Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in 𝐿²

Kazarian, K.; Zink, Robert

doi:10.1090/S0002-9947-97-02034-5

Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in $L^2$
HTML articles powered by AMS MathViewer

by K. S. Kazarian and Robert E. Zink PDF

Trans. Amer. Math. Soc. 349 (1997), 4367-4383 Request permission

Abstract:

About fifty years ago, R. P. Boas and Harry Pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed, bounded, nonnegative measurable function. In subsequent years, several variations and extensions of this theorem have been given by a number of other investigators, and this program is continued here. A mildly surprising corollary of one of the results is that the trigonometric and Walsh systems can be multiplicatively transformed into quasibases for $L^{1}[0,1]$.

References

Stefan Banach, Théorie des opérations linéaires, Éditions Jacques Gabay, Sceaux, 1993 (French). Reprint of the 1932 original. MR 1357166
Ben-Ami Braun, On the multiplicative completion of certain basic sequences in $L^{p},$ $1<p<\infty$, Trans. Amer. Math. Soc. 176 (1973), 499–508. MR 313777, DOI 10.1090/S0002-9947-1973-0313777-9
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
Yves Meyer, Les nouvelles intégrales singulières de Calderón, Séminaire Bourbaki (1978/79), Lecture Notes in Math., vol. 770, Springer, Berlin, 1980, pp. Exp. No. 528, pp. 57–65 (French). MR 572417
Bernard R. Gelbaum, Notes on Banach spaces and bases, An. Acad. Brasil. Ci. 30 (1958), 29–36. MR 98974
Casper Goffman and Daniel Waterman, Basic sequences in the space of measurable functions, Proc. Amer. Math. Soc. 11 (1960), 211–213. MR 112028, DOI 10.1090/S0002-9939-1960-0112028-4
Stefan Kaczmarz and Hugo Steinhaus, Theorie der Orthogonalreihen, Monografje Matematyczne, Warszawa–Lwów, 1935; reprint, Chelsea, New York, 1951.
K. S. Kazarjan, The multiplicative completion of basic sequences in $L_{p}$, $1\leq p<\infty$, to bases in $L_{p}$, Akad. Nauk Armjan. SSR Dokl. 62 (1976), no. 4, 203–209 (Russian, with Armenian summary). MR 430670
K. S. Kazarjan, The multiplicative complementation of some incomplete orthonormal systems to bases in $L^{p},\ 1\leq p<\infty$, Anal. Math. 4 (1978), no. 1, 37–52 (Russian, with English summary). MR 481910, DOI 10.1007/BF01904857
K. S. Kazarian, On bases and unconditional bases in the spaces $L^{p}(\mu ),$ $1\leq p<\infty$, Studia Math. 71 (1981/82), no. 3, 227–249. MR 667313
K. S. Kazarjan, The multiplicative completion of certain systems, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 13 (1978), no. 4, 315–351, 353 (Russian, with English and Armenian summaries). MR 526648
K. S. Kazaryan, Multiplicative completion of uniformly bounded orthonormal systems to bases in $L^{p}$, $1<p<\infty$, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 18 (1983), no. 5, 344–361 (Russian, with English and Armenian summaries). MR 733353
K. S. Kazaryan, Improving a theorem of R. Boas and H. Pollard on multiplicative completion, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 25 (1990), no. 4, 409–412, 415 (Russian); English transl., Soviet J. Contemporary Math. Anal. 25 (1990), no. 4, 98–101 (1991). MR 1115785
K. S. Kazarjan, Summability of generalized Fourier series and Dirichlet’s problem in $L^p(d\mu )$ and weighted $H^p$-spaces $(p>1)$, Anal. Math. 13 (1987), no. 3, 173–197 (English, with Russian summary). MR 905124, DOI 10.1007/BF02115933
J. J. Price and Robert E. Zink, On sets of functions that can be multiplicatively completed, Ann. of Math. (2) 82 (1965), 139–145. MR 177085, DOI 10.2307/1970566
Ivan Singer, Bases in Banach spaces. II, Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. MR 610799, DOI 10.1007/978-3-642-67844-8
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
A. A. Talaljan, The representation of measurable functions by series, Russian Math. Surveys 15 (1960), no. 5, 75–136. MR 0125401, DOI 10.1070/RM1960v015n05ABEH001115
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776