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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    The journal of Fourier analysis and applications 5 (1999), S. 193-201 
    ISSN: 1531-5851
    Keywords: Primary 30D15 ; 42C15 ; Secondary 30D10 ; 42C30 ; Paley-Wiener space ; entire functions of exponential type ; exponential frames ; discrete norms ; sampling theorem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is well known that for certain sequences {tn}n∈ℤ the usual Lp norm ∥·∥p in the Paley-Wiener space PW τ p is equivalent to the discrete norm ‖f‖p,{tn}:=(∑ n=−∞ ∞ |f(tn)|p)1/p for 1 ≤ p = 〈 ∞ and ‖f‖∞,{tn}:=sup n∈ℤ|f(tn| for p=∞). We estimate ∥f∥p from above by C∥f∥p, n and give an explicit value for C depending only on p, τ, and characteristic parameters of the sequence {tn}n∈ℤ. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Weinheim : Wiley-Blackwell
    Berichte der deutschen chemischen Gesellschaft 123 (1990), S. 1733-1737 
    ISSN: 0009-2940
    Keywords: Dithiolactones ; Electroreduction ; Haloalkanedithioates ; Thioacetals, semicyclic ; Chemistry ; Inorganic Chemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Intramolecular cyclization occurs on electroreduction of the methyl -haloalkanedithioates 1 in methanol. The expected semicyclic thioacetals 2 are formed according to an ECEH mechanism. The biheterocycles 12 result from dimerization of intermediate radicals whereas the semicyclic ketene dithio-acetals 13 are formed by non-electrochemical processes. The 2-(methylthio)thiolanes 2 a and 16 are obtained with much higher yields and better selectivity by co-electroreduction of the -dithio-lactones 15 in the presence of dimethyl sulfate.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Weinheim : Wiley-Blackwell
    Berichte der deutschen chemischen Gesellschaft 123 (1990), S. 2065-2068 
    ISSN: 0009-2940
    Keywords: Benzo[c]thiophenes ; Oxidation ; Quinones ; Sulfines ; Diels-Alder reaction ; Chemistry ; Inorganic Chemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Unexpected Diels-Alder and Oxidation Reactions of Benzo[c]thiophenesThe oxidation of thiophenes by peracids to form thiophene S-oxides could not be carried out with benzo[c]thiophenes. With the sterically hindered compound 1a the quinones 2 and 3 were produced whereas the anisyl-substituted derivative 1b formed the sulfine 5. Significantly different behaviour of 1a and 1b was also observed for the Diels-Alder reaction with N-phenylmaleimide. The unexpected and unusual adduct 6 was the only product formed from 1a, with low yield though, whereas the normal Diels-Alder adduct was obtained from 1b.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Weinheim : Wiley-Blackwell
    Berichte der deutschen chemischen Gesellschaft 125 (1992), S. 127-141 
    ISSN: 0009-2940
    Keywords: Dithiopivaloates, S-aryl, S-benzyl ; Electroreduction ; Chemistry ; Inorganic Chemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Electroreduction of Organic Compounds, 19.  -  Formation of Benzoanellated Sulfur Heterocycles by Intramolecular Cathodic Cyclization of Dithiocarboxylic EstersCathodic reduction of aryl (3) and benzyl (5) dithiopivaloates and related dithioesters with leaving groups at the benzene ring or a side chain yield the sulfur heterocycles 23, 30 - 32, and 34 depending on the nature of the starting material and the reaction conditions. In case of the α-oxo-dithioester 22, thioindigo (44) is formed.  -  The corresponding thioamides 14, 16, and 18 show a strong tendency to reductive dehalogenation but the S,N-heterocycles 38 and 39 are also formed in minor amounts.  -  The formation of the rearranged products 29 - 32, and 34 is discussed in terms of C,S-splitting in the primarily formed radical anion and subsequent C,C-coupling of the fragments and follow-up reactions.
    Additional Material: 7 Tab.
    Type of Medium: Electronic Resource
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