Fourier transform versus Hilbert transform

Certain relations between the Fourier transform of a function and the Hilbert transform of its derivative are revealed. They concern the integrability/non-integrability of both transforms. Certain applications are discussed.     

Salem’s Conditions in the Nonperiodic Case

Salem’s necessary conditions for a trigonometric series to be the Fourier series of an integrable function are generalized to the nonperiodic case for functions in the Wiener algebra. Applications of the obtained result are given.     

An Interplay of Multidimensional Variations in Fourier Analysis

We compare the Fourier integral of a function of bounded variation and the corresponding trigonometric series, generated by that function, in the multidimensional case. Several known notions of bounded variation are used and a new one is introduced. The obtained results are applied to integrability of multidimensional trigonometric series.     

On the representation of a function as an absolutely convergent Fourier integral

We obtain new sufficient conditions for the representability of a function by an absolutely convergent Fourier integral in ℝ ^d . These conditions are given in terms of the simultaneous behavior of a function and its derivatives at ∞. We test the sharpness of the conditions using well-known examples.     

Molecular cloning and biochemical characterization of a family-9 endoglucanase with an unusual structure from the gliding bacteria <Emphasis Type="Italic">Cytophaga hut chinsonii</Emphasis>

Cytophaga hutchinsonii was originally isolated from sugarcane piles. This microorganism therefore probably produces an array of enzymes allowing it to digest cellulosic substrates. C. hutchinsonii thus represents a rich source of potentially effective cellulase enzymes that can be harnessed for conversion of biomass to simple sugars. These sugars can then be used as feedstock for ethanol production or other chemical syntheses. In this study, we report the PCR cloning of an endoglucanase gene (Cel9A) from C. hutchinsonii using degenerated primers directed at the catalytic domain. Alignment of the amino acids sequence revealed that Cel9A has a gene structure totally different from the other known cellulose degraders. The most striking feature of this cloned protein is the absence of a cellulose-binding domain (CBD), which to date was believed to be imperative in cellulose hydrolysis. Consequently, the Cel9A gene, encoding β-1,4 endoglucanase from C. hutchinsonii was over-expressed in Escherichia coli with a His-Tag based expression vector. The resulting polypeptide, with a molecular mass of 105 KDa, was purified from cell extracts by affinity chromatography on cellulose. Mature Cel9A was optimally active at pH 5.0 and 45°C. The enzyme efficiently hydrolyzes carboxymethyl-cellulose (CMC). Analysis of CMC and filter paper hydrolysis suggests that Cel9A is a nonprocessive enzyme with endo-cellulase activities.     

A multiplier method with automatic limitation of penalty growth

This paper presents a multiplier method for solving optimization problems with equality and inequality constraints. The method realizes all the good features that were foreseen by R. Fletcher for this type of algorithm in the past, but which suffers from none of the drawbacks of the earlier attempts.Research sponsored by the National Science Foundation (RANN) Grant ENV76-04264 the National Science Foundation Grant ENG73-08214-A01 and the Sweden-American Foundation.     

Selective bromination of perylene diimides under mild conditions

A novel method for the bromination of perylene diimides, PDI (1), under mild conditions is reported. Variation of the reaction conditions allows mono- and dibromination of PDIs to afford 2 and 3 (these can be separated through standard procedures) or exclusive dibromination to afford 3. Pure 1,7 regioisomers are obtained through repetitive crystallization. The structure of 1,7-3b was elucidated by a single-crystal X-ray analysis. The facility of the bromination reaction, which decreases in the order 1a > 1b > 1c, depends on PDI aggregation propensities. Monobrominated PDIs were utilized for the syntheses of novel unsymmetrical piperidinyl (4a and 4b) and trimethylsilylethynyl derivatives (5a and 5b). Computational studies (DFT) on imide substituent rotation in PDIs reveal that in the case of bulky groups there is a restricted rotation leading to isomers, in agreement with our experimental results. An aromatic core twist in PDIs bearing one and two bromine substituents was also investigated by DFT.