## Some new Fourier and Jackson-Nikol’skii type inequalities in unbounded orthonormal systems.(English)Zbl 07528442

Summary: We consider the generalized Lorentz space $$L_{\psi,q}$$ defined via a continuous and concave function $$\psi$$ and the Fourier series of a function with respect to an unbounded orthonormal system. Some new Fourier and Jackson-Nikol’skii type inequalities in this frame are stated, proved and discussed. In particular, the derived results generalize and unify several well-known results but also some new applications are pointed out.

### MSC:

 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series 42B05 Fourier series and coefficients in several variables 42C15 General harmonic expansions, frames 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 26D15 Inequalities for sums, series and integrals 26D20 Other analytical inequalities 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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