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The variation of general Fourier coefficients of Lipschitz class functions. (English) Zbl 1499.42127

This work investigates variational behavior of Fourier coefficients of Lipschitz functions \(f\) (\(\Vert f\Vert_{{\mathrm{Lip}} 1}=\Vert f\Vert_\infty+\sup_{x\neq y}\vert f(x)-f(y)\vert /\vert x-y\vert \)) with respect to general orthonormal systems (ONSs) on \([0,1]\). Given an ONS \(\{\varphi_n\}\) on \([0,1]\), and real-valued \(f\in L^2[0,1]\), let \(C_n(f)=\int_0^1 f(x)\,\varphi_n(x)\, dx\) denote the \(n\)-th Fourier coefficient of \(f\) with respect to \(\{\varphi_n\}\) and, for a sequence \(s=\{s_n\}\subset\mathbb{R}\) set \(P_n(x,s)=\sum_{k=1}^n (\varphi_k(x)-\varphi_{k+1}(x))\, s_k\) and \(R_n(s)=\frac{1}{n}\sum_{i=1}^{n-1}\int_0^{\frac{i}{n}} P_n(x,s)\, dx\). Denote by \(W_p\) the space of sequences \(\{a_n\}\) satisfying \(\sum_{k=1}^\infty \vert a_k-a_{k+1}\vert ^p<\infty\). Write \(W_1=W\). Let \(\Delta\) denote the sequences \(\{t_n\}\) with values in \(\{0,\pm 1\}\). In the case \(p=1\), Theorem. 1 states that if \(\{\varphi_n\}\) is an ONS on \([0,1]\), if \(\{C_n(1)\}\in W\) and \(R_n(t)=0(1)\) whenever \(t=\{t_n\}\in\Delta\), then for any \(f\) in the Lipschitz class \( {\mathrm{Lip}}\, 1\), \(\{C_n(f)\}\in W\). Theorem 2 states, conversely, that if for some \(t\in \Delta\), \(\limsup_{n\to\infty} R_n(t)=+\infty\) then there is a \(g\in {\mathrm{Lip}} \, 1\) such that \(\{C_n (g)\}\notin W\). In the case \(p>1\), Theorem 3 states that if \(\{C_n(1)\}\in W_p\), and if for arbitrary \(a=\{a_n\}\in\ell_q\) (\(q=p/(p-1)\)), \(R_n(a)=0(1)\), then \(\{C_n(f)\}\in W_p\) whenever \(f\in {\mathrm{Lip}}\, 1\). Conversely, if there is a \(b=\{b_n\}\in \ell_q\) such that \(\limsup_{n\to\infty} R_n(b)=+\infty\), then there is a \(g\in {\mathrm{Lip}}\, 1\) such that \(\{C_n(g)\} \notin W_p\). Theorem 5 shows that given a quickly growing increasing sequence, subsequences of ONS’s can be constructed such that \(\{C_{n_k}(f)-C_{n_{k+1}}(f)\}\) is absolutely summable against the sequence, whenever \(f\in {\mathrm{Lip}}\, 1\). Theorems 6–9 show that if \(\{\varphi_n\}\) is either the trigonometric system \(\sqrt{2}\cos (2\pi nx)\) or the Haar wavelet system on \([0,1]\) (suitably indexed), then for any \(t\in\Delta\), \(R_n(t)=O(1)\) so Theorem 1 applies: whenever \(f\in {\mathrm{Lip}}\,1\), \(\{C_n(f)\}\in W\).

MSC:

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