Geodesic
On the Representation of Measurable Functions by Absolutely Convergent Orthogonal Spline Series
Trudy Matematicheskogo Instituta imeni V.A. Steklova, 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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     author = {G. G. Gevorkyan},
     title = {On the {Representation} of {Measurable} {Functions} by {Absolutely} {Convergent} {Orthogonal} {Spline} {Series}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {73--82},
     publisher = {mathdoc},
     volume = {319},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2022_319_a5/}
}
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G. G. Gevorkyan. On the Representation of Measurable Functions by Absolutely Convergent Orthogonal Spline Series. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 73-82. http://geodesic.mathdoc.fr/item/TM_2022_319_a5/