The Hilbert transform

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1 (February 14, 217) The Hilbert transform Paul arrett garrett/ [This document is garrett/m/fun/notes /hilbert transform.pdf] 1. The principal-value functional 2. Other descriptions of the principal-value functional 3. Uniqueness of odd/even homogeneous distributions 4. Hilbert transforms 5. Eamples Appendi: uniqueness of equivariant functionals 8. Appendi: distributions supported at {} 9. Appendi: the snake lemma Formulaically, the Cauchy principal-value functional η is ηf = principal-value functional of f = P.V. d = ε + >ε d This is a somewhat fragile presentation. In particular, the apparent integral is not a literal integral! The uniqueness proven below helps prove plausible properties like the Sokhotski-Plemelj theorem from [Sokhotski 1871], [Plemelji 198], with a possibly unepected leading term: d = iπ f() + P.V. ε + + iε d See also [elfand-silov 1964]. Other plausible identities are best certified using uniqueness, such as ηf = 1 2 f( ) d (for f C 1 () L 1 ()) using the canonical continuous etension of f( ) ηf = f() e 2 d at. Also, (for f C o () L 1 ()) The Hilbert transform of a function f on is awkwardly described as a principal-value integral (Hf)() = 1 π P.V. f(t) t dt = 1 π f(t) ε + t >ε t dt with the leading constant 1/π understandable with sufficient hindsight: we will see that this adjustment makes H etend to a unitary operator on L 2 (). The formulaic presentation of H makes it appear to be a convolution with (a constant multiple of) the principal-value functional. The fragility of these presentations make eistence, continuity properties, and range of applicability of H less clear than one would like. 1

2 Paul arrett: The Hilbert transform (February 14, 217) 1. The principal-value functional The principal-value functional η is better characterized as the unique (up to a constant multiple) odd distribution on, positive-homogeneous of degree as a distribution (see below). This characterization allows unambiguous comparison of various iting epressions closely related to the principal-value functional. Further, the characterization of the principal-value functional makes discussion of the Hilbert transform less fragile and more convincing. Again, let ηf = principal-value functional of f = P.V. d = ε + >ε d We prove that η is a tempered distribution: [1.1] Claim: For f S, for every h >, Thus, u is a tempered distribution: η(f) h d + 2h sup f () h η(f) sup(1 + 2 ) + sup f () (implied constant independent of f) Proof: Certainly η(f) = h d + ε + ε< h By the Mean Value Theorem, = f() + f (ξ ) for some ξ between and. Thus, ε< h d = ε< h d f() d + f(ξ) d = + f(ξ) d ε< h ε< h because 1/ is odd and f() is even. Thus, ε< h d = ε< h f(ξ) d ε< h f(ξ) d 2h sup ε< h f () 2h sup f () h The latter is independent of ε. With < h 1, d = d + d h 1 h 1 d + 1 d ( ) d + 2h sup 1 h h sup( ) d + 2h sup sup(1 + 2 ) Of course, sup h f () sup f (), giving the corollary. 2

3 Paul arrett: The Hilbert transform (February 14, 217) To make the degree of a positive-homogeneous function ϕ agree with the degree of the integration-against-ϕ distribution u ϕ, due to change-of-measure, we find that (u ϕ t)(f) = u ϕ (f t 1 ) = u ϕ () f(t 1 ) d = u ϕ (t) d(t) = t u ϕ t (f) Thus, for agreement of the notion of homogeneity for distributions and (integrate-against) functions, the dilation action u u t on distributions should be (u t)(f) = 1 t u(f t 1 ) (for t >, test function f, and distribution u) Parity is as epected: u is odd when u( f( )) = u( ) for all test functions f, and is even when u( f( )) = +u( ) for all test functions f. [1.2] Claim: The principal-value distribution η is positive-homogeneous of degree 1, and is odd. Proof: The ε th integral in the it definition of η is itself odd, by changing variables: η(f ( 1)) = f( ) d = ε + ε ε + ε d( ) = d = d = η(f) ε + ε ε + ε as claimed. For the degree of homogeneity, for t > and test function f, (η t)(f) = 1 t η(f 1 t ) = 1 t f( t ) d = 1 ε + ε t = 1 t ε + t ε t d(t) = 1 t ε + ε d() = 1 t η(f) ε + ε/t d() That is, η is homogeneous of degree Other descriptions of the principal-value functional [2.1] Claim: Let ϕ be any even function in L 1 () C 1 (), with ϕ() = 1. Then P.V. d = f() ϕ() where ( f() ϕ() ) / is etended by continuity at =. d (for f C o () L 1 (), for eample) Proof: By the demonstrated odd-ness of the principal-value integral, it is on the even function ϕ. Etending f() ϕ() by continuity at, as claimed. f() ϕ() = ε + ε d = ϕ() d + f() ε + ε f() ϕ() d ε + ε 3 d = P.V. d +

