Towards direct $L^2$-bounds for maximal partial sums of Walsh--Fourier series: The case of dyadic partial sums

The study of pointwise convergence of Fourier series eventually culminated in the much-celebrated Carleson-Hunt theorem [10,22], cf., [15,24,32] and [5,27] for later perspectives. The result led to further investigations addressing convergence of Fourier series and integrals in higher dimensional Euclidean settings, e.g., [2,3,9,30] and more recently [28,33]; convergence in topological group settings, e.g., [13,23,37] and more recently [4,16,17,35]; spaces on which maximal partial sum operators are bounded, e.g., [29,36] and convergence of expansions in other orthogonal functions, e.g., [6,20,25,31] including orthogonal polynomials or special functions, e.g., [11,12]; and other related questions.

The concept of lacunary partial sums1 has provided corresponding convergence results in each of these settings, including [1,4,7,14,17] and others. In 1924 Kolmogorov [26] established almost-everywhere convergence of lacunary partial sums of Fourier series in L 2 [0, 1] by means of comparison with Cesàro means. That approach did not consider any norm bound on a maximal partial sum operator.

In this work we consider dyadic partial sums S 2 N of expansions in Walsh functions. Almost-everywhere convergence of Walsh-Fourier series of functions in L 2 [0, 1] was established by Billard [8] shortly after Carleson’s work appeared (cf., Hunt [21]). It can be regarded as the first extension of Carleson’s methods to a different setting. The fact that Walsh functions are characters of the Cantor group allowed Gosselin [19] to extend Carleson’s approach to Vilenkin groups that have a parallel structure. Subsequent work by Gosselin [18] extended C. Fefferman’s approach to almost everywhere convergence of Fourier series in [15] to the Walsh setting. The script was flipped starting with Thiele’s work in 2000 [38] when the Walsh setting was seen to provide a somewhat cleaner context for a phase space (wavepacket) approach to questions of boundedness of a family of operators that included partial sum operators and certain multilinear singular integral operators, cf., Muscalu et al. [34]. The approach to uniform boundedness of linearized partial sum operators outlined here is fundamentally different from prior approaches and uses special properties of Walsh functions.

Our goal here is in one sense much more modest than the aforementioned works: we seek concrete bounds on operator norms of a restricted family of dyadic partial sum operators in the Walsh setting. On the other hand the ultimate goal (not achieved here) is a sharp uniform L 2 → L 2 bound on this family of operators, which is somewhat new in the study of Fourier series where maximal partial sum bounds are initially established on a different space (L(log + L) 1+δ in Carleson’s work) with L 2 bounds following by interpolation. The program at hand seeks a direct L 2 bound on dyadic partial sums, using optimization methods to identify families of matrices corresponding to dyadic partial sums that should have maximal norms. We outline very briefly at the end a different dilation approach that should also extend to provide direct L 2 estimates for general (not necessarily dyadic) maximal partial sums of Walsh expansions.

In the case of dyadic partial sums of Walsh-Fourier series we conjecture an explicit optimal bound on a class of linearized dyadic partial sum operators, provide arguments for the plausibility of the conjecture and support these arguments with concrete numerical evidence. The partial sum estimates are applied to dyadic step functions that are constant on dyadic intervals k/2 N , (k + 1)/2 N , k = 0, . . . , 2 N -1. This allows rephrasing boundedness of partial sum operators in terms of boundedness of a family of what we refer to as (dyadically) truncated Walsh-Hadamard (DTWH) matrices.

Here is an outline of the presentation. In Sect. 2 we review the Walsh functions and show that on dyadic step functions on [0, 1], certain linearized Walsh-Fourier (dyadic) partial sum operators can be represented in terms of DTWH matrices. We then state a conjecture (Conj. 1) that would provide a sharp bound on norms of maximal dyadic partial sum operators as well as a second (looser) conjecture (Conj. 3) on uniform norm bounds for linearized Walsh-Fourier partial sum operators that are not-necessarily dyadic. In Sect. 3 we formalize a secondary conjecture (Conj. 6) based on branching properties of the sets of columns of DTWH matrices that would lead to a proof of Conj. 1. In Sect. 4 we state a specific case (Conj. 8) of Conj. 6, outline a reduction to this special case, and provide evidence of its validity. In Sect. 5 we outline a path to fill in further details needed to prove Conj. 8 from which the main conjecture, Conj. 1, would follow. Finally, in Sect. 6 we discuss general truncations and out-line very broadly steps that are needed to prove Conj. 3. A proposition supporting Conj. 8 that is stated in Sect. 4 is proved in Appendix. A.

When referring to the norm of a m

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