Local Convergence of the Continuous and Semi-Discrete Wavelet Transform in <em>L^p</em>(ℝ)

• Main Authors: Jaime Navarro-Fuentes, Salvador Arellano-Balderas, Oscar Herrera-Alcántara
• Format: Article
• Language: English
• Published: MDPI AG 2021-03-01
• Series: Mathematics
• Subjects: admissibility condition; the continuous wavelet transform; inversion formula; semi-discrete wavelet transform; tight frames
• Online Access: https://www.mdpi.com/2227-7390/9/5/522

The smoothness of functions <i>f</i> in the space <inline-formula><math display="inline"><semantics><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></semantics></math></inline-formula> is studied through the local convergence of the continuous wavelet transform of <i>f</i>. Additionally, we study the smoothness of functions in <inline-formula><math display="inline"><semantics><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> by means of the local convergence of the semi-discrete wavelet transform.