4 Paul arrett: The Hilbert transform (February 14, 217) The leading term in the following might be unepected: [2.2] Claim: (Sokhotski-Plemelj) d = iπ f() + P.V. ε + + iε d Proof: Aiming to apply the previous claim, ε + + iε d = ε + f() iε 1/(1 + 2 ) d + f() d ε + + iε For < ε < 1, the right-most integral is evaluated by residues, moving the contour through the upper half-plane: 1/(1 + 2 ) + iε d = 2πi es =i 1 (1 + 2 )( + iε) = 2πi 1 (i + i)(i + iε) π i = iπ as claimed. [2.3] Claim: With f continuously differentiable at, thereby etending f( ) by continuity at, 1 f( ) 2 d = P.V. d (for f C1 () L 1 ()) Proof: For test function f, since f( ) is continuous as, using the odd-ness of the principal-value functional, 1 f( ) 2 d = 1 2 f( ) d = ε + ε ε + ε d as claimed. 3. Uniqueness of odd/even homogeneous distributions After some clarification of notions of parity and positive-homogeneity, we show that there is a unique distribution of a given parity and positive-homogeneity degree. [3.1] Claim: For a positive-homogeneous distribution u of degree s, d du is again positive-homogeneous of degree s, and with the same parity. Proof: For test function f, (( d d u) t)(f) = 1 t ( d d u)(f t 1 ) = 1 t ( d d u)( f t 1 ) = 1 t u( d d ( (f t 1 ))) = 1 t u(f t 1 + t 1 (f t 1 )) = 1 t u(f t 1 + ( f ) t 1 )) = 1 t u((f + f ) t 1 ) = (u t)(f + f ) = (u t)( d d (f)) = ts ( u( d d (f))) = t s ( d d u)(f) 4

5 Paul arrett: The Hilbert transform (February 14, 217) as asserted. Preservation of parity is similar: writing f () = f( ) for functions f, and u (f) = u(f ) for distributions u, ( d d u) (f) = ( d d u)(f ) = ( d d u)( f ) = u( d d ( f )) = u( d d (( f) )) as claimed. = u( ( d d ( f)) ) = u ( d d ( f)) = (( d d )u )(f) The Euler operator d d is the infinitesimal form of dilation f(t). This is manifest in the following: [3.2] Corollary: d du = s u for positive-homogeneous u of degree s. Proof: Differentiate the distribution-valued equality u t = t s u with respect to t and set t = 1: the righthand side is s u. For the left-hand side, for test function f, since t u t is a smooth distribution-valued function of t, ( t (u t))(f) = t ((u t)(f)) = t (1 t u(f 1 t )) = u( t (1 t f 1 t )) = u ( 1 t 2 (f 1 t ) t 2 (f 1 t ) ) Evaluating at t = 1 gives ( (s u)(f) = u f f ) ( d ) = u(f) ( u)(f ) = u(f) + d ( u) (f) as claimed. = u(f) + (1 u + u )(f) = ( d d u ) (f) [3.3] Theorem: Up to constant multiples, the principal-value distribution η is the unique odd distribution of positive-homogeneity degree 1. More generally, ecept for s = n with ε = ( 1) n+1 with n = 1, 2,..., there is a unique (up to constant multiples) distribution of parity ε and of positive-homogeneity degree s. Proof: Fi parity ε = ±, and let V be the set of test functions of parity ε vanishing to infinite order at. This is the closure, in the space D ε = D ε () of test functions of parity ε, of the space of test functions on with support not containing and of parity ε. On Cc (, + ) there is a unique continuous functional on Cc (, + ) positive homogeneous of degree s (as distribution), by the appendi, given by integration against s. iven a choice of parity ε, there is a unique etension to a continuous linear functional vs ε on Do, ε and of positive-homogeneity degree s. Dualize the short eact sequence V D ε D ε /V to a short eact sequence (invoking Hahn-Banach for eactness at the right-most joint) (D ε /V ) D ε V The quotient D ε /V is identifiable with germs of smooth functions of parity ε supported at modulo germs of smooth functions vanishing identically at of parity ε. The dual is the collection of distributions supported at of parity ε, which by the theory of Taylor-Maclaurin series (see appendi) consists of finite linear combinations of Dirac δ and its derivatives, or parity ε. Of course, δ is even, δ is odd, and so on. 5

