the north eastern part of the Adriatic Sea). Some geomorphologic indicators with possible influences on sea level changes, such as sea bottom morphology, the submarine zone sediments disposition, karsification depths, cave development depths and others have been analysed. The present land, coastal, and submarine relief in the Kvarner region was formed as a consequence of tectonic movements, different lithology, climatic changes, and sea level fluctuations during the late geological history, but also because of the erosional amd accumulation processes caused by these changes. The remnants of the older morphogenetic phases have mostly been destroyed. The riverbeds were cut into the bedrock to -60 m beacuse of the sea level fall during the Würm. The carbonate rocks were karstified 70 m below the recent sea level. The present sediment distribution in the Kvarner submarine zone is a consequence of several morphogentic phases correlated with the sea level changes during the Late Pleistocene. Due to much lower sea level in some periods of the Würm glacial the connections between the Rijeka Bay, Kvarnerić, Vinodol and Velebit channels and the open Adriatic were reduced or even interrupted. In the depressions of the Rijeka Bay, Vinodol, and Velebit Channel during periods of lower sea level, lakes could be formed and only the Kvarnerić Bay could have remained connected to the Central Adriatic basin. In this period the Cres – Lošinj archipelago elevations were a morphologic barrier that divided open Adriatic sedimentation zone from the relatively isolated Kvarner region. Therefore the sea bottom is considerably deeper eastern (-70 to -90 m) than western of the Cres and Lošinj islands (-40 to -50 m). That is also the reason for the bottom sediments western of Cres – Lošinj archipelago to be mainly sandy. while those on eastern side to be more silty. Due to the lower sea level, the terrigenous material deposition occurred deeper and farther away from the present coastline, in areas where terrigenous influence is neglibible today. According to the traces of the ancient riverbeds in the Kvarnerić region the sea level at the peak of Würm was more than 100 m below present position. Marine terraces on -45 m and tidal notches on -19 m might have been formed during periods of the slower sea level rise or even its temporal stagnation during the Würm-Holecene transgression.U radu se prikazuje utjecaj promjene morske razine tijekom posljednjih 150 000 godina na morfogenetska zbivanja u području Kvarnera (sjeveroistočni dio Jadranskog mora). Analizirani su oni geomorfološki indikatori koji bi mogli ukazivati na promjene morske razine: morfologija dna, raspored dna, raspored sedimenata u podmorju, dubine okršavanja, dubine razvoja spilja i slično. Recentno kopno, obalni i podmorski reljef u području Kvarnera su posljedica tektonskih pokreta, različitog litološkog sastava, klimatskih promjena i fluktuacije morske razine tijekom mlađe geološke prošlosti, ali također i posljedica erozijskih i akumulacijskih procesa prouzročenih tim promjenama. Tragovi starijih morfogenetskih etapa uglavnom su uništeni. Zbog snižavanja morske razine tijekom virma korita vodotoka usječena su u podlogu do -60 m. Karbonatne stijene su okršene 70 m ispod današnje razine mora. Današnji raspored sedimenata u podmorju Kvarnera posljedica je nekoliko morfogenetskih etapa sukladnih promjenama morske razine tijekom gornjeg pleistocena. Tijekom niže razine mora u nekim razdobljima virmskog glacijala veze između Riječkog zaljeva, Kvarnerića i Vinodolskog i Vinodolskog kanala i otvorenog Jadrana bile su reducirane ili čak prekinute. U razdobljima niže morske razine u depresijama Riječkog zaljeva, Vinodolskog i Vinodolskog kanala mogla su preostati jezera, a jedino je Kvarnerić mogao ostati povezan s morskim bazenom srednjeg Jadrana. U tom razdoblju uzvišenja Cresko-lošinjskog otočja bila su morfološka barijera koja je odvajala sedimentaciju u zoni otvorenog Jadrana od relativno izoliranog akvatorija Kvarnera. Zato su dubine mora znatno veće istočno od Cresa i Lošinja (-70 do -90 m) od onih na zapadu (-40 do -50 m), te su sedimenti zapadno od Cresko-lošinjskog arhipelaga pretežno pjeskoviti, a istočno više muljeviti. Zbog obalne linije koja je bila niža i dalje od današnje, taloženje krupnozrnatih terigenih sedimenata zbivalo se dublje i dalje od recentne obalne crte, u području gdje je utjecaj terigene sedimentacije danas zanemariv. sudeći prema usječenim dolinama vodotoka kojim su tekli do bazena Kvarnerića, more je na vrhuncu virma bilo preko 100 m niže. Tragovi marinskih terasa na -45 m i plimskih potkapina na -19 m mogli bi biti posljedica sporijeg rasta ili čak stagnacije morske razine tijekom virmsko-holocenske transgresije

