SWhat changed
Weierstrass guarantees that a continuous curve can be approximated by polynomials, but does not give one useful speed for all continuous curves. Jackson adds a regularity condition: the change in the function must be bounded by a constant times the change in its input. For such a Lipschitz function on a compact interval, he constructs polynomials of degree at most n whose largest error is bounded by a constant divided by n. His route first smooths the curve with a fourth-power sinc kernel, then replaces that kernel by a polynomial while tracking error and degree. This supplies a quantitative approximation capability.
Scientific scope
Weierstrass guarantees that a continuous curve can be approximated by polynomials, but does not give one useful speed for all continuous curves. Jackson adds a regularity condition: the change in the function must be bounded by a constant times the change in its input. For such a Lipschitz function on a compact interval, he constructs polynomials of degree at most n whose largest error is bounded by a constant divided by n. His route first smooths the curve with a fourth-power sinc kernel, then replaces that kernel by a polynomial while tracking error and degree. This supplies a quantitative approximation capability. The constant depends on the function and interval; the theorem does not give the same rate for all continuous functions, a sharp best constant or convergence of ordinary Fourier partial sums. The linked archival scan repeats printed page 18 and omits page 19. Its theorem statement is available, but the scan has this gap in the construction. Other dissertation extensions are outside this contribution.
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 constant depends on the function and interval; the theorem does not give the same rate for all continuous functions, a sharp best constant or convergence of ordinary Fourier partial sums. The linked archival scan repeats printed page 18 and omits page 19. Its theorem statement is available, but the scan has this gap in the construction. Other dissertation extensions are outside this contribution.
Contributors
- The named authors in the research record · SCIENCE: Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred.
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