{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:00:59Z","timestamp":1760058059089,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,18]],"date-time":"2025-03-18T00:00:00Z","timestamp":1742256000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007446","name":"King Khalid University","doi-asserted-by":"publisher","award":["RGP.2\/82\/45","PNURSP2025R337"],"award-info":[{"award-number":["RGP.2\/82\/45","PNURSP2025R337"]}],"id":[{"id":"10.13039\/501100007446","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Princess Nourah bint Abdulrahman University","award":["RGP.2\/82\/45","PNURSP2025R337"],"award-info":[{"award-number":["RGP.2\/82\/45","PNURSP2025R337"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper seeks to present the fundamental features of the category of conditional exponential convex functions (CECFs). Additionally, the study of continuous CECFs contributes to the characterization of convolution semigroups. In this context, we expand our focus to include a much broader class of Gaussian processes, where we define the generalized Fourier transform in a more straightforward manner. This approach is closely connected to the method by which we derived the Gaussian process, utilizing the framework of a Gelfand triple and the theorem of Bochner\u2013Minlos. A part of this work involves constructing the reproducing kernel Hilbert spaces (RKHS) directly from CECFs.<\/jats:p>","DOI":"10.3390\/axioms14030223","type":"journal-article","created":{"date-parts":[[2025,3,18]],"date-time":"2025-03-18T04:34:43Z","timestamp":1742272483000},"page":"223","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Conditional Exponential Convex Functions on White Noise Spaces"],"prefix":"10.3390","volume":"14","author":[{"given":"Ahmed. 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Top."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Berezansky, Y.M., and Kondratiev, Y.G. (1995). Spectral Methods in Infinite Dimensional Analysis, Kluwer.","DOI":"10.1007\/978-94-011-0509-5"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hida, T. (1980). Brownian Motion, Springer.","DOI":"10.1007\/978-1-4612-6030-1"},{"key":"ref_6","unstructured":"Colombeau, J.F. (2011). Elementary Introduction to New Generalized Functions, Elsevier. [1st ed.]."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Aliukov, S., Alabugin, A., and Osintsev, K. (2022). Review of Methods, Applications and Publications on the Approximation of Piecewise Linear and Generalized Functions. Mathematics, 10.","DOI":"10.3390\/math10163023"},{"key":"ref_8","unstructured":"Kuo, H.H. (1996). White Noise Distribution Theory, CRC Press. [1st ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s40072-021-00200-2","article-title":"An improved characterisation of regular generalised functions of white noise and an application to singular SPDEs","volume":"10","author":"Grothaus","year":"2022","journal-title":"Stoch. PDE Anal. Comput."},{"key":"ref_10","unstructured":"Colombeau, J.F. (1984). New Generalized Functions and Multiplication of Distributions, Elsevier."},{"key":"ref_11","first-page":"351","article-title":"Linear partial differential operators and generalized distributions","volume":"6","year":"1965","journal-title":"Ark. F\u00f6r Mat."},{"key":"ref_12","unstructured":"Gelfand, I.M., and Shilov, G.E. (1964). Generalized Function, Academic Press, Inc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1017\/S0027763000017852","article-title":"On quantum theory in terms of white noise","volume":"68","author":"Hida","year":"1977","journal-title":"Nagoya Math. J."},{"key":"ref_14","first-page":"926","article-title":"Exact solutions for stochastic generalized Hirota-Satsuma coupled KdV equations","volume":"49","author":"Ghany","year":"2011","journal-title":"Chin. J. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Holden, H., \u00d8sendal, B., Ub\u00f8e, J., and Zhang, T. (2010). Stochastic Partial Differential Equations, Springer Science + Business Media, LLC.","DOI":"10.1007\/978-0-387-89488-1"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Lighthill, J. (1958). Introduction to Fourier Analysis and Generalized Function, Cambridge University Press.","DOI":"10.1017\/CBO9781139171427"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"33286","DOI":"10.3934\/math.20241588","article-title":"An efficient numerical scheme in reproducing kernel space for space fractional partial differential equations","volume":"9","author":"Liu","year":"2024","journal-title":"AIMS Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"100453","DOI":"10.1016\/j.padiff.2022.100453","article-title":"Time-dependent generalized normal and gamma distributions as solutions of higher-order partial differential equations","volume":"6","author":"Serdyukov","year":"2022","journal-title":"Partial Differ. Equations Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1007\/BF01090792","article-title":"Nuclear spaces of functions of infinitely many variables","volume":"25","author":"Berezansky","year":"1973","journal-title":"Ukr. Math. J."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Hida, T. (1975). Analysis of brownian functionals. Stochastic Systems: Modeling, Identification and Optimization, I, Springer. Number 13 in Carleton Mathematical Lecture Notes.","DOI":"10.1007\/BFb0120763"},{"key":"ref_21","first-page":"581","article-title":"On strongly negative definite functions for the product of commutative hypergroups","volume":"71","author":"Ghany","year":"2011","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_22","first-page":"300","article-title":"Characterisation of convolution semigroups","volume":"4","author":"Shazly","year":"1987","journal-title":"Indian J. Theor. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Paulsen, V.I., and Raghupathi, M. (2016). An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, Cambridge University Press.","DOI":"10.1017\/CBO9781316219232"},{"key":"ref_24","unstructured":"Ghojogh, B., Ghodsi, A., Karray, F., and Crowley, M. (2021). Reproducing Kernel Hilbert Space, Mercer\u2019s Theorem, Eigenfunctions, Nystr\u00f6m Method, and Use of Kernels in Machine Learning: Tutorial and Survey. arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.apnum.2023.02.011","article-title":"Sparse machine learning in Banach spaces","volume":"187","author":"Xu","year":"2023","journal-title":"Appl. Numer. 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