Spectral Analysis of Abstract Functions

Previous article Next article Spectral Analysis of Abstract FunctionsYu. A. RozanovYu. A. Rozanovhttps://doi.org/10.1137/1104027PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractIn this paper a spectral theory is developed for abstract functions similar to the well-known spectral theory for stationary random processes.[1] A. N. Kolmogorov, Stationary sequences in Hilbert's space, Bolletin Moskovskogo Gosudarstvenogo Universiteta. Matematika, 2 (1941), 40pp, 6 MR0009098 0063.03291 Google Scholar[2] M. G. Krein, On an extrapolation problem of A. N. Kolmogorov, Doklady Akad. Nauk SSSR, 46 (1944), 306–309 0063.03356 Google Scholar[3] J. L. Doob, Stochastic Processes, IL, Moscow, 1956, (Translated into Russian.) Google Scholar[4] I. M. Gel'fand, Abstrakte Funktionen and lineare Operatoren, Mat. Sb., 4 (1938), 235–238 Google Scholar[5] E. Hills, Functional Analysis and Semigroups, IL, Moscow, 1951, (Translated into Russian.) Google Scholar[6] N. Wiener, The Fourier Integral, Cambridge University Press, 1933 0006.05401 Google Scholar[7] S. Bochner, Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse, Math. Ann., 108 (1933), 378–410 10.1007/BF01452844 MR1512856 0007.10803 CrossrefGoogle Scholar[8] P. Halmos, Measure Theory, IL, Moscow, 1953, (Translated into Russian.) Google Scholar[9] A. Blanc-Lapierre and , Robert Fortet, Théorie des fonctions aléatoires. Applications à divers phénomènes de fluctuation, Masson et Cie, Paris, 1953xvi+693 MR0061780 0051.35702 Google Scholar[10] Harald Cramér, A contribution to the theory of stochastic processes, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, 329–339 MR0044771 0044.33702 Google Scholar[11] Kari Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung, Ann. Acad. Sci. Fennicae. Ser. A. I. 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A., 39 (1953), 302–307 MR0055619 0059.11801 CrossrefGoogle Scholar[13] Salomon Bochner, Harmonic analysis and the theory of probability, University of California Press, Berkeley and Los Angeles, 1955viii+176 MR0072370 0068.11702 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Estimating a joint probability distribution model of fluctuating wind speeds of monsoons from field-measured wind speed dataJournal of Wind Engineering and Industrial Aerodynamics, Vol. 227 Cross Ref Probability distribution estimation for harmonisable loads and responses of linear elastic structuresProbabilistic Engineering Mechanics, Vol. 68 Cross Ref Signals Featuring Harmonics With Random Frequencies – Spectral, Distributional and Ergodic PropertiesIEEE Transactions on Signal Processing, Vol. 69 Cross Ref On Generalized Derivative Sampling Series Expansion23 August 2019 Cross Ref Stochastic Processes and Spectral Analysis for Hilbert C^* C ∗ -Module-Valued Maps3 November 2015 | Bulletin of the Malaysian Mathematical Sciences Society, Vol. 41, No. 1 Cross Ref Kramer's generalized sampling of stochastic processes Cross Ref A Spectral Representation of a Class of Nonstationary Processes30 November 2011 | Journal of Statistical Theory and Practice, Vol. 5, No. 3 Cross Ref Locally stationary harmonizable complex improper stochastic processes31 August 2010 | Journal of Time Series Analysis, Vol. 32, No. 1 Cross Ref On Sharp Bounds for Remainders in Multidimensional Sampling Theorem1 September 2007 | Sampling Theory in Signal and Image Processing, Vol. 6, No. 3 Cross Ref Moment problem with contractive solutions: the regular case2 May 2007 | Proceedings of the American Mathematical Society, Vol. 135, No. 9 Cross Ref Irregular Sampling of Generalized Harmonizable ProcessesStochastic Analysis and Applications, Vol. 22, No. 5 Cross Ref Stochastic Processes Cross Ref Covariance analysis and associated spectra for classes of nonstationary processesJournal of Statistical Planning and Inference, Vol. 100, No. 2 Cross Ref The kolmogorov isomorphism theorem and extensions to some nonstationary processes Cross Ref Optimized estimation of moments for nonstationary signalsIEEE Transactions on Signal Processing, Vol. 45, No. 5 Cross Ref Almost periodic Harmonizable processesGeorgian Mathematical Journal, Vol. 3, No. 3 Cross Ref The aftermath of Cramér's work on stochastic processesScandinavian Actuarial Journal, Vol. 1995, No. 1 Cross Ref Trace measures of a positive definite bimeasureJournal of Multivariate Analysis, Vol. 40, No. 1 Cross Ref Consistent statistical estimate of spectral measure discrete component for a class of random processesJournal of Nonparametric Statistics, Vol. 2, No. 1 Cross Ref On the product of two harmonizable time seriesStochastic Processes and their Applications, Vol. 38, No. 2 Cross Ref GRAPHICAL METHODS FOR DETERMINING THE PRESENCE OF PERIODIC CORRELATIONJournal of Time Series Analysis, Vol. 12, No. 4 Cross Ref Periodically Correlated Processes and Their Stationary DilationsA. 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