Geodesic
On the Representation of Measurable Functions by Absolutely Convergent Orthogonal Spline Series
Informatics and Automation, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 73-82.

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We show that if $\{f_n(t)\}_{n=-m+2}^{\infty }$ is an orthonormal system in $L^2[0,1]$ consisting of splines of order $m$ with dyadic rational knots and $f(t)$ is an a.e. finite measurable function, then, first, there exists a series with respect to this system that converges absolutely a.e. to this function and, second, for any $\varepsilon >0$ the function $f(t)$ can be changed on a set of measure less than $\varepsilon $ so that the corrected function has a uniformly absolutely convergent Fourier series with respect to this system.
Keywords: spline of order $m$, absolutely convergent series, representation of functions, correction of functions. 
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G. G. Gevorkyan. On the Representation of Measurable Functions by Absolutely Convergent Orthogonal Spline Series. Informatics and Automation, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 73-82. http://geodesic.mathdoc.fr/item/TRSPY_2022_319_a5/

[1] Arutyunyan F.G., “O ryadakh po sisteme Khaara”, DAN ArmSSR, 42:3 (1966), 134–140

[2] Ciesielski Z., “Spline bases in function spaces”, Approximation theory: Proc. Conf. (Poznan, 1972), D. Reidel Publ. Co., Dordrecht, 1975, 49–54 | DOI | MR

[3] Ciesielski Z., “Constructive function theory and spline systems”, Stud. math., 53:3 (1975), 277–302 | DOI | MR

[4] DeVore R.A., Lorentz G.G., Constructive approximation, Grundl. Math. Wiss., 303, Springer, Berlin, 1993 | MR

[5] Franklin Ph., “A set of continuous orthogonal functions”, Math. Ann., 100 (1928), 522–529 | DOI | MR

[6] Gevorkyan G.G., “O predstavlenii izmerimykh funktsii absolyutno skhodyaschimisya ryadami po sisteme Franklina”, DAN ArmSSR, 83:1 (1986), 15–18

[7] G. G. Gevorkyan, “On the convergence of Franklin series to $+\infty $”, Math. Notes, 106:3–4 (2019), 334–341 | DOI | MR

[8] Gevorkyan G.G., Keryan K.A., Poghosyan M.P., “Convergence to infinity for orthonormal spline series”, Acta math. Hung., 162:2 (2020), 604–617 | DOI | MR

[9] Gundy R.F., “Martingale theory and pointwise convergence of certain orthogonal series”, Trans. Amer. Math. Soc., 124:2 (1966), 228–248 | DOI | MR

[10] Haar A., “Zur Theorie der orthogonalen Funktionensysteme. (Erste Mitteilung.)”, Math. Ann., 69:3 (1910), 331–371 | DOI | MR

[11] S. V. Konyagin, “Limits of indeterminacy of trigonometric series”, Math. Notes, 44:6 (1988), 910–920 | DOI | MR

[12] Luzin N.N., Integral i trigonometricheskii ryad, Gostekhizdat, M.; L., 1951 | MR

[13] V. A. Skvortsov, “Differentiation with respect to nets and the Haar series”, Math. Notes, 4:1 (1968), 509–513 | DOI | MR

[14] Talalyan A.A., Arutyunyan F.G., “O skhodimosti ryadov po sisteme Khaara k $+\infty $”, Mat. sb., 66:2 (1965), 240–247