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\(M\)-positive sets in \(\mathbb{R}^d\). (English. Russian original) Zbl 1490.42029

Math. Notes 111, No. 4, 643-647 (2022); translation from Mat. Zametki 111, No. 4, 631-635 (2022).
The authors study an orthogonal Walsh-type basis for the space \(L_2(U),\) where \(U\) is a compact self-similar set in \(\mathbb R^d\) playing the role of the unit interval on the half-line. The characteristic functions of such self-similar sets are known to be the scaling functions used for constructing wavelets in \(L_2(\mathbb R^d).\) The authors define Walsh type functions on a set of \(M\)-positive vectors and show that they form an orthogonal basis for the space \(L_2(U).\)

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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