On the Regularity of Wavelets

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I. INTRODUCTION
I f f is a trigonometric polynomial with f ( 0 ) = 1 then the infinite product defines an entire function F (e.g., see [6,Lemma 2.31)Here and in the following the arguments of our functions are always real numbers.The function g, can be made unique by additional conditions which, however, are not needed in this correspondence because we only use I gN( z) I .If we denote the functions of (1.1) corresponding to f,,,, g, by F, and G,, respectively, then (1.2) shows that As in [2, p. 981, (4.28)] (there is a misprint in that formula: the power 1 + a has to be replaced by a ) , let a,,, (the "regularity index" of the Fourier transform of F,) be the supremum of all 0 such that Then the problem is to find upper and lower bounds of a,.For instance, it is known that a2 2 0.5 (see [2, p. 9841).In Section 111, we improve this to 0.51 < a , < 0.53.
(1. 6) In a remark on p. 984 of [2] we find the value a2 = 2 -(log (1 + &))/(log 2) = 0.55 . . . .According to [3] this value corresponds to a different definition of the regularity index a*.There a2 is the supremum of all a such that the Fourier transform of F2 belongs to the space Lip-a of functions f satisfying In particular, one is interested in the asymptotic behavior of a , as N tends to infinity.It is known that a, , " p. 9831.However, the limit of a, , " given on p. 983 of [2] has turned out to be wrong [3].In Section IV of this paper, we will prove that

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This result does not depend on the particular choice of the regularity index, i.e., it remains the same if the sequence 0 1 ~ is replaced by some other sequence G, , , such that aN -GN is a bounded sequence.The value of the limit (1.7) was found independently by another method by A. Cohen and J. P. Come.Some results of A. Cohen will appear in [l].Since the estimations are delicate at various points it is certainly a good idea to have two different proofs of (1.7).Our estimates of a N are elementary.They are based on a simple proposition that is proved in Section 11.
We refer to Y. Meyer's book [5] as a general reference on wavelet theory and to the two papers [4] that deal in-depth with the regularity of scaling functions.The author thanks I. Daubechies for sending him a friendly letter with informations about her paper and the work of A. Cohen and J. P. Come.

A BASIC PROPOSITION
The definition (1.3) of PN shows that P N ( y ) 2 1 for 0 I y I 1 .
Hence we find that which yields the rough estimate aN 5 N -1.To find better bounds we will use the following result.
Proposition I : Let N be a fixed positive integer, and let h > 0.
a) If there is a positive constant C such that Since H is increasing this implies that there is a positive constant C , such that Integration by parts gives, for z 2 1 ,

5)
Now (1.5) implies Np > 0, and thus (2.4) and (2.5) give fi -N + log (2A)/log 2 5 0. This proves (2.2). b)Let C, := m a x o r f r 2 n 1 G ( t ) 1. Then (2.3) shows that H(2,2a) I CC2(2h)m.Since H is increasing this yields Now (2.5) and I F ( z ) I = I F( -z ) I shows that this implies (2.2) with the inequality reversed.U Proposition 1 is closely related to the estimates used in the Appendix of [2].The lower bound for the regularity index that is found in [l]  From Proposition 1 follows that the regularity index 01 satisfies a 2 N -1 -b -E ( E > 0), in agreement with Cohen's result.

LOWER AND UPPER BOUNDS FOR .d2
We now apply Proposition 1 to find lower and upper bounds for the regularity index a,.Let us write g in place of g , in this section.Since we can take g ( z ) = P + qe", where p : = f ( 1 + &), q : = i(la), ( 3 .1 ) we obtain I g ( z ) l 2 9 ?g ( z ) = p + q c o s z 4 2

