Alexander GONCHAROV, Yasemin ŞENGÜL TEZEL

Logarithmic dimension and bases in Whitney spaces

We give a formula for the logarithmic dimension of the generalized Cantor-type set K. In the case when the logarithmic dimension of K is smaller than 1, we construct a Faber basis in the space of Whitney functions E(K) .

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[1] Alpan G, Goncharov A. Two measures on Cantor sets. Journal of Approximation Theory 2014; 186: 28-32.
[2] Arslan B, Goncharov A, Kocatepe M. Spaces of Whitney functions on Cantor-type sets. Canadian Journal of Mathematics 2002; 54 (2): 225-238.
[3] Bessaga C. Nuclear Fréchet spaces without bases. I. Variation on a theme of Djakov and Mitiagin. Bulletin de l’Académie Polonaise des Sciences Série des Sciences Mathématiques, Astronomiques et Physiques 1976; 24 (7): 471-473.
[4] Best E. A closed dimensionless linear set. Proceedings of the Edinburgh Mathematical Society 1939; 6 (2): 105-108.
[5] Carleson L. Selected Problems on Exceptional Sets. Van Nostrand Mathematical Studies No. 13D. Princeton, USA: Van Nostrand Co. 1971.
[6] Conway JB. Functions of One Complex Variable II. Berlin, Germany and New York, USA: Springer-Verlag, 1996.
[7] DeVore RA, Lorentz GG. Constructive Approximation. Berlin, Germany and New York, USA: Springer-Verlag, 1993.
[8] Domański P, Vogt D. The space of real-analytic functions has no basis, Studia Mathematica 2000; 142 (2): 187-200.
[9] Erdös P, Gillis J. Note on the transfinite diameter. Journal of London Mathematical Society 1937; 12 (3): 185-192.
[10] Falconer KJ. The Geometry of Fractal Sets. Cambridge, UK: Cambridge University Press, 1985.
[11] Goncharov A. Extension via interpolation, Orlicz centenary volume. II, 43–49. Polish Academy of Sciences Institute Mathematical, Warsaw, Poland: Banach Center Publications, 68: 2005.
[12] Goncharov A. Bases in the spaces of C ∞-functions on Cantor-type sets, Constructive Approximation 2006; 23: 351-360.
[13] Goncharov A, Ural Z. Mityagin’s extension problem. Progress report, Journal of Mathematical Analysis and Applications 2017; 448 (1): 357-375.
[14] Mitjagin BS. Approximate dimension and bases in nuclear spaces. Uspekhi Matematicheskikh Nauk 1961; 16 (4) (100): 63-132.
[15] Zobin N, Mityagin BS. Examples of nuclear Fréchet spaces without basis. Functional Analysis 1974; 84: 35-47.
[16] Tsuji M. Potential theory in modern function theory. Reprinting of the 1959 original. New York, USA: Chelsea Publishing Co., 1975.
[17] Vogt D. A nuclear Fréchet space of C ∞-functions which has no basis, Note di Matematica 2005/06; 25 (2): 187-190.
[18] Whitney H. Analytic extensions of differentiable functions defined in closed sets. Transactions of the American Mathematical Society 1934; 36 (1): 63-89.