Crvenica u području Kvarnera: geomorfološki uvjeti postanka

Red soil or terra rossa is the most common soil type developed on carbonate rocks in the Kvarner area. In most cases, terra rossa has polygenetic origin and was formed by mixing of insoluble residue of carbonate rocks with weathered and eroded loess and flysch sediments. Climatic and glacioeustatic changes and tectonic movements during Quaternary have influenced recent position of terra rossa and its mineralogical composition. The recent position of terra rossa is a consequence of erosion. In the areas where tectonic movements were more pronounced, because of slope erosion and resedimentation-colluviation processes terra rossa is found in isolated depressions in the form of thicker depostits. Examples are northern part of Cres island and southern slopes of the Učka mountain. In the areas where tectonic movements were less pronounced terra rossa covers larger areas in the form of thinner deposits. An example is the western part of the island Krk.Crvenica ili terra rossa je najzastupljeniji pedološki tip odnosno pokrivač na karbonatnim stijenama kvarnerskog područja. Crvenica je najčešće poligenetske prirode i tada je nastala miješanjem netopljivih ostataka karbonatnih stijena s trošnim i erodiranim naslagama lesa i fliša. Na minerološki sastav i današnji položaj crvenice utjecale su klimatske i glacioeustatičke promjene, kao i tektonski pokreti tijekom kvartara. Današnji raspored crvenice posljedica je erozije. Tamo gdje su tektonski pokreti bili izraženi, terra rossa se zbog erozije padina i resedimentiranja-koluvacije nalazi u izoliranim depresijama u obliku debljih nakupina. Primjeri su sjeverni dio otoka Cresa i padine južne Učke. Tamo gdje su tektonski pokreti bili slabo izraženi, terra rossa je male debljine, ali pokriva velike površine. Primjer je zapadni dio otoka Krka

On the origin of both a recent and submerged tombolo on Prvić Island in the Kvarner area (Adriatic Sea, Croatia)

This paper analyzes the origins of two tombolos (one recent and another fossil/submerged) on the southwestern coast of Prvić Island, which is located in the Kvarner area in the northeastern part of the Adriatic Sea. A recent tombolo on Cape Pipa was formed by the erosion of Quaternary deposits and Palaeogene siliciclastic rocks. The submerged tombolo is much bigger, clearly visible on the sea bed toward the southwest. The conditions for forming a submerged and recent tombolo have occurred during a slow rise and then stagnation of sea level of the Adriatic Sea in the Holocene. The sea  flooded the fossil tombolo probably in the final part of a period of rapid sea-level rise at the beginning of the Holocene when a large proportion of the Quaternary sediments were eroded. Waves from the northwest (tramuntana) and southeast (jugo) refract and diffract around the tombolo. The nourishment of the beach body happens permanently on both sides of the cape. Accumulated sediments are protected by resistant rocky blocks of breccia on the peak of Cape Pipa, acting as a natural tombolo. Due to the fact that wave directions are perpendicular to the beach coastline, they do not generate longshore currents that would erode sediment in beach bodies around Cape Pipa. Therefore, the recent tombolo is probably stable in the present climatic and oceanographic conditions in the Kvarner area.</p