6 Paul arrett: The Hilbert transform (February 14, 217) Let T = d d s, and consider the very small (vertical) comple of short eact sequences (D ε /V ) (D ε /V ) T D ε D ε T V T V The corresponding long (co-) homology sequence is ker T ker T D ker T (D ε/v ) (Dε/V ) ε V T ((D ε /V ) ) D ε T (D ε) V T (V ) The functional vs ε is in ker T V, and we hope to etend it uniquely to a functional in ker T D ε. It would suffice that ker T (Dε/V ) = and T ((D ε/v ) ) = (D ε /V ), that is, T is an isomorphism on distributions supported on {} of parity ε and positive-homogeneity degree s. The only failures are s = n with ε = ( 1) n+1, because ( d ) d ( n) δ (n) = (for n = 1, 2,...) For the principal-value functional, although s = 1 does occur as a pole for parity +1, the parity is ε = 1, so there is no obstruction to the etension, and it is unique. For e(s) > 1, let u ε s be the distributions given by integration against s for ε = +1 and sgn s for ε = 1. For e(s) > 1, these are locally integrable, and of respective parities ε = ±1 and of positivehomogeneity degree s. [3.4] Corollary: The distributions u ε s have meromorphic continuations to tempered-distribution-valued functions of s C, with only simple poles. For ε = +1, the poles are at n = 1, 3, 5,..., with residues constant multiples of δ (n). For ε = 1, the poles are at n = 2, 4, 6,..., with residues constant multiples of δ (n). The meromorphic continuations maintain the positive-homogeneity and parity. For s 1, u ε s = 1 d. For s = 1, s+1 d u ε s+1 u 1 = 2 δ and ( ) s u+1 s s= = u 1 1 Proof: The functionals v ε s on V are entire, since there is no convergence issue for test functions vanishing to infinite order at. Thus, at pole n, s n (s + n)u ε s vanishes on test functions vanishing to infinite order at. Thus, the residue s n (s + n)u ε s is a distribution supported on {}. It has the same parity ε, and is positive-homogeneous of degree n, in the distributional sense. Thus, there is no pole unless ε = ( 1) n, for n =, 1, 2,.... For e(s) 1, the integrate-against functionals u ε s given by s for ε = +1 and sgn s for ε = 1 are of (distributional) positive-homogeneity degree s, and parity ε. For each e(s) 1, u ε s is the unique etensions of the distributions v ε s, up to constants possibly depending on s. By the vector-valued form of the identity principle from comple analysis, the same homogeneity and parity properties hold for any meromorphic continuation. From d d uε s = s u ε s, = η d d u ε s = 1 s u ε s 6 = s u ε s 1

