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The results bridges the theoretical gap in implementing the derivative sampling theorems for band-limited signals in the LCT domain. Both complex analysis and real analysis techniques are established to derive the convergence analysis. The truncation error is also investigated and rigorous estimates for it are given. Nevertheless, the convergence rate is<jats:inline-formula><jats:alternatives><jats:tex-math>$$O(1\/\\sqrt{N})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>O<\/mml:mi><mml:mo>(<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:msqrt><mml:mi>N<\/mml:mi><\/mml:msqrt><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, which is slow. 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