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      "statement": "# Haar's dyadic orthogonal system with localized convergent expansions\n\nHaar constructs a complete orthonormal system of localized dyadic step functions on [0,1]. In its specified natural order, Fourier-Haar expansions converge uniformly for continuous functions and almost everywhere for Lebesgue-integrable functions. The proof uses localized positive averaging kernels; dense recursive partitions yield related systems.\n\n## Scientific scope\n\nThe basis functions are discontinuous step functions: this is not a Schauder basis made of continuous elements of C[0,1], nor a claim that arbitrary reordering preserves convergence. No general wavelet optimality, unconditionality in every space or the paper's later Sturm–Liouville results is included.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nAlfred Haar\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Zur Theorie der orthogonalen Funktionensysteme, Erste Mitteilung](https://history-of-approximation-theory.com/fpapers/haar1910.pdf)\n",
      "summary": "Haar constructs a complete orthonormal system of localized dyadic step functions on [0,1]. In its specified natural order, Fourier-Haar expansions converge uniformly for continuous functions and almost everywhere for Lebesgue-integrable functions. The proof uses localized positive averaging kernels; dense recursive partitions yield related systems.",
      "title": "Haar's dyadic orthogonal system with localized convergent expansions"
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    "what_changed": "Haar constructs a complete orthonormal system of localized dyadic step functions on [0,1]. In its specified natural order, Fourier-Haar expansions converge uniformly for continuous functions and almost everywhere for Lebesgue-integrable functions. The proof uses localized positive averaging kernels; dense recursive partitions yield related systems.",
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