7 Paul arrett: The Hilbert transform (February 14, 217) Thus, u ε s = 1 d s+1 d u ε s+1. Since u ε s+1 is a holomorphic distribution-valued function in e(s) > 2, this etends u ε s to e(s) > 2, ecept for possible pole at s = 1. By induction, u ± s has a meromorphic continuation to all of C ecept for possible (simple) poles at negative integers. As above, the only possible residues are distributions supported at, which can only occur at non-positive even integers, or non-positive odd integers, depending on parity. For ε = +1, the relation d d u ε s = s u ε s 1 evaluated at s = yields the residue of u +1 s at 1, namely, a constant multiple of δ. For ε = 1, there is no pole, and evaluation of the relation at s = just gives d d1 =. Thus, we differentiate in s before evaluation: the relation u ε s = s u ε s 1 gives ) u ( +1 s = s s= s s= u+1 s = ( ) s u 1 s 1 = (u 1 s 1 s + s s= s u 1 s ) = u 1 1 s= Of course, u +1 s = s = ( s log ) s s= s s= This completes the discussion. s= = log [3.5] emark: In particular, the meromorphic continuations are tempered distributions. That is, all positivehomogeneous distributions are tempered. Also, away from, homogeneous distributions are given locally by smooth functions. 4. Hilbert transforms The most immediate description of the Hilbert transform Hf is (Hf)(y) = 1 π P.V. f(y) y dy = 1 π ε + ε f(y) y dy From the previous discussion of the principal-value functional η, Hf eists at least as a point-wise function. In fact, with (L y ) = f( y), y L y f is a smooth S -valued function, so f η(l y f) is in C (), but growth properties are unclear. The usual heuristic is so f(y) P.V. dy = (η f)() y Hf = 1 π η f (for f S ()) ranting this for a moment, taking Fourier transform would seem to give (Hf) = 1 η f π The principal-value functional η is a tempered distribution, so its Fourier transform makes sense at least as a tempered distribution. [4.1] Claim: The Fourier transform of an odd/even distribution of positive-homogeneity degree s is odd/even of positive-homogeneity degree (s + 1). Proof: Since Schwartz functions are dense, the interaction of dilation and Fourier transform can be eamined via functions with point-wise values and Fourier transforms defined by literal integrals, although none of these can be homogeneous. It is immediate by changing variables that parity is preserved. With (f t)() = f(t), for functions f, (f t) (ξ) = e 2πiξ f(t) d = 1 e 2πiξ/t f(t) d = 1 t t ( f 1 t )(ξ) 7

8 Paul arrett: The Hilbert transform (February 14, 217) That is, f t = 1 t f 1 t Thus, without trying to use pointwise sense of a tempered distribution, as claimed. (û t)(f) = û( 1 t f 1 t ) = 1 t u ( (f 1 t ) ) = 1 t = 1 (t t u f ) t ( ( 1) ) s u ( t f) = t (s+1) u( f) = t (s+1) û(f) = 1 t (u 1 )( f) t [4.2] Corollary: The principal-value distribution η, has Fourier transform has degree and of odd parity. In particular, η = iπ sgn. Proof: By uniqueness of distributions of a given degree and parity, it suffices to evaluate η and sgn on a given odd Schwartz function, for eample, e π2. On one hand, ( ) η(e π2 ) = η (e π2 ) ( ) = η i 1 e π2 ) On the other, sgn e π2 d = e π2 d = 2 Thus, η = iπ sgn. = π 1 e d = i 1 e π2 e π2 d = 2 = π 1 Γ(1) = π 1 With the factors of π cancelling, this suggests that we could prove [4.3] Corollary: Hf = ( i sgn f ) [... iou...] d = i 1 e π2 d = i 1 2 e d π2 = e π d [4.4] Corollary: The Hilbert transform etends by continuity to an isometric isomorphism of L 2 to itself. Proof: Assuming the corollary, since Fourier tranform is an isometry of L 2 to itself, and multiplication by sgn is an isometry, the Hilbert transform is an L 2 -isometry on S L 2, and etends by continuity to L 2. [4.5] Corollary: H H = f( ) for f S or f L 2. But what meaning should convolution of f S with η S have? [iou] [5.1] Eample: H sin = [5.2] Eample: Since ch[ 1,1] sin 2πξ = πξ, sin 2π H π = 5. Eamples ( ) ( i sgn ŝin = i sgn δ1 δ ) ( 1 = δ 1 + δ ) 1 = cos 2i 2 ( ) ( ) e 2πiξ e 2πiξ i sgn ch [ 1,1] = i ch [ 1,] ch [,1] ) = i 2πiξ 8 = cos 2πξ 1 πξ

