Abstract
We characterize, for finite measure spaces, those orthonormal bases with the following positivity property: if f is a non-negative function, then the partial sums in the expansion of f are non-negative. The bases are necessarily generalized Haar functions and the partial sums are a martingale closed on the right by f.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barret, H.H., Denny, J.L., Wagner, R.F., and Myers, K.J. (1995).Objective assessment of image quality, II. Fisher information, Fourier crosstalk, and figures of merit for task performance,J. Opt. Soc. Am. A 12(5), 834–852.
Daubechies, I. (1992).Ten Lectures on Wavelets, CBMS-NSF Series in Appl. Math., SIAM Publ., Philadelphia.
Durrett, R. (1991).Probability: Theory and Examples, Wadsworth & Brooks/Cole, Pacific Grove, CA.
Karatzas, I. and Shreve, S.(1991).Brownian Motion and Stochastic Calculus, 2nd ed., Springer-Verlag, New York.
Loève, M. (1963).Probability Theory, 3rd ed., D. van Nostrand, Princeton, NJ.
Shiryayev, A.N. (1984).Probability, Springer-Verlag, New York.
Author information
Authors and Affiliations
Additional information
Communicated by Lawrence A. Shepp
Rights and permissions
About this article
Cite this article
Denny, J.L., Abbey, C.K. Partial sums of orthonormal bases preserving positivity—And martingales. The Journal of Fourier Analysis and Applications 4, 151–157 (1998). https://doi.org/10.1007/BF02475986
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02475986