Abstract In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemarié wavelet system of nonnegative integer order, n . We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to $$s < n+1$$ s < n + 1 , which includes values of s where the Battle-Lemarié system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case $$s=n+1$$ s = n + 1 . This builds off of earlier work by G. Garrigós, A. Seeger, and T. Ullrich, where they proved the case $$n=0$$ n = 0 , i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemarié system to give an unconditional basis of the Triebel-Lizorkin spaces.

Abstract The study of some signal processing problems within Bayesian frameworks and semigroups theory, in the case where the Banach space under consideration may be nonseparable, was discussed in the paper published in 2022 by two of the authors. For the Cauchy problem for the diffusion equation, which is a typical example of evolution equation, there was studied the question of what can be saved in the study of the problem in the case with the initial data in the nonseparable classical Morrey space. In this paper we develop this approach to an essentially more general case of generalized Morrey spaces with general weights. The study is based on emersion of such spaces into weighted Lebesgue spaces with Muckenhoupt weights.

Abstract We show that the method in the recent work by Roncal, Shrivastava, and Shuin can be adapted to show that certain $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -improving bounds in the interior of the boundedness region for the bilinear spherical or triangle averaging operator imply sparse bounds for the corresponding lacunary maximal operator, and that $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -improving bounds in the interior of the boundedness region for the single-scale maximal bilinear spherical averaging operator implies sparse bounds for the corresponding full maximal operator. More generally we show that the proof applies for bilinear convolutions with compactly supported finite Borel measures that satisfy appropriate $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -improving and continuity estimates. This shows that the method used by Roncal, Shrivastava, and Shuin can be adapted obtain sparse bounds for a general class of bilinear operators that are not of product type, for a certain range of $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> exponents.