Abstract
There are three results proved in this paper. The first one characterizes the Hölder classes in Orlicz spaces by the coefficients of the orthogonal spline expansions of the Franklin type. The second one gives a sharp estimate for the correlation of two random variables obtained as a composition of two Borel functions with the components of a given two-dimensional Gaussian vector. The third one is obtained with the help of the first two and it states that the Wiener measure is concentrated on the Banach space of Hölder functions with exponent 1/2 but in the norm of the Orlicz spaceL *M withM(t)=expt(t 2)−1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Boor De (1981):On a max-norm bounded for the least-squares spline approximation. In: Approximation and Function Spaces. (Proc. International Conference, Gdańsk, August, 27–31, 1979) (Z. Ciesielski, ed.). Warsaw: PWN; Amsterdam: North-Holland, pp. 163–175.
Z. Ciesielski (1963):Properties of the orthnormal, Franklin system. Studia Math.,23:141–157.
Z. Ciesielski (1966):Properties of the orthonormal Franklin system, II. Studia Math.,27:290–323.
Z. Ciesielski (1975):Constructive function theory and spline systems. Studia Math.,53:277–302.
Z. Ciesielski (1985):Haar orthogonal functions in analysis and probability. In: Colloquia Mathematica Societatis János Bolyai, vol. 49 (Proc. Alfred Haar Memorial Conference, Budapest). Amsterdam: North-Holland, pp. 25–56.
Z. Ciesielski (1991):Modulus of Smoothness of the Brownian Paths in the L p Norm. Proc. Conf. Approximation Theory, Varna, Bulgaria.
Z. Ciesielski, J. Domsta (1972):Construction of an orthonormal basis in C m(Id) and W mp (Id). Studia Math.,41:211–224.
Z. Ciesielski, J. Domsta (1972):Estimates for the spline orthonormal functions and for their derivatives. Studia Math.,44:315–320.
X. Fernique (1970):Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris Sér. A-B,270:A1698-B1699.
W. B. Johnson, G. Schechtman, J. Zinn (1985):Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab.,13:234–253.
M. A. Kracnoselskii, Ya. B. Rutitskii (1958): Convex Functions and Orlicz, Spaces. Moscow: State Publ. Phys.-Math. Literature.
P. Levy (1954): Le Mouvement Brownien. Paris: Gauthier-Villars.
J. Lindenstrauss, L. Tzafriri (1979): Classical Banach, Spaces, II. Berlin: Springer-Verlag.
S. Mazur (1929):Une remarque sur l'homéomorphie des champs fonctionels. Studia Math.,1:83–85.
N. Wiener (1930):Generalized harmonic analysis. Acta Math.,55:117–258.
N. Wiener (1933): Fourier Integral and Certain of Its Applications. Cambridge: Cambridge University Press.
Wu Garidi (1991):On approximation by polynomials in Orlicz spaces. J. Approx. Theory and Its Appl.,7(3):97–110.
Author information
Authors and Affiliations
Additional information
Communicated by Ronald A. DeVore.
Rights and permissions
About this article
Cite this article
Ciesielski, Z. Orlicz spaces, spline systems, and brownian motion. Constr. Approx 9, 191–208 (1993). https://doi.org/10.1007/BF01198003
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01198003