Remarks on the Rademacher-Menshov Theorem Christopher Meaney Abstract We describe Salem s proof of the Rademacher-Menshov Theorem, which shows that one constant works for all orthogonal expansions in all L -spaces. By changing the emphasis in Salem s proof we produce a lower bound for sums of vectors coming from bi-orthogonal sets of vectors in a Hilbert space. This inequality is applied to sums of columns of an invertible matrix and to Lebesgue constants. Keywords: Orthogonal expansion, Rademacher-Menshov Theorem, Bessel s inequality, bi-orthogonal, Lebesgue constants. MSC (000): 4C15, 46C05. Received 1 August 006 / Accepted 18 October 006. 1 Introduction Here we give an exposition of Salem s proof [14] of the Rademacher-Menshov Theorem. Although it is more elaborate than some proofs and over sixty years old, Salem s method makes it clear that one constant works for all orthogonal expansions in all L -spaces. Furthermore, some of the inequalities used in the proof lead to a general inequality concerning bi-orthogonal sets of vectors in Hilbert spaces (Proposition in the next section.) In recent work [] with Leonardo Colzani and Elena Prestini, we used the universal nature of the constant in the Rademacher-Menshov Theorem [13, 11] to produce some almost-everywhere convergence results for inverse Fourier Transforms. Theorem 1 below contains the basic idea used in that work. I am grateful to the organizers of the Murramarang conference for the opportunity to participate. 100
Statement of Main Results Proposition.1. There is a positive constant C with the following property. For every positive measure space (, µ), for every n 1, and for every finite set {F 1,..., F n } of orthogonal functions in L (, µ), the maximal function M(x) = max F 1 m n j (x) (.1) has norm ( 1/ M C log (n + 1) F j ). (.) Suppose H is a Hilbert space, with inner-product written as v, w. We say that two sets of vectors {v 1,..., v n } and {w 1,..., w n } are bi-orthogonal when v j, w k = 0, j k. Proposition.. There is a positive constant c with the following property. For every Hilbert space H and every pair of bi-orthogonal sets {v 1,..., v n } and {w 1,..., w n } in H, (log n) min v k, w k c max w m max 1 k n 1 m n 1 k n k v j. (.3) These are proved in Section 4. In the next section we give some applications. 3 Consequences 3.1 Almost everywhere convergence. Suppose that (, µ) is a positive measure space and that L (, µ) = n=1h n is an orthogonal decomposition into closed subspaces H n. Let P n be projection onto H n. Each function f L (, µ) has an orthogonal expansion P n f, n=1 101
which converges to f in norm. The partial sum operators are S N f(x) = N P n f(x), N 1, x. n=1 Proposition.1 says that max S mf 1 m N C log(n + 1) S N f, N 1, f L (, µ). Define the maximal function S f(x) = sup S N f(x). N 1 This is dominated by two pieces, ( ) S f(x) sup S mf(x) + sup max S nf(x) S mf(x). m 0 m 0 m n< m+1 We can apply the Cauchy-Schwarz inequality to control the dyadic piece, as on pages 80 81 of [1], m k ( m ) P n f(x) = P n f(x) 1 k k k P n f(x). n= n= k 1 +1 This implies that sup S mf m 0 c (log(n + 1)) P n f. n=1 n= k 1 +1 For the other term, notice that if we have a non-negative sequence (a m ) m=1 then sup a m a m. m 1 m=1 We can use Proposition.1 to show that ( ) sup max S nf S mf C m 0 m n< m+1 m=0 m+1 1 (log ( m + 1)) n= m P n f. Combining these facts gives the general form of the Rademacher-Menshov Theorem. 10
