We present results on the expansion of elements of the spaces $L_p(0,1)$, $0 <p <1 $, into systems of functions that are contractions and shifts of one function. New algorithms are designed for the expansion of functions into Fourier-type series in these systems with integer coefficients. The expansions produced by the proposed methods have the property of image compression, i.e., a large number of expansion coefficients are zeros. The results may be of interest to specialists in the transmission and processing of digital information and to other researchers who have a need to expand nonintegrable functions into series in systems of contractions and shifts of one function.
Keywords: functional systems of contractions and shifts of one function of the space $L_p (0,1)$ with $0<p<1$, Fourier-type series with integer coefficients, digital processing, information transmission
Received August 21, 2024
Revised November 4, 2024
Accepted November 18, 2024
Vadim Ivanovich Filippov, Dr. Phys.-Math. Sci., Prof., Povolzhsky Institute of Management named after P.A. Stolypin, Saratov, 410031 Russia, email: 888vadim@mail.ru
REFERENCES
1. Filippov V.I., Oswald P. Representation in Lp by series of translates and dilates of one function. J. Approx. Theory, 1995, vol. 82, no. 1, pp. 15–29.
2. Filippov V.I. Representation systems obtained using translates and dilates of a single function in multidimensional spaces $E_{\varphi}$. Izv.: Math., 2012, vol. 76, no. 6, pp. 1257–1270. doi: 10.1070/IM2012v076n06ABEH002623
3. Filippov V.I. On generalization of Haar system and other function systems in spaces $E_{\varphi}$. Russian Math. (Iz. VUZ), 2018, vol. 62, no. 1, pp. 76–81. doi: 10.3103/S1066369X18010115
4. Filippov V.I. Fourier-type series with integer coefficients in systems of contractions and shifts of a single function in spaces $L_p,\, p\ge 1$. Russian Math. (Iz. VUZ), 2019, vol. 63, no. 6, pp. 51–57. doi: 10.3103/S1066369X19060069
5. Filippov V.I. Algorithmization of integer decomposition of elements of spaces $ L_p, 0 < p < 1$, using software in the form of the Matlab software package. Vestnik SGTU im. GagarinaYu.A., 2021, vol. 90, no. 3, pp. 52–61 (in Russian).
6. Fekete M. Über die verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z., 1923, bd. 17, pp. 228–249. doi: 10.1007/BF01504345
7. Al’per S.Ya. On the approximation of functions on closed sets by polynomials with entire coefficients. Izv. Akad. Nauk SSSR. Ser. Mat., 1964, vol. 28, no. 5, pp. 1173–1186 (in Russian).
8. Borodin P.A., Shklyaev K.S. Density of quantized approximations. Russian Math. Surv., 2023, vol. 78, no. 5, pp. 797–851. doi: 10.4213/rm10115e
9. Benedetto J.J., Njeunje F.O.N. Haar approximation from within for $L_p({\mathbb R}^d),$ $0<p<1$. Sampl. Theory Signal Process. Data Anal., 2021, vol. 19, art. no. 2. doi: 10.1007/s43670-020-00001-z
10. Benedetto J.J. Noise reduction in terms of the theory of frames. Wavelet Anal. Appl., 1998, vol. 7, pp. 259–284. doi: 10.1016/S1874-608X(98)80010-1
11. Arestov V.V. On inequalities of S. N. Bernstein for algebraic and trigonometric polynomials. Sov. Math., Dokl., 1979, vol. 20, pp. 600–603.
12. Arestov V.V. The Szegö inequality for derivatives of a conjugate trigonometric polynomial in $L_0$. Math. Notes, 1994, vol. 56, no. 5–6, pp. 1216–1227. doi: 10.1007/BF02266689
13. Golubov B.I. On the existence of bases from shifts of functions in homogeneous spaces. Zb. prats’ In-tu matematiki NAN Ukraini, 2008, vol. 5, no. 1, pp. 104–112 (in Russian).
14. Rudin W. Functional analysis, New York, McGraw-Hill Book Comp., 1973, 407 p. Translated to Russian under the title Funktsional’nyi analiz, Moscow, Mir Publ., 1975, 450 p.
15. Yoshida K. Functional analysis. Berlin, Heidelberg, Springer, 1965, 458 p. doi: 10.1007/978-3-662-25762-3 . Translated to Russian under the title Funktsional’nyy analiz. Moscow, Mir Publ., 1967, 624 p.
16. Ul’yanov P.L. On Haar series. Mat. Sb. (N.S.), 1964, vol. 63(105), no. 3, pp. 356–391 (in Russian).
Cite this article as: V.I. Filippov. Integer expansion of elements from spaces of nonintegrable functions into Fourier-type series in systems of contractions and shifts of one function. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 286–300.
