Ultradifferentiable classes of entire functions.

Area of Science:

• Analysis
• Functional Analysis
• Harmonic Analysis

Background:

• Ultradifferentiable functions are crucial in analysis.
• Standard growth and regularity requirements are often violated in certain function classes.
• Weight sequences play a key role in defining function spaces.

Purpose of the Study:

• To investigate ultradifferentiable function classes defined by small weight sequences.
• To establish connections between these classes and weighted spaces of entire functions.
• To generalize existing results and explore applications in operator theory.

Main Methods:

• Analysis of function classes defined by weight sequences.
• Utilizing associated weight functions and conjugate weight sequences.
• Generalizing results from the small Gevrey setting.
• Applying function classes to study operator boundedness on Hilbert spaces.

Main Results:

• Demonstrated that ultradifferentiable function classes can be viewed as weighted spaces of entire functions.
• Generalized results from the small Gevrey setting to arbitrary convenient families of sequences.
• Showed the applicability of these function classes in detecting boundedness of normal linear operators on Hilbert spaces.

Conclusions:

• The study provides a novel perspective on ultradifferentiable functions via weight sequences.
• Established a link between small sequences and dual sequences, broadening the theoretical framework.
• The findings have implications for understanding operator behavior in evolution equation problems.