Theorem 3.1. There is a positive constant α so that for all f L (, µ), ( 1/ S f α (log(n + 1)) P n f ). n=1 If the right hand side is finite, then f(x) = lim N S Nf(x), almost everywhere on. Remark 3.1. This method was used in [, 9, 10]. 3. Invertible Matrices. Suppose we equip C n with its usual inner product. If A is an invertible n n matrix with complex entries then the equation A 1 A = I can be viewed as saying that the columns of A and the rows of A 1 form a pair of bi-orthogonal sets in C n. In this case, Proposition. gives the following result. Theorem 3.. Suppose that {a 1,..., a n } are the columns of an n n invertible matrix with complex entries A and that {b 1,..., b n } are the rows of A 1. Then log n c max b j max a 1 + + a m, 1 j n 1 m n where c is a positive constant independent of n and A. 3.3 Lebesgue Constants This example follows the methods of another paper of Salem [15]. Suppose that {φ 1,..., φ n } is an orthonormal subset of L (, µ) consisting of essentially bounded functions, with φ j M, 1 j n. Define the maximal function Φ(x) = max φ j (x) 1 m n φ j (x). 103
If Φ(x) = 0 then φ j (x) = 0 for 1 j n and so the set of places where Φ(x) = 0 can be discarded from without any effect on our calculations. Notice that for all x where Φ(x) 0, we have m φ j(x) Φ(x), 1 m n. Φ(x) On the set where Φ(x) 0, define g j = φ j / Φ and h j = φ j Φ. These give bi-orthogonal sets {g 1,..., g n } and {h 1,..., h n } in L (, µ). Furthermore, g j, h k = δ jk, g j Φ 1 and h j M Φ 1. Proposition. says that log n cm Φ 1. Theorem 3.3. There is a positive constant β with the following property. Suppose that {φ 1,..., φ n } is an orthonormal set in L (, µ) consisting of essentially bounded functions, with M = max 1 j n φ j. Then max 1 m n φ j β log(n)/m. 1 Remark 3.. This is a weak form of an inequality conjectured by Littlewood. For much stronger results in the case of characters on compact abelian groups see [8, 6]. For other orthonormal systems see [7, 3]. The inequality here can also be viewed as a special case of Theorem 1 of Olevskiĭ s book [1]. 4 Proofs Recall Bessel s inequality for orthogonal vectors in a Hilbert space (page 531 of [5].) Suppose that {v 1,..., v n } is an orthogonal set of non-zero vectors in 104
a Hilbert space H. Then {v 1 / v 1,..., v n / v n } is an orthonormal set in H and for every vector w H we have 4.1 Proof of.1 w, v j v j w. (4.1) Here we rework the proof published by Salem [14] in 1941 in a slightly more abstract setting. 4.1.1 The general set up. Suppose that H is a Hilbert space. Now let V = L (, µ) H be the Hilbert space of H-valued µ-measurable square-integrable functions on. Let {v 1,..., v n } and {w 1,..., w n } be a bi-orthogonal pair of subsets of H and define some elements of V by multiplying terms, p k (x) = F k (x)w k, 1 k n. Then {p 1,..., p n } is an orthogonal subset of V and (4.1) states that P, p k V p k V P V, P V. (4.) Let f 1 f f n f n+1 = 0 be a decreasing sequence of characteristic functions of measurable subsets of. For G L (, µ) define an element of V by P G (x) = G(x) f j (x)v j. (4.3) The Abel transformation lets us rewrite this as P G (x) = G(x) f k (x)σ k, where σ k = k v j and f k = f k f k+1, for 1 k n. Notice that { f 1,..., f n } is a set of characteristic functions of mutually disjoint subsets 105
of. For each x, at most one of the terms f k (x) is non-zero. In particular, P G (x) H = G(x) f k (x) σ k H. Integrating over gives P G V G max σ k H. 1 k n Combining this with (4.), we have F k G f k dµ vk, w k F k w k H G max v 1 + + v k H. (4.4) 1 k n 4.1. A specific case. Following Salem, let us now assume that H = L (0, 1) and v k (t) = t sin (πkt) and w k (t) = sin (πkt) / t, 0 < t < 1, k 1. The usual estimates on Lebesgue constants (page 67 in [16]) show that w k H A log (k + 1) and v 1 + + v k H B log (k + 1), 1 k n. The constants A and B are independent of k and n. Furthermore, v k, w k = 1, 1 k n. For this choice of H, inequality (4.4) becomes 1 log(k + 1) F k G f k dµ F k AB G log(n + 1), and the constant AB is independent of, µ, and n. Moving the logarithm term from the left hand side gives F k G f k dµ F k AB G (log(n + 1)). (4.5) 106