= p + -( e -" + e ' Z ) = : h , ( z ) . ( 3 . 2 )
Hence, It is easy to see that a,, = p m .We now use to obtain where bjm = ajZm,,,, j = -1 , 0, 1.To find bjm we use that By accident, this upper bound for a2 is the exact value for a differently defined regularity index, see [2, p. 9841.However, we will now show that the inequality (3.4) is strict.
First we sharpen the estimate (3.2).Write g ( z ) = x + iy, x , y , z E R .Then (3.1) shows that x + iy lies on a circle centered at p with radius -4.Hence,

( P -q )
This yields

1% AI a 2 s 1 -log 2 ( 3 . 9 )
Write We now proceed to find lower bounds for a z .Using the Cauchy-Schwarz inequality we find The latter equality follows from I g ( z ) I = 2 -+ ( e i z + e-'").

Z
In the special case f( z) = cos z we obtain This method is used in[2] to solve dilation equations of multiresolution analysis by functions with compact support.The solutions (called scaling functions) are Fourier transforms of infinite products of the type previous shown.Then wavelet bases of L 2 ( R ) can be defined in terms of these scaling functions.One i s interested to find Manuscript received December 20, 1991; revised May 25, 1991.The author is with the Department of Mathematical Sciences, University IEEE Log Number 9105029. of Wisconsin-Milwaukee, P.O.Box 413, Milwaukee, WI 53201.scaling functions with compact support as smooth as possible which then leads to the problem to investigate the behavior of F for R 3 z + CO.In this correspondence, we treat this problem for the special trigonometric polynomials f = f,, N E N , given in [2, Section 41.These polynomials are products N fN(Z) = ( 3 1 + e l z ) ) gN(Z), where g, is a trigonometric polynomial satisfying lgN(z)lz = p N ( s i n 2 ( ; z ) there is a positive constant C such that (2.1) holds with the inequality reversed then (2.2) holds with the inequality reversed.Proof: a) Since N is fixed during the proof let us agree to suppress the subscript N of the various functions.Define e ( z ) := I s i n ( i z ) I N I G ( z ) 1 and N ( z) := / ' 6 ( t ) dt. 0 Then G ( 2 t ) = g ( t ) G ( t ) yields H(2,27r) = 2 m L 2 T6 ( 2 m t ) dtNow I G ( t ) I 2 1 and assumption(2.1)shows that H(2,2a) 2 C(2A),.
is an immediate consequence of the proposition: If B , = supzeRrIJm~l I g(2'x) 1, b, = log B, / m log 2 and b = inf, b,, then the left-hand side of (2.1) is I CB, 5 C(2b+c)m.

(3. 10 )Lemma 2 :
This is the lower bound for a2 given in [2, p. 9941.We now improve this lower bound by using an inequality similar to (3.5).If g ( z ) = x + Q , x , y , z E R then If f l y I 1 and z : = q 2 ( y ) then which yields Hence, (4.1) holds for m = M + 1, which completes the proof. 0The polynomial PN of (1.3) satisfies the inequality 1 g( z ) 1 5 h3( z ) := a + b( eiz + e-") + c(e2iz + 1 where Proof: From Using the matrix (3.8) again we find (see [2, Proposition 4.51) it follows that P N ( i ) = 2N-1.Since PN is increasing on [0, 11, we conclude that P J y ) 5 2N-l for O 5 y 5 +.If +I y I 1 then where h is the spectral radius of A .Then Proposition 1 b) yields N -1 + j N -1 N -l + j j = O IV. THE ASYMPTOTIC BEHAVIOR OF THE SEQUENCE aN We first derive a lower bound for aN that is based on the Lemma I: Let p:[O, 11 + [0, 11 be a function such that following two lemmas.= (2y)N-+3N( ; ) = ( 4 y y .U We apply Lemma 1 to and p ( y ) p ( 4 y ( l -y ) ) 5 +, on f I y l 1.
We prove (4.1) by induction on m.If m = 1, 2 then (4.1) is true.Assume that (4.1) holds for all m I M .If 0 I y I and z : =4 ( Y ) thenWe now provide an upper bound for aN based on the following Lemma 3: For all m, N E N , lemma.