Abstract
Based on the theory of fractal functions, in previous papers, the first author introduced fractal versions of functions in \({\mathcal {L}}^p\)-spaces, associated fractal operator and some related notions. More recently, it has been realized that the fractalization of a Lebesgue integrable function can be viewed as an internal binary operation, termed fractal convolution, of the germ function and a parameter map. In the current note, we continue to study this fractal convolution with a different viewpoint in mind. In particular, we consider both left and right partial fractal convolution operators on \({\mathcal {L}}^p\)-spaces. As an application, we obtain bases and frames consisting of fractal functions by exploring fractal convolutions and their intriguing links with the perturbation theory of Schauder bases and frames. Thus, the theory of fractal functions and theory of bases and frames seem to come together very nicely via fractal convolution. Also established along the way are results involving bases and frames obtained by using partial fractal convolutions with the null function.
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References
Barnsley, M.F., Demko, S.: Iterated function systems and the global construction of fractals. Proc. R. Soc. Lond. Ser. A 399, 243–275 (1985)
Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986)
Barnsley, M.F.: Fractals Everywhere. Academic Press Inc., New York (1988)
Casazza, P.G., Christensen, O.: Perturbation of operators and applications to frame theory. J. Fourier Anal. Appl. 3(5), 543–557 (1997)
Casazza, P.G., Freeman, D., Lynch, R.G.: Weaving Schauder frames. J. Approx. Theory 211, 42–60 (2016)
Christensen, O.: Frame perturbations. Proc. Am. Math. Soc. 123(4), 1217–1220 (1995)
Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2002)
Heil, C.E.: A Basis Theory Primer. Birkhäuser, New York (2011)
Hutchinson, J.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)
Mandelbrot, B.: The Fractal Geometry of Nature. W.H. Freeman, New York (1983)
Massopust, P.: Fractal Functions, Fractal Surfaces, and Wavelets, 2nd edn. Academic Press, New York (2016)
Navascués, M.A.: Fractal polynomial interpolation. Z. Anal. Anwend. 25(2), 401–418 (2005)
Navascués, M.A.: Fractal trigonometric approximation. Electron. Trans. Numer. Anal. 20, 64–74 (2005)
Navascués, M.A.: Fractal approximation. Complex Anal. Oper. Theory 4(4), 953–974 (2010)
Navascués, M.A.: Fractal bases of \(L_p\) spaces. Fractals 20(2), 141–148 (2012)
Navascués, M.A.: Fractal functions of discontinuous approximation. J. Basic Appl. Sci. (Math.) 10, 173–176 (2014)
Navascués, M.A., Massopust, P.: Fractal convolution: a new operation between functions. Fract. Calc. Appl. Anal. 22(3), 619–643 (2019)
Poumai, K.T., Rathore, G.S., Bhati, S.: A note on frames in Banach spaces. Int. J. Comp. Appl. Math. 12(2), 631–638 (2017)
Thim, J.: Continuous nowhere differentiable functions. Masters Thesis, Lulea (2003)
Viswanathan, P., Chand, A.K.B.: Fractal rational functions and their approximation properties. J. Approx. Theory 185, 31–50 (2014)
Viswanathan, P., Navascués, M.A., Chand, A.K.B.: Associated fractal functions in \({\cal{L}}^p\) spaces and in one-sided uniform approximation. J. Math. Anal. Appl. 433, 862–876 (2016)
Viswanathan, P., Navascués, M.A.: A fractal operator on some standard spaces of functions. Proc. Edinburgh Math. Soc. 60, 771–786 (2017)
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Communicated by Christopher Heil.
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Navascués, M.A., Viswanathan, P. & Mohapatra, R. Convolved fractal bases and frames. Adv. Oper. Theory 6, 42 (2021). https://doi.org/10.1007/s43036-021-00138-1
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DOI: https://doi.org/10.1007/s43036-021-00138-1