You're absolutely right! Several wavelet properties can be successfully transferred to quaternion spaces. Here are some key characteristics that hold promise:

• Localization: Wavelets offer the ability to analyze signals in both time and frequency domains. This property can be extended to quaternions, allowing you to pinpoint specific features within quaternion-valued data.
• Multiscale Analysis: Decomposing a signal into different scales (coarse and fine) is a core wavelet function. This multiscale analysis can be adapted to quaternion spaces, enabling you to extract information at varying levels of detail.
• Orthogonality or Biorthogonality: Wavelet bases can be orthogonal (wavelets are perpendicular in a mathematical sense) or biorthogonal (dual wavelets have a specific relationship). These properties can be translated to quaternion spaces, leading to efficient signal reconstruction.
• Vanishing Moments: Wavelets can have vanishing moments, meaning they cancel out polynomials of a certain degree. This property can be adapted to quaternions, allowing for better representation of specific data types.

However, it's important to consider some potential challenges:

• Non-Commutativity: Unlike real or complex numbers, quaternions don't commute (changing the order of multiplication matters). This non-commutative nature might require adjustments to traditional wavelet constructions.
• Computational Complexity: Quaternion operations can be more computationally expensive compared to real or complex numbers. This might lead to increased processing time when applying wavelets to quaternion spaces.

Here's some guidance for your exploration:

• Research Papers: Explore research papers on quaternion wavelet transforms. Look for studies that discuss the transferability of specific wavelet properties (e.g., vanishing moments, orthogonality) to quaternions.
• Software Libraries: Some libraries exist for quaternion-valued signal processing. Investigate if they offer wavelet transform functionalities specific to quaternions.
• Consult an Expert: If you're working on a specific application, consider consulting a mathematician or signal processing expert familiar with quaternion wavelets. They can provide tailored advice based on your needs.

Remember, research in this field is ongoing. New wavelet constructions and algorithms might overcome the challenges associated with non-commutativity and computational complexity.