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Fourier coefficients of cusp forms (1978)

Bruggeman, R. W.

Springer

Inventiones mathematicae 45 (1978), S. 1-18

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ISSN: 1432-1297

Keywords: Cusp forms ; Fourier coefficients ; Modular group

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract Let ψ1,ψ2,ψ3,... be an orthonormal basis of the space of cusp forms of weight zero for the full modular group. Let be the Fourier series expansion. The following theorem is proved: Let σ∈(1/4, 1/2); letf be a holomorphic function on the strip |Res|≦σ, satisfyingf(−s)=f(s) and $$f(s) = \mathcal{O}(|\tfrac{1}{4} - s^2 |^{ - 2} |cos \pi s|^{ - 1} )$$ on this strip; letm andn be non-zero integers, then $$\sum\limits_{j = 1}^\infty {f(s_j )\bar \gamma _{jm} \gamma _{jn} } $$ converges and is equal to $$\begin{gathered} - (2\pi i)^{ - 1} \int\limits_{\operatorname{Re} s = 0} {f(s)c_{00} ( - s)c_{0|m|} (s)c_{0|n|} (s)ds} \hfill \\ + (2\pi i)^{ - 1} (4\pi |m|)^{ - 1} \int\limits_{\operatorname{Re} s = 0} {f(s)c_{mn} (s)2sds} \hfill \\ - \delta _{mn} (2\pi i)^{ - 1} (4\pi |m|)^{ - 1} \int\limits_{\operatorname{Re} s = 0} {f(s)\sin \pi s2sds.} \hfill \\ \end{gathered} $$ The functionsc 00(s) andc 0|m|(s) are coefficients occurring in the Fourier series expansion of the Eisenstein series; the functionc mn(s) is a coefficient in the Fourier series expansion of a Poincaré series. The theorem is applied to obtain some asymptotic results concerning the Fourier coefficients γjn. Under additional conditions on the functionf the formula in the theorem is modified in such a way that the Fourier coefficients of holomorphic cusp forms appear.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF01406220

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Optimization of an HPLC peroxyoxalate chemiluminescence detection system for some dansyl amino acids (1990)

Baeyens, W. ; Bruggeman, J. ; Dewaele, C. ; [et al.]

New York : Wiley-Blackwell

Journal of Bioluminescence and Chemiluminescence 5 (1990), S. 13-23

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ISSN: 0884-3996

Keywords: Chemiluminescence ; chromatography ; dansyl amino acids ; bis(2,4,6-trichlorophenyl)oxalate ; TCPO ; bis(2,4-dinitrophenyl)oxalate ; DNPO ; peroxyoxalates ; Chemistry ; Biochemistry and Biotechnology

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Biology , Chemistry and Pharmacology

Notes: Bis(2,4,6-trichlorophenyl) oxalate (TCPO)-hydrogen-peroxide-generated chemiluminescence (CL) of four dansyl amino acids has been used as a model system for the optimization of a detection system in reversed-phase high-performance liquid chromatography. Dansylated alanine, glutamic acid, methionine, and norleucine were subjected to peroxyoxalate induced CL in a static system and in a flow system under various conditions with respect to TCPO (ethyl acetate) and hydrogen peroxide (acetone) concentrations, solvent composition and flow, using a two-pump or a one-pump post-column reagent system. From the CL-decay curve, the influence on the emission signal from the total flow rate in the detector was investigated. Special attention was focused on the mixing of the LC eluate and the reagent in order to combine an efficient collection of the emitted light using a 74μI flow cell (originally 10μI in the fluorescence detector) with minimal extra column band broadening. Therefore, a capillary fused-silica tubing of about 100μm i.d. was inserted against the end-frit of the column and brought through a mixing tee, in which the solutions of TCPO and hydrogen peroxide were added. The column end tubing ended in the flow cell and the LC eluate and the reagents were mixed when entering the flow-cell. Average detection limits (S/N=2) of 200fmol injected dansylated amino acid could be reached. A comparison is made between the use of TCPO and DNPO (bis (2, 4-dinitrophenyl) oxalate).

Additional Material: 9 Ill.