9 [5.3] Eample: H s [... iou...] [... iou...] Paul arrett: The Hilbert transform (February 14, 217) Appendi: uniqueness of equivariant functionals [7.1] Theorem: Let be a topological group. Let V Cc o () be a quasi-complete locally conve topological vector space of comple-valued functions on stable under left and right translations, and containing an approimate identity {ϕ i }. Then there is a unique right -invariant continuous functional on V (up to constant multiples): it is f f(g) dg (with right translation-invariant measure) Proof: Let T g f(y) = f(yg) be right translation. With right translation-invariant measure dg on, since ϕ i(g) dg = 1 and ϕ i is non-negative, ϕ i (g) dg is a probability measure on the (compact) support of ϕ i. Thus, for any f V, we have a V -valued elfand-pettis integral T ϕi f = ϕ i (g) T g f dg closure of conve hull of {ϕ i (g)f : g } V The assumption V Cc o () is meant to entail that the translation action of on V is continuous, meaning that V V by g f T g f is (jointly) continuous. By continuity, given a neighborhood N of in V, we have T ϕi f f + N for all sufficiently large i. That is, T ϕi f f. For a right-invariant (continuous) functional u on V, ( ) u(f) = u g ϕ i (h) f(gh) dh i This is ( ) u g f(hg) ϕ i (h 1 ) dh ( ) = u g f(h) ϕ i (gh 1 ) dh by replacing h by hg 1. By properties of elfand-pettis integrals, and since f is guaranteed to be a compactly-supported continuous function, we can move the functional u inside the integral: the above becomes f(h) u ( g ϕ i (gh 1 ) ) dh Using the right -invariance of u the evaluation of u with right translation by h 1 gives f(h) u(g ϕ i (g)) dh = u(ϕ i ) f(h) dh By assumption the latter epressions approach u(f) as i. For f so that the latter integral is non-zero, we see that the it of the u(ϕ i ) eists, and then we conclude that u(f) is a constant multiple of the indicated integral with right Haar measure. [7.2] Corollary: In the situation of the theorem, for a continuous group homomorphism χ : C, suppose that V is stable under multiplication by the function χ. Then there is a unique continuous functional λ on V such that λ(t g f) = χ(g) λ(f) for all g. Proof: Let u be the -invariant functional of the theorem, and put λ(f) = u(χ f). 9

10 Paul arrett: The Hilbert transform (February 14, 217) 8. Appendi: distributions supported at {} [8.1] Theorem: A distribution u with support {} is a (finite) linear combination of Dirac s δ and its derivatives. ecall: the support of a distribution u is the complement of the union of all open sets U n such that u(f) = (for f D K with compact K U) Proof: Since the space D of test functions on n is D = co K D K, it suffices to classify u in D K with support {}. A continuous linear map T from a it of Banach spaces (such as D K ) to a normed space (such as C) factors through a itand, under the mild assumption that the images of the it in the itands are dense. Thus, there is an order k such that u factors through C k K = {f C k (K) : f (α) vanishes on K for all α with α k} Fi a smooth compactly-supported function ψ identically 1 on a neighborhood of, bounded between and 1, and (necessarily) identically outside some (larger) neighborhood of. For ε > let ψ ε () = ψ(ε 1 ) Since the support of u is just {}, for all ε > and for all f D( n ) the support of f ψ ε f does not include, so u(ψ ε f) = u(f) Thus, for some constant C (depending on k and K, but not on f) ψ ε f k = sup K sup (ψ ε f) (α) () C sup α k i k sup sup j i ε j ψ (j) (ε 1 ) f (i j) () For f vanishing to order k at, that is, f (α) () = for all multi-indices α with α k, on a fied neighborhood of, by a Taylor-Maclaurin epansion, for some constant C and, generally, for α th derivatives with α k, For some constant C ψ ε f k C sup sup i k j i C k+1 f (α) () C k+1 α Thus, for all ε >, for smooth f vanishing to order k at, Thus, u(f) = for such f. ε j ε k+1 i + j C ε k+1 i C ε k+1 k = C ε u(f) = u(ψ ε f) C ε That is, u is on the intersection of the kernels of δ and its derivatives δ (α) for α k. enerally, 1

