What is recorded here?
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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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.
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People and systems
Alfred Haar
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.
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English catalogue narrative
Haar's dyadic orthogonal system with localized convergent expansions
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.
Scientific scope
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.
Reading status
This 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.
People and systems
Alfred Haar
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.