Haar's dyadic orthogonal system with localized convergent expansions

What changed

Scientific scope

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. The 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.

Evidence

The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The 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.

Contributors

The named authors in the research record · SCIENCE: Source-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.

Zur Theorie der orthogonalen Funktionensysteme, Erste Mitteilung

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