4.1.3 Controlling the maximal function. Define an integer-valued function m(x) on by { } m(x) = min m : F k (x) = M(x), x, and let f k be the characteristic function of the subset {x : m(x) k}. For each x there is the partial sum S m(x) (x) = m(x) F k (x) = f k (x)f k (x). For an element G L (, µ), Cauchy-Schwarz gives G(x)S m(x) (x) dµ(x) = F k F k G f k dµ F k ( ) 1/ ( F k F k G f k dµ F k (4.6) ) 1/ Using inequality (4.5) gives G(x)S m(x) (x) dµ(x) ( 1/ AB G log(n + 1) F k ). This is true for all G L (, µ) and so it follows that M = ( 1/ S m( ) C log(n + 1) F k ).. (4.7) This completes the proof of Proposition.1. For alternative proofs, see.3.1 on page 79 of [1] and Chapter 8 of [4]. 4. Menshov s Result In 193 Menshov [11] showed that the logarithm term in Proposition.1 is best possible. The following is taken from page 55 of [4]. 107
Lemma 4.1. There is a positive constant c 0 so that for every n there is an orthonormal subset {ψ1 n, ψ n,..., ψn} n in L (0, 1) for which the set { } j x [0, 1] ; max ψk n (x) 1 j n > c 0 n log(n) has Lebesgue measure greater than 1/4. Notice that this means that the maximal function Ψ n (x) = max j 1 j n ψn k (x) satisfies Ψ n n (log(n)) c 0 4 and yet n ψ n j = n. 4.3 Proof of Proposition. We use the set {ψ n 1, ψ n,..., ψ n n} as the orthonormal set in Salem s proof of Proposition.1. Keeping the earlier notation, fix a function G on [0, 1] for which G(x) = 1 and G(x)S m(x) (x) = Ψ n (x) 0. Since Ψ n is nonnegative, inequality (4.6) becomes Ψ n 1 ( 1 ) 1/ n G(x)f k (x)ψk n(x) dx. Put this back into inequality (4.4) to get Ψ n 1 n Lemma 4.1 shows that and so c 0 (log(n)) 16 min 1 k n v k, w k max 1 m n w m H Ψ n 1 n 0 max 1 k n v 1 + + v k H. c (log(n)) 0 16 min v k, w k max w m 1 k n 1 m n H max v 1 + + v k H. 1 k n This completes the proof of Proposition.. 108
References [1] György Alexits, Convergence problems of orthogonal series, Translated from the German by I. Földer. International Series of Monographs in Pure and Applied Mathematics, Vol. 0, Pergamon Press, New York, 1961. [] Leonardo Colzani, Christopher Meaney, and Elena Prestini, Almost everywhere convergence of inverse Fourier transforms, Proc. Amer. Math. Soc. 134 (006), no. 6, 1651 1660. [3] Saverio Giulini, Paolo M. Soardi, and Giancarlo Travaglini, A Cohen type inequality for compact Lie groups, Proc. Amer. Math. Soc. 77 (1979), no. 3, 359 364. [4] B. S. Kashin and A. A. Saakyan, Orthogonal series, Translations of Mathematical Monographs, vol. 75, American Mathematical Society, Providence, RI, 1989. [5] Anthony W. Knapp, Basic real analysis, Cornerstones, Birkhäuser Boston Inc., Boston, MA, 005. [6] S. V. Konyagin, On the Littlewood problem, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no., 43 65, 463. [7] C. Markett, Cohen type inequalities for Jacobi, Laguerre and Hermite expansions, SIAM J. Math. Anal. 14 (1983), no. 4, 819 833. [8] O. Carruth McGehee, Louis Pigno, and Brent Smith, Hardy s inequality and the L 1 norm of exponential sums, Ann. of Math. () 113 (1981), no. 3, 613 618. [9] Christopher Meaney, On almost-everywhere convergent eigenfunction expansions of the Laplace-Beltrami operator, Math. Proc. Cambridge Philos. Soc. 9 (198), no. 1, 19 131. [10] Christopher Meaney, Detlef Müller, and Elena Prestini, A. e. convergence of spectral sums on Lie groups, to appear in Ann. l Inst. Fourier. [11] D. Menchoff, Sur les séries de fonctions orthogonales. (Premiére Partie. La convergence.), Fundamenta math. 4 (193), 8 105. 109
[1] A. M. Olevskiĭ, Fourier series with respect to general orthogonal systems, Springer-Verlag, New York, 1975. [13] H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (19), 11 138. [14] Raphaël Salem, A new proof of a theorem of Menchoff, Duke Math. J. 8 (1941), 69 7. [15] Raphaël Salem, On a problem of Littlewood, Amer. J. Math. 77 (1955), 535 540. [16] Antoni Zygmund, Trigonometric series. nd ed. Vols. I, II, Cambridge University Press, New York, 1959. Christopher Meaney, Department of Mathematics, Macquarie University, North Ryde, NSW 109 Australia. chrism@maths.mq.edu.au 110