11 Paul arrett: The Hilbert transform (February 14, 217) [8.2] Proposition: A continuous linear function λ V vanishing on the intersection of the kernels of a finite collection λ 1,..., λ n of continuous linear functionals on V is a linear combination of the λ i. Proof: The linear map q : V C n by v (λ 1 v,..., λ n v) is continuous since each λ i is continuous, and λ factors through q, as λ = L q for some linear functional L on C n. We know all the linear functionals on C n, namely, L is of the form L(z 1,..., z n ) = c 1 z c n z n (for some constants c i ) Thus, λ(v) = (L q)(v) = L(λ 1 v,..., λ n v) = c 1 λ 1 (v) c n λ n (v) epressing λ as a linear combination of the λ i. 9. Appendi: the snake lemma The snake lemma produces a long eact sequence from a short eact sequence of complees, where in this contet a comple is a chain of homomorphisms... f 2 M 1 f 1 M f M 1 f 1... where Imf i ker f i1 for all indices. [1] For a slightly specialized version [2] of the snake lemma, consider a short eact sequences of very small complees of the form A A, B B, and C C, with the complees themselves now written vertically: A B T A B T C C T We assume that the maps T : A A, T : B B, and T : C C make the squares in the latter diagram commute. In the following, one should consider the improbability of defining some sort of map from any part of C to any part of A, given short eact A B C : [9.1] Claim: There is a long eact sequence ker T A ker T B ker T C A T A B T B C T C [1] There is also an assertion of naturality, which we ignore here. See [Weibel 1994] as general reference for these and related ideas. [2] I first saw this small but important eample in [Casselman 1993]. 11

12 Paul arrett: The Hilbert transform (February 14, 217) Proof: The most serious issue, which is all we address, is defining the connecting homomorphism δ : ker T C A/T A. For eample, the eactness of the left half and of the right half of the long sequence are relatively elementary. [3] iven c ker T C, we want to determine a A well-defined module T A, to eventually fit into the long eact sequence. Let b c for some b B. Since T c = C and the squares commute, T b under the map B C. Thus, by eactness of A B C, there is a A mapping to T b. Declare a = δc. To make δ well-defined, note that the precise ambiguity in choice of a A is by T A. Bibliography [Bedrosian 1962] E. Bedrosian, A product theorem for Hilbert transforms, Memoranudm M-3439-P, December 1962, and Corp., Santa Monica, CA. [Calderon-Zygmund 1952] A.P. Calderon, A. Zygmund, On the eistence of certain singular integrals, Acta Math. 88 (1952), [Casselman 1993] W. Casselman, Etended automorphic forms on the upper half-plane, Math. Ann. 296 (1993), [Chaudhury 212] K. N. Chaudhury, L p -boundedness of the Hilbert transform, arxiv: v9 [cs.it] 2 Oct 212. [Fefferman 1971] C. Fefferman, Characterization of bounded mean oscillation, Bull. AMS 77 (1971), [Fefferman-Stein 1972] C. Fefferman, E. Stein, H p spaces of several variables, Acta Math. 129 (1972), [arrett 216a] P. arrett, Vector-valued integrals, garrett/m/fun/notes /vector-valued-integrals.pdf [arrett 216b] P. arrett, Holomorphic vector-valued functions, garrett/m/fun/notes /holomorphic-vector-valued.pdf [elfand-silov 1964] I.M. elfand,.e. Silov, eneralized Functions, I: Properties and Operators, Academic Press, NY, [Pichorides 1972] S. K. Pichorides, On the best values of the constants in the theorems of M. iesz, Zygmund, and Kolmogorov, Studia Math. 44 (1972), [Plemelj 198] J. Plemelj, Ein Egänzungssatz zur Cauchyschen Integraldarstellung analytischer Funktionen, andwerte betreffend, Monatsch. Math. Phys. 19 (198), [Plemelj 198b] J. Plemelj, iemannsche Funktionenscharen mit gegebener Monodromiegruppe, Monatsch. Math. Phys. 19 (198), [iesz 1928] M. iesz, Sur les fonctions conjugées, Math. Z. (1928), [Sokhotskii 1873] Y.W. Sokhotskii, On definite integrals and functions used in series epansion, (ussian?) St. Petersburg, [Wiki 215] theorem, accessed 8 Oct 215. [Stein 197] E. Stein, Singular integrals and Differentiability Properties of Functions, Princeton U. Press, [3] The eactness of those halves also follows from the slightly-less elementary fact that right adjoints are left eact, and left adjoints are right eact. 12

13 Paul arrett: The Hilbert transform (February 14, 217) 197. [Weibel 1994] C. Weibel, An introduction to homological algebra, Cambridge Univ. Press, [Wiki 214] Wikipedia, Hilbert transform, transform, 2 Dec 214. [Woodward-Davies 1952] P.M. Woodward, I.L. Davies, Information theory and inverse probability in telecommunication, Proc. Inst. Elec. Engrs. Part III 99 (1952), [Zygmund 1968] A. Zygmund, Trigonometric Series I, II, Cambridge,