This research establishes a formal theoretical foundation for prompt engineering, demonstrating how LLMs can approximate smooth functions through well-designed prompts.

• Mathematical rigor: Applies approximation theory to explain why certain prompt strategies work better than others
• Performance optimization: Provides a framework for systematically designing effective prompts rather than relying on trial and error
• Engineering applications: Enables more predictable LLM behavior for critical engineering systems requiring reliability and precision
• Future development: Creates a foundation for developing more advanced prompt engineering techniques

For engineering teams, this framework transforms prompt creation from an art to a science, allowing for more systematic design of LLM interactions in technical applications.

A Theoretical Framework for Prompt Engineering: Approximating Smooth Functions with